text:
\section{Introduction}

Exploratory Factor Analysis (EFA) has been widely used to learn the latent structure underlying multivariate data. A major problem in EFA is to identify an interpretable factor structure among infinitely many equivalent solutions that give the same data distribution, where two equivalent solutions  differ by a rotation transformation \citep[see chapters 10-12,][]{mulaik2009foundations}. Mathematically, one aims to find a sparse solution for which many entries of the loading matrix are exactly or approximately zero, so that each factor can be interpreted based on a small number of manifest variables whose loadings on the factor are not close to zero. This idea dates back to the seminal discussion on simple factor structure in \citet{thurstone1947multiple}.

Methods for obtaining sparse loading structures can be classified into two categories -- rotation and regularised estimation methods. A rotation method  involves two steps. In the first step, an estimate of the loading matrix is obtained. Typically, but not necessarily, a maximum likelihood estimator is used in this step \citep{bartholomew2011latent},  under some arbitrary but mathematically convenient constraints that avoid the rotational indeterminacy. In the second step, one rotates the estimated loading matrix to minimise a certain loss function that measures the sparsity level of the rotated solution\footnote{Rotation methods that solve maximisation problems can always be reformulated to solve minimisation problems. }, where a smaller loss function value indicates a more sparse solution. Different rotation methods have been proposed that differ by first, whether the factors are allowed to be correlated and second the loss function for measuring sparsity. A rotation method is called an orthogonal rotation when the factors are constrained to be uncorrelated, and an oblique rotation, otherwise. Different loss functions have been proposed for orthogonal and oblique rotations, including
varimax \citep{kaiser1958varimax}, oblimin \citep{jennrich1966rotation},
geomin \citep{yates1987multivariate}, simplimax \citep{kiers1994simplimax}, componentwise loss \citep{jennrich2004rotation,jennrich2006rotation}, among many others.
Among the existing rotation methods, we draw attention to the monotone concave Component Loss Functions \citep[CLFs;][]{jennrich2004rotation,jennrich2006rotation} due to their desired theoretical properties and superior performance in recovering sparse loading matrices. Specifically,  \citep{jennrich2004rotation,jennrich2006rotation}
provided some theoretical guarantees to the CLFs when the true loading matrix has a perfect simple structure and further found that the CLFs are often more accurate in recovering sparse loading matrices than other rotation methods, under both orthogonal and oblique settings.

In recent years, several regularised estimation methods have been proposed for EFA \citep[e.g.,][]{trendafilov2014simple, yamamoto2017graphical, jin2018approximated, geminiani2021single}.
Slightly different from rotation methods,
a regularised estimation method simultaneously performs parameter estimation and model selection. It introduces a LASSO-type sparse-inducing regularisation term into the loss function for parameter estimation, where the regularisation term  imposes sparsity on the estimated loading. It typically obtains a sequence of candidate models by varying the weight of the regularisation term in the loss function. The final model is chosen from the candidate models, often using an information criterion.

This paper proposes a new family of oblique
rotations based on component-wise $L^p$ loss functions, for $0 < p\leq 1$. The proposed loss functions are shown to be special cases of monotone concave CLFs and thus share the same theoretical properties. We note that \citet{jennrich2004rotation,jennrich2006rotation} considered the $L^1$ loss function but not the
$L^p$ loss functions with  $p<1$. With the proposed rotations, we answer several previously unaddressed questions regarding rotation and regularised estimation methods. First, we establish the statistical consistency of the rotated solution. More specifically, we provide conditions under which the rotated solution converges to the true sparse loading matrix as the sample size goes to infinity. These conditions also provide insights into the choice of $p$.
Seemingly straightforward, this consistency result requires some refined analysis and, to our best knowledge, such results have not been established for other rotation methods. In particular, the theoretical results for the CLFs in \citet{jennrich2004rotation,jennrich2006rotation} were established with respect to the population loading matrix rather than its estimate. Second, do regularised estimation methods outperform rotation methods, or vise versa? To obtain some insights into
this question, we show that the proposed rotation method can be viewed as the limiting case of a
regularised estimator when the weight of the regularisation term converges to zero.  In addition,
to compare the two methods in terms of their performance in model selection, we develop a hard-thresholding procedure that conducts
model selection based on a rotated solution. Through computational complexity analysis and simulation studies, it is found that rotation achieves similar statistical accuracy as regularised estimation, but with a lower computational cost.
Third, monotone concave CLFs, including the proposed $L^p$ loss functions, are not smooth everywhere. Consequently, the traditional gradient projection algorithms  are no longer applicable.
\citet{jennrich2004rotation,jennrich2006rotation}
bypassed the computational issue by replacing a CLF with a  smooth approximation, but also pointed out potential issues with this treatment.
We propose an Iteratively Reweighted Gradient Projection (IRGP) algorithm that may better solve this nonsmooth optimisation problem.  Finally, uncertainty quantification for the rotated solution
affects the interpretation of the factors and, thus, is vital in EFA. However, the delta method, which is used to obtain confidence intervals for rotation methods with a smooth objective function \citep{jennrich1973standard}, is not applicable due to the non-smoothness of the current loss functions. That is, the delta method requires the loss function to be smooth at the true loading matrix, which is not satisfied for monotone concave CLFs. We tackle this problem by developing a post-selection inference procedure that gives asymptotically valid confidence intervals for loadings in a rotated solution.  The proposed method is evaluated and compared with regularised estimation and traditional rotation methods via simulation studies. The proposed method is further illustrated by an application to big-five personality assessment.

\iffalse
Exploratory factor analysis \citep[EFA;][]{mulaik2009foundations} comprises probabilistic dimension-reducing techniques to deduce the relationships between a set of observed variables, called manifest variables, and a smaller set of latent variables, referred to as factors. Highly popular in the social and behavioral sciences, factor analysis models are usually estimated by maximum likelihood \citep{joreskog1967some, lawley1940vi}. A desirable feature to aid interpretation of the obtained solution is a sparse loading matrix with many small entries and only a few large loadings. As in \citet{thurstone1947multiple}, the simplest structure would be that where each manifest variable is loading on one single factor. This objective is traditionally addressed by factor rotation, either through orthogonal or oblique rotation. The former seeks a solution of the loading matrix such that the factors are independent, whereas the latter allows the factors to be correlated which is typically more realistic. There are many analytic oblique rotation criteria to find a sparse loading matrix. Among them, promax \citep{hendrickson1964promax} and direct quartimin \citep{jennrich1966rotation} are the most popular oblique rotation methods and available in most statistical software, and geomin \citep{yates1987multivariate} and simplimax \citep{kiers1994simplimax} are considered two of the best performing methods \citep{jennrich2006rotation}. See \citet{browne2001overview} for an overview.

As analytic rotation methods will produce many small, non-zero factor loadings, recent attention has been turned to regularized estimation of the loading matrix \citep[e.g.,][]{trendafilov2014simple, yamamoto2017graphical, jin2018approximated, geminiani2021single}. The focus of such approach has been on soft-thresholding approaches by introducing the least absolute shrinkage and selection operator \citep[LASSO;][]{tibshirani1996regression} as a penalty function in the likelihood function. By construction, it performs both model estimation and variable selection, and in contrast to rotation-based methods the small loadings do not have to be manually set to 0 (i.e., hard-thresholding). To overcome the computational complexity of regularized estimation, \citet{jin2018approximated} approximated the penalized maximum likelihood solution, considering orthogonal EFA, and \citet{geminiani2021single} introduced a trust-region algorithm for regularized estimation of the loading matrix.
\fi

The rest of the paper is structured as follows. Section~\ref{sec:meth} proposes  $L^p$ criteria for oblique rotation, and draws a connection with regularised estimation. Section~\ref{sec:inf} discusses statistical
inferences based on the proposed rotation method and establishes their asymptotic properties, and Section~\ref{sec:comp} develops an iteratively reweighted gradient projection algorithm for solving the optimisation problem associated with the proposed rotation criteria. The proposed method is evaluated via simulation studies in Section~\ref{sec:sim} and an application to big-five personality assessment in Section~\ref{sec:real}. We conclude this paper with discussions on the limitations of the proposed method and future directions in Section \ref{sec:conc}. Proof of the theoretical results and additional details about the simulation and real application
are given in the supplementary material.

\iffalse
Section 2 introduces the $L^p$ criteria for oblique rotation. It furthermore presents a theoretical result connecting the obtained solution under the $L^p$ rotational family with the regularized estimation solution. Section 3 outlines the hard-thresholding procedure used for model selection and presents the statistical consistency of the proposed estimator. In Section 4, procedures for drawing statistical inference on the sparse true loading matrix are developed. This is in Section 5 followed by a description of the computational methods developed for the optimization problem. Section 6 presents a simulation study and Section 7 introduces a case study based on the Big 5 data set. The paper concludes with a discussion. All proofs are found in the appendix.
\fi

\section{\texorpdfstring{$L^p$}{} Rotation Criteria}\label{sec:meth}
\subsection{Problem Setup}

We consider an exploratory linear factor model with $J$ indicators and $K$ factors given by
\begin{equation}\label{eq:1}
    \mathbf{X}|\boldsymbol{\xi} \sim \mathcal{N}( \boldsymbol{\Lambda}\boldsymbol{\xi} , \mathbf{\Omega} ),
\end{equation}
where $\mathbf{X}$ is a $J$-dimensional vector of manifest variables, $\boldsymbol{\Lambda} = (\lambda_{jk})_{J\times K}$ is the loading matrix,  $\boldsymbol{\xi}$ is a $K$-dimensional vector of common factors, and $\boldsymbol\Omega = (\omega_{ij})_{J\times J}$ denotes the residual covariance matrix. It is assumed that the common factors are normally distributed with variances fixed to 1, i.e,
$
\boldsymbol{\xi} \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{\Phi}),
$
where $\boldsymbol{\Phi}\in \mathbb{R}^{K \times K}$ has diagonal entries $\phi_{kk}$, $k = 1, \ldots, K$, equal to 1 and is symmetric positive definite, denoted by $\boldsymbol{\Phi} \succ 0$. The manifest variables are assumed to be conditionally independent given $\boldsymbol{\xi}$, i.e., the off-diagonal entries of $\mathbf{\Omega}$ are set to 0. To simplify the notation, we use
$\boldsymbol{\theta} = (\boldsymbol{\Lambda}, \boldsymbol{\Phi}, \boldsymbol{\Omega})$ to denote all of the unknown parameters. The model in \eqref{eq:1} implies the marginal distribution of $\mathbf X$
\begin{equation}\label{eq:2}
\mathbf{X} \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma(\theta)}),
\end{equation}
where
$
\boldsymbol{\Sigma(\theta)} = \boldsymbol{\Lambda} \boldsymbol{\Phi} \boldsymbol{\Lambda}' + \boldsymbol{ \Omega}.
$
Without further constraints, the parameters in \eqref{eq:2} are not identifiable due to rotational indeterminacy. That is, two sets of parameters $\boldsymbol \theta$ and
$\tilde\boldsymbol \theta = (\tilde{\boldsymbol{\Lambda}}, \tilde{\boldsymbol{\Phi}}, \tilde{\boldsymbol{\Omega}})$ give the same distribution for $\mathbf X$ if
$\boldsymbol{\Lambda} \boldsymbol{\Phi} \boldsymbol{\Lambda}' = \tilde{\boldsymbol{\Lambda}} \tilde{\boldsymbol{\Phi}} \tilde{\boldsymbol{\Lambda}}'$
and
$\boldsymbol{\Omega} = \tilde{\boldsymbol{\Omega}}$. Note that the normality assumptions above are not essential. They are adopted for the ease of writing, and the development in the current paper does not rely on these normality assumptions. Throughout this paper, we assume that the number of factors $K$ is known.

An oblique rotation method is a two-step procedure. In the first step, one obtains an estimate of the model parameters, under the constraints that $\boldsymbol{\Phi} = \mathbf I$ and other arbitrary but mathematically convenient constraints that fix the rotational indeterminacy. Note that due to the rotational indeterminacy, one can always constrain $\boldsymbol{\Phi} = \mathbf I$ and absorb the dependence between the factors into the loading matrix $\boldsymbol \Lambda$. The estimate can be obtained by any reasonable estimator for factor analysis, such as the least-square \citep{joreskog1972factor}, weighted-least-square \citep{browne1984asymptotically}, and maximum likelihood estimators \citep{joreskog1967some}. We denote this estimator by $\hat{\boldsymbol{\theta}} = (\hat{\mathbf{A}}, \mathbf{I},\hat{\boldsymbol{\Omega}})$.  In the second step, one finds an oblique rotation matrix $\hat{\mathbf T}$, such that the rotated loading matrix $\hat{\mathbf{\Lambda}}= \hat{\mathbf{A}} \mathbf{\hat{T}}^{{\prime}-1}$ minimises a certain loss function $Q$ that measures the sparsity level of a loading matrix. The functional form of $Q$ will be proposed in the sequel. Here, an oblique rotation matrix $\mathbf T$ satisfies that $\mathbf T$ is invertible and  $(\mathbf{T'} \mathbf{T})_{kk} = 1, \, k=1, \ldots, K$. Consequently, any rotated solution $(\hat{\mathbf{A}} \mathbf{ {T}}^{{\prime}-1}, \mathbf{T'} \mathbf{T}, \hat{\boldsymbol{\Omega}})$ is still in the parameter space and gives the same distribution for $X$. More precisely, we let
\begin{align}\label{eq:13}
\mathcal{M}=&\{ \mathbf{T} \in \mathbb{R}^{K \times K} : rank(\mathbf{T}) = K, \, (\mathbf{T'} \mathbf{T})_{kk} = 1, \, k=1, \ldots, K \}
\end{align}
be the space for oblique rotation matrices, where $rank(\cdot)$ gives the rank of a matrix. Then the oblique rotation problem involves solving the optimisation
\begin{equation}\label{eq:rotation}
\mathbf{\hat{T}} \in \operatornamewithlimits{arg\,min}_{ \mathbf{T} \in \mathcal{M}} Q(\hat{\mathbf{A}} \mathbf{T'}^{-1}),
\end{equation}
and the rotated solution is given by $(\hat{\mathbf{A}} \hat{\mathbf{{T}}}^{{\prime}-1}, \hat{\mathbf{T}}' \hat{\mathbf{T}}, \hat{\boldsymbol{\Omega}})$. Equivalently, the rotated loading matrix $\hat{\boldsymbol\Lambda}$ satisfies

\begin{equation}\label{eq:rotation2}
  (\hat{\boldsymbol\Lambda}, \hat{\boldsymbol\Phi}) \in \operatornamewithlimits{arg\,min}_{\boldsymbol\Lambda, \boldsymbol\Phi} Q(\boldsymbol\Lambda), \mbox{~such that~} \boldsymbol\Lambda \boldsymbol\Phi\boldsymbol\Lambda' = \hat{\mathbf A}\hat{\mathbf A}'.
\end{equation}

As explained in Remark~\ref{rmk:indet}, the minimiser of \eqref{eq:rotation} or equivalently that of \eqref{eq:rotation2} is not unique.

\begin{remark}\label{rmk:indet}
Let $\mathcal{D}_1$ be the set of all $K\times K$ permutation matrices and $\mathcal{D}_2$ be the set of all $K\times K$ sign flip matrices.
For any  $\mathbf D_1 \in \mathcal{D}_1$, $\mathbf D_2\in \mathcal D_2$, and $K\times K$ matrix $\mathbf T$,
$\mathbf T \mathbf D_1$ is a matrix whose columns are a permutation of those of $\mathbf T$ and,
$\mathbf T\mathbf D_2$ is a matrix whose $k$th column is either the same as the $k$th column of $\mathbf T$ or  the $k$th column of $\mathbf T$ multiplied by $-1$.  Let $\hat{\mathbf T}$ be one solution to the optimisation problem   \eqref{eq:rotation}. It is easy to check that $\hat{\mathbf T} \mathbf  D_1 \mathbf  D_2$ also minimises the objective function \eqref{eq:rotation}, for any $\mathbf  D_1\in \mathcal{D}_1$ and $\mathbf  D_2\in \mathcal{D}_2$. The resulting loading matrix is equivalent to
$\hat{\boldsymbol\Lambda}$ up to a column permutation and column sign flips.

\end{remark}

We conclude the problem setup with two remarks.

\begin{remark}
The rotation problem not only applies to the linear factor model, but also other settings, such as item factor analysis \citep{reckase2009multidimensional, chen2019joint, chen2021item} and machine learning models \citep{rohe2020vintage}.
\end{remark}

\begin{remark}

Although the current paper focuses on oblique rotations, we note that the proposed criteria can be easily extended to orthogonal rotation, as the latter can be viewed as a special case of the former when $\boldsymbol\Phi$ is fixed to be an identify matrix. That is, given a loss function $Q$, orthogonal rotation solves the problem

$$\min_{\boldsymbol\Lambda} Q(\boldsymbol\Lambda), \mbox{~such that~} \boldsymbol\Lambda \boldsymbol\Lambda' = \hat{\mathbf A}\hat{\mathbf A}'.$$

\end{remark}

\iffalse
which refer to the problem that $ (\boldsymbol{\Lambda}(\mathbf{T'})^{-1},\mathbf{T}'\mathbf{T} ,\mathbf{\Omega} ) $ and $ (\boldsymbol{\Lambda},\mathbf{I},\mathbf{\Omega})$ result in the same distribution for $\mathbf{X}$ as long as $\mathbf{T}'\mathbf{T} $ is a vaild candidate for $\boldsymbol{\Phi}$, i.e., when the diagonal entries of $\mathbf{T}'\mathbf{T}$ all take value 1. On the other hand, this gives us the opportunity to find a simpler and more interpretable structure, assuming that each manifest variable is accounted by only a few common factors. That is to say, the true loading matrix is sparse.
\fi

\subsection{Proposed Rotation Criteria}\label{subsec:propose}

\citet{jennrich2004rotation,jennrich2006rotation} proposed a family of monotone concave CLFs for the choice of $Q$ in \eqref{eq:rotation}, taking the form

\begin{equation}\label{eq:CLC_jennrich}
Q(\mathbf{\Lambda}) =\sum_{j=1}^J\sum_{k=1}^K h(|\lambda_{jk}|),
\end{equation}
where $\mathbf{\Lambda} = (\lambda_{jk})_{J\times K}$ and $h$ is a concave and monotone increasing function
that maps from $[0,\infty)$ to $[0,\infty)$.
This family of loss functions is appealing for several reasons. First, a CLF takes a simple form that does not involve products of loadings and their higher-order polynomial terms. Second, the monotone concave CLFs have desired properties. In particular, \citet{jennrich2006rotation} proved that a monotone concave CLF is minimized by loadings with perfect simple structure when such loadings exist.  Third, simulation studies in
\citet{jennrich2004rotation,jennrich2006rotation} showed that these loss functions tend to outperform traditional rotation methods, e.g.(promax, simplimax, quartimin and geomin) when the true loading matrix is sparse.

Two examples of $h$ are given in  \citet{jennrich2004rotation,jennrich2006rotation},
including the linear CLF
where $h(|\lambda|) = |\lambda|$
and the basic CLF
where $h(|\lambda|) = 1 - \exp(-|\lambda|)$. However, there does not exist a full spectrum of monotone concave CLFs for dealing with true loading matrices with different sparsity levels.
To fill this gap, we propose a general family of
monotone concave CLFs that are named the $L^p$ CLFs. More specifically, for each value of $p \in (0,1]$, the loss function takes the form

\begin{equation}\label{eq:CLC_Lp}
Q_p(\mathbf{\Lambda}) =\sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|^p,
\end{equation}

Proposition~\ref{prop:concave} below shows that this choice of $h$ yields a monotone concave CLF.

\begin{proposition}\label{prop:concave}
The absolute value function $h(x) = |x|^p$, $p \in (0, 1]$ is monotonically increasing and concave on the interval $[0, \infty)$.

\end{proposition}

Under very mild regularity conditions, any $L^p$ CLF is uniquely minimized by a loading matrix of perfect simple structure when such loading matrix exist,
where we say the minimiser is unique when all the minimisers of the loss function are equivalent up to column permutation and sign flip transformations (see Remark~\ref{rmk:indet} for these transformations).
We summarise this result in Proposition~\ref{prop:simple} below. This result improves Theorem~1 of \citet{jennrich2006rotation}, as the uniqueness of the perfect simple structure is not established in \citet{jennrich2006rotation} for the $L^1$-criterion.

\begin{proposition}\label{prop:simple}
Suppose that the true loading matrix $\boldsymbol\Lambda^*$ has perfect simple structure, in the sense that each row has at most one nonzero entry. Further suppose that $\boldsymbol\Lambda^*$ is of full column rank, i.e.,  $rank(\boldsymbol\Lambda^*)=K$. Then for any oblique rotation matrix $\mathbf T\in \mathcal{M}$,
$$Q_p(\boldsymbol\Lambda^* \mathbf{T'}^{-1}) \geq Q_p(\boldsymbol\Lambda^*),$$
where the two sides are equal if and only if
$\mathbf{T'}^{-1} = \mathbf D_1\mathbf D_2$  for  $\mathbf D_1 \in \mathcal{D}_1$ and $\mathbf D_2\in \mathcal D_2$.

\end{proposition}

Why do we need the loss functions with $p <1$, given that the choice of $p=1$ is already available in \citet{jennrich2004rotation,jennrich2006rotation}? This is because different $L^p$ CLFs were found to behave differently when the true loading matrix  does not have a perfect simple structure but still contains many zero loadings. Such a loading structure is more likely to be recovered by an $L^p$ CLF when $p<1$ than the $L^1$ CLF.
In what follows, we elaborate on this point. Let
$\mathbf{\Lambda}^*$ be the true sparse loading matrix and $\boldsymbol{\Phi}^*$ be the corresponding covariance matrix for the factors. For the true loading matrix $\mathbf{\Lambda}^*$ to be recovered by an $L^p$ CLF, a minimum requirement is that
\begin{equation}\label{eq:minimum}
Q_p(\mathbf{\Lambda}^*) = \min_{\mathbf{\Lambda}} Q_p(\mathbf{\Lambda}), \mbox{~such that there exists~}  \boldsymbol{\Phi} \succ 0, \phi_{kk}=1, k = 1, ..., K,  \boldsymbol{\Lambda}^* \boldsymbol{\Phi}^* \boldsymbol{\Lambda}^{*'} = \boldsymbol{\Lambda} \boldsymbol{\Phi} \boldsymbol{\Lambda}'.
\end{equation}
In other words, $\boldsymbol{\Lambda}^*$ needs to be a stationary point of $Q_p$.
Figure~\ref{fig:lp} gives the plots for $|x|^p$ with different choices of $p$ and their derivatives when $x > 0$.
We note that when $p<1$ the derivative of $|x|^p$ converges to infinity as $x$ approaching zero. The smaller the value of $p$, the faster the convergence speed is.
On the other hand, when $p=1$, the derivative of $|x|$ takes the value one for any $x > 0$. Therefore,  when $\boldsymbol{\Lambda}^*$ is sparse but does not have a perfect simple structure, it is more likely to be a stationary point of $Q_p$ for $p<1$ than $Q_1$. We illustrate this point by a numerical example, where  $\boldsymbol{\Lambda}^*$ is given below  and
$\boldsymbol{\Phi}^*$ is set to be an identity matrix
$$\boldsymbol{\Lambda}^{*'} = \left(\begin{array}{ccccccc}

    1.20&    0 &   0.15&    0&    0.25&    1.05&    0.18 \\
     0&    0.27&    0&    1.04&    0.15&    1.29&    0.11
\end{array} \right) .$$
Figure~\ref{fig:contour} shows the contour plots of $Q_p(\mathbf{\Lambda})$ for $\mathbf{\Lambda}$ satisfying the constraints in \eqref{eq:minimum}, with $p = 0.5$ and $1$, respectively.  It is not difficult to show that when $K=2$, $\mathbf{\Lambda}$ satisfying the constraints in \eqref{eq:minimum} has two degrees of freedoms, which are parameterised by $(\theta_1, \theta_2)$ in each plot. The point $(0,0)$, which is indicated by a black cross,
corresponds to $\mathbf{\Lambda} =\mathbf{\Lambda}^*$, and the point indicated by a red point  corresponds to the $\mathbf{\Lambda}$ matrix such that $Q_p(\mathbf{\Lambda})$ is minimised.  As we can see, when $p=0.5$, the loss function is minimised by $\mathbf{\Lambda}^*$. On the other hand, when $p=1$, the minimiser of the loss function does not contain as many zeros as $\mathbf{\Lambda}^*$.

Finally, we remark that due to the singularity of $L^p$ function near zero when $p<1$, the optimisation for $Q_p$ tends to be more challenging with a smaller value of $p$. This is also reflected by the contour plots in Figure~\ref{fig:contour}, where we see
$Q_{0.5}$ is very non-convex, even around the minimiser. On the other hand, $Q_{1}$ seems locally convex near the minimiser.  Therefore, although the $L^p$-rotation with $p<1$ may be better at recovering sparse loading matrices, its computation is more challenging.

\begin{comment}
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{L_p.png}
\end{figure}
\end{comment}
\begin{figure}[t]
\centering
     \begin{subfigure}{.45\textwidth}
         \centering
         \includegraphics[width=.9\linewidth]{L_p.pdf}
         \caption{}

     \end{subfigure}
     \hfill
     \begin{subfigure}{.45\textwidth}
         \centering
         \includegraphics[width=.9\linewidth]{dL_p.pdf}
         \caption{}

     \end{subfigure}
\caption{Panel (a): Plots of $|x|^p$, for different choices of $p$. Panel (b): Plots of the derivative of  $|x|^p$, for different choices of $p$. }
\label{fig:lp}
\end{figure}

\begin{figure}[t]
\centering
     \begin{subfigure}{.45\textwidth}
         \centering
         \includegraphics[width=.9\linewidth]{p=0.5.png}
         \caption{}

     \end{subfigure}
     \hfill
     \begin{subfigure}{.45\textwidth}
         \centering
         \includegraphics[width=.9\linewidth]{p=1.png}
         \caption{}

     \end{subfigure}
        \caption{Plots of contours of $|\boldsymbol{\Lambda}^{*}\mathbf{T}^{-1'}|^p$, where $\mathbf{T}=[ \cos(\theta_1),\sin(\theta_2);\sin(\theta_1),\cos(\theta_2)]$. Panel (a): $p=0.5$. Panel (b): $p=1$.}
        \label{fig:contour}
\end{figure}

 \subsection{Connection with regularised estimation }

The proposed rotation criteria have a close connection with regularised estimators for EFA. In what follows, we establish this connection. Recall that the proposed procedure relies on an initial estimator of the loading matrix for which $\boldsymbol\Phi$ is constrained to be an identity matrix. We further require it to be an $M$-estimator, obtained by minimising a certain loss function, denoted by $L(\boldsymbol{\Sigma(\theta)})$. Note that all the popular EFA estimators are all $M$-estimators. For instance, when the maximum likelihood estimator is used, then  {the loss function to be minimised is}
$$L(\boldsymbol{\Sigma(\theta)}) = \log \text{det} (\boldsymbol{\Sigma(\theta)}) +  \text{tr}(\boldsymbol{\Sigma(\theta)}^{-1} \boldsymbol{S}),$$
where $
\boldsymbol{S} = (\sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^\top)/N
$ is the sample covariance matrix.

Now we introduce an $L^p$ regularised estimator based on the loss function  $L(\boldsymbol{\Sigma(\theta)})$ in the form
\begin{equation}\label{eq:regul}
\hat \boldsymbol \theta_{\gamma,p} \in \operatornamewithlimits{arg\,min}_{\boldsymbol \theta} L(\boldsymbol{\Sigma(\theta)}) + \gamma  \sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|^p,
\end{equation}
where $\gamma > 0$ is a tuning parameter and the covariance matrix $\boldsymbol \Phi$ is estimated rather than constrained to be an identity matrix. We note that the minimiser of \eqref{eq:regul} is also not unique due to column permutation and sign flips similar to the non-uniqueness of optimisation \eqref{eq:rotation}. We denote the set of minimisers as
$$\hat{\mathcal{C}}_{\gamma,p} = \operatornamewithlimits{arg\,min}_{\boldsymbol \theta} L(\boldsymbol{\Sigma(\theta)}) + \gamma  \sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|^p.$$
Note that the regularisation term takes the same form as the $L^p$ CLF. It is used to impose sparsity on the estimate of the loading matrix. When $p=1$, it becomes a LASSO regularised estimator that has been considered in for example \citet{choi2010penalized}, \citet{hirose2014estimation, hirose2015sparse}, \citet{jin2018approximated} and \citet{geminiani2021single}.
The regularised estimator \eqref{eq:regul} is similar in spirit to $L^p$-regularised regression \citep{mazumder2011sparsenet,lai2011unconstrained,zheng2017does}, where the $L^p$ regularisation with $p<1$ has been shown to  better recover sparse signals under high-dimensional linear regression settings while computationally more challenging \citep{zheng2017does}.

As summarised in Proposition~\ref{prop:regul} below, the proposed $L^p$ rotation solution can be viewed as a limiting case of the $L^p$-regularised estimator when the tuning parameter $\gamma$ converges to zero.

\begin{proposition}\label{prop:regul}

Consider a fixed $p \in (0,1]$ and a fixed dataset. Suppose that for any sufficiently small $\gamma > 0$, $\hat{\mathcal{C}}_{\gamma,p}$ only contains $n = 2^KK!$ elements that are equivalent up to column permutation and sign flips of the loading matrix, where $K!$ denotes $K$ factorial that counts the number of all possible permutations and $2^K$ gives the total number of sign flip transformations. Furthermore, assume that for any sufficiently small $\gamma > 0$, one can label the elements of $\hat{\mathcal{C}}_{\gamma,p}$, denoted by $\hat{\boldsymbol \theta}_{\gamma, p}^{(i)}$,  $i=1, ..., n$, such that there exists a sufficiently small constant $\delta > 0$,
$\hat{\boldsymbol \theta}_{\gamma, p}^{(i)}$ is a continuous and bounded function of $\gamma$ in $(0, \delta)$, for each $i$. Then the limit
$$\hat{\boldsymbol \theta}_{0, p}^{(i)} = (\hat{\boldsymbol\Lambda}_{0,p}^{(i)}, \hat{\boldsymbol\Phi}_{0,p}^{(i)}, \hat{\boldsymbol\Omega}_{0,p}^{(i)}) = \lim_{\gamma \rightarrow 0} \hat{\boldsymbol \theta}_{\gamma, p}^{(i)} $$
exists, and $\hat{\boldsymbol \theta}_{0, p}^{(i)}$ satisfies that
$(\hat{\boldsymbol\Lambda}_{0,p}^{(i)}, \hat{\boldsymbol\Phi}_{0,p}^{(i)})$ solves the optimisation problem \eqref{eq:rotation2} and  $\hat{\boldsymbol\Omega}_{0,p}^{(i)} = \hat{\boldsymbol\Omega}$, where $\hat{\boldsymbol{\theta}} = (\hat{\mathbf{A}}, \mathbf{I},\hat{\boldsymbol{\Omega}})$ minimises the loss function $L(\boldsymbol{\Sigma(\theta)})$.

\end{proposition}

We now discuss the implications of this connection. First, if we have a numerical solver for the regularised estimator \eqref{eq:regul}, then one can obtain an approximate solution to the $L^p$-rotation problem \eqref{eq:rotation2} by using a sufficiently small tuning parameter $\gamma$.  Second, thanks to this connection, the choice between regularised estimation and rotation becomes the choice of the tuning parameter in regularised estimation. Note that the tuning parameter $\gamma$ corresponds to a bias-variance trade-off in estimating the model parameters $\boldsymbol \theta$. As $\gamma$ increases, the bias of the regularised estimator increases and the variance decreases. In applications where the sample size is large relative to the number of model parameters, the optimal choice of the tuning parameter is often close to zero.
In that case, it is a good idea to use the rotation method, as the regularised estimator under the optimal tuning parameter may not be substantially more accurate than the rotation solution and searching for the optimal tuning parameter can be computationally costly. The computation of these methods will be discussed in Section~\ref{sec:comp}.

\section{Statistical Inference and Asymptotic Theory}\label{sec:inf}

\subsection{Estimation Consistency}

We establish the statistical consistency of the proposed estimator based on the $L^p$ rotation.  Suppose the true parameter set
that we aim to recover is
$(\mathbf{\Lambda^*},\mathbf{\Phi^*}, \mathbf{\Omega^*})$, where the true loading matrix $\mathbf{\Lambda^*}$ is sparse.
To emphasise the dependence on the sample size, we attach the sample size $N$ as a subscript to the initial estimator in the first step of the rotation method; that is,
$\hat{\boldsymbol{\theta}}_N = (\hat{\mathbf{A}}_N, \mathbf{I},\hat{\boldsymbol{\Omega}}_N)$. We require the initial estimator to be consistent, in the sense that

\begin{enumerate}
\item[C1.]   $\mathbf{\hat{A}}_N  \mathbf{\hat{A}}_N' \overset{pr}{\to} \mathbf{\Lambda^*\Phi ^*}\mathbf{\Lambda^* }'$ and $  \hat{\boldsymbol{\Omega}}_N \overset{pr}{\to} \boldsymbol{\Omega}^*$, where the notation ``$\overset{pr}{\to}$" denotes convergence in probability.
\end{enumerate}

This requirement easily holds when the linear factor model is correctly specified and the loss function $L(\boldsymbol{\Sigma(\theta)})$ is reasonable (e.g., the negative log-likelihood). In addition, we require that the EFA model is truly a $K$-dimensional model, in the sense that condition C2 holds.

\begin{enumerate}
    \item[C2.]  $rank(\mathbf{\Lambda^*\Phi ^*}\mathbf{\Lambda^* }') =K$.
\end{enumerate}

For the  the $L^p$ rotation estimator to be consistent, for a specific value of $p \in (0, 1]$, we further require that the true loading matrix uniquely minimises the $L^p$ CLF, in the sense of condition C3 below.
\begin{enumerate}
\item[C3.]

$( \boldsymbol\Lambda^*, {\boldsymbol\Phi}^*) \in \operatornamewithlimits{arg\,min}_{\boldsymbol\Lambda, \boldsymbol\Phi} Q_p(\boldsymbol\Lambda)  \mbox{~such that~} \boldsymbol\Lambda \boldsymbol\Phi\boldsymbol\Lambda' = \boldsymbol\Lambda^* \boldsymbol\Phi^*\boldsymbol\Lambda^{*'}.$ In addition, for any other $( \boldsymbol\Lambda^\dagger, {\boldsymbol\Phi}^\dagger) \in \operatornamewithlimits{arg\,min}_{\boldsymbol\Lambda, \boldsymbol\Phi} Q_p(\boldsymbol\Lambda) \mbox{~such that~} \boldsymbol\Lambda \boldsymbol\Phi\boldsymbol\Lambda' = \boldsymbol\Lambda^* \boldsymbol\Phi^*\boldsymbol\Lambda^{*'}$, there exist $\mathbf D \in \mathcal D_1$ and $\tilde{\mathbf D} \in \mathcal D_2$, such that
${\boldsymbol\Lambda}^\dagger \mathbf D\tilde{\mathbf D} = \boldsymbol\Lambda^*$
and $\tilde{\mathbf D}^{-1} \mathbf  D^{-1}{\boldsymbol\Phi}^\dagger (\mathbf D^{-1})'(\tilde{\mathbf  D}^{-1})' = \boldsymbol\Phi^*$. Recall that $\mathcal D_1$ and $\mathcal D_2$ are the sets of column permutation and sign flip transformations, respectively, which are given in Remark~\ref{rmk:indet}.
\end{enumerate}

Condition C3 tends to hold when the true loading matrix contains many zeros, as the $L^p$ loss function is a good approximation to the $L^0$ function that counts the number of non-zero elements. In particular, according to Proposition~\ref{prop:simple},  condition C3 is guaranteed to hold when $\boldsymbol\Lambda^*$ has a perfect simple structure, i.e., if it has at most one nonzero loading in each row. As discussed in Section~\ref{subsec:propose}, this condition is more likely to hold for a smaller value of $p$, when there are cross loadings.
Conditions C1 through C3 guarantee the estimation consistency of the $L^p$ rotation estimator, up to column permutation and sign flips. This result is summarised in Theorem~\ref{thm:consistency} below.

\begin{theorem}\label{thm:consistency}
Suppose that for a given $p\in (0, 1]$ conditions C1 through C3 hold. Then there exist $\mathbf D_{N} \in \mathcal D_1$ and
$\tilde{\mathbf D}_{N} \in \mathcal D_2$, such that
$\hat{\boldsymbol\Lambda}_{N,p} \mathbf D_{N}\tilde{\mathbf D}_{N} \overset{pr}{\to} \boldsymbol\Lambda^*$ and $\tilde{\mathbf D}_{N}^{-1} \mathbf  D_{N}^{-1}\hat{\boldsymbol\Phi}_{N,p} (\mathbf D_{N}^{-1})'(\tilde{\mathbf  D}_{N}^{-1})' \overset{pr}{\to} \boldsymbol\Phi^*$, where
\begin{equation*
  (\hat{\boldsymbol\Lambda}_{N,p}, \hat{\boldsymbol\Phi}_{N,p}) \in \operatornamewithlimits{arg\,min}_{\boldsymbol\Lambda, \boldsymbol\Phi} Q_p(\boldsymbol\Lambda), \mbox{~such that~} \boldsymbol\Lambda \boldsymbol\Phi\boldsymbol\Lambda' = \hat{\mathbf A}_N\hat{\mathbf A}_N'.
\end{equation*}
\end{theorem}

\subsection{Model Selection}

The interpretation of the factors relies on the sign pattern of the loading matrix, so that each factor can be interpreted based on the associated manifest variables and their directions (positive or negative associations).
Learning this sign pattern is a model selection problem. A regularised estimator may seem advantageous as it yields simultaneous parameter estimation and model selection. We note that, however, model selection can be easily achieved with a rotation method, using a Hard-Thresholding (HT) procedure. Similar HT procedures have been proven successful in the model selection for linear regression models
\citep{meinshausen2009lasso}.

More precisely, let $\boldsymbol\Gamma^* = \left(\mbox{sgn}(\lambda_{jk}^*)\right)_{J\times K}$ denote the true sign pattern of $\boldsymbol\Lambda^*$, where $\mbox{sgn}(x)$ returns the sign of a scalar satisfying that
$$\mbox{sgn}(x) = \left\{\begin{array}{ccc}
    1 & \mbox{~if~} x > 0,  \\
    0 & \mbox{~if~} x = 0,  \\
    -1& \mbox{~if~} x < 0.
\end{array}\right.$$
Given the $L^p$ rotation estimator $\hat{\boldsymbol\Lambda}_{N,p} = \left(\hat \lambda_{jk}^{(N,p)}\right)_{J\times K}$,
the HT procedure estimates the pattern of $\boldsymbol\Gamma^*$ by
$\hat{\boldsymbol\Gamma}_{N,p} = \left(\mbox{sgn}(\hat \lambda_{jk}^{(N,p)}) \times 1_{\{|\hat \lambda_{jk}^{(N,p)}|>c\}}\right)_{J\times K}$, where $c > 0$ is a pre-specified threshold.  If the threshold $c$ is chosen properly, then $\hat{\boldsymbol\Gamma}_{N,p} $ consistently estimates
$\boldsymbol\Gamma^*$. We state this result in Theorem~\ref{thm:selection} below.

\begin{enumerate}
    \item[C4.] The threshold $c$ lies in the interval $(0, c_0)$, where
    $c_0 = \min\{ |\lambda_{jk}^*|: \lambda_{jk}^*\neq 0\}$.
\end{enumerate}

\begin{theorem}\label{thm:selection}
Suppose that for a given $p\in (0, 1]$ conditions C1 through C4 hold. Then there exist  $\mathbf D_{N} \in \mathcal D_1$ and $\tilde{\mathbf D}_{N} \in \mathcal D_2$, such that the probability
$P(\hat{\boldsymbol\Gamma}_{N,p}\mathbf D_{N}\tilde{\mathbf D}_{N} = \boldsymbol\Gamma^*)$
converges to 1 as the sample size $N$ goes to infinity.
\end{theorem}

In practice, the value of $c_0$ is unknown and thus cannot be used for choosing the threshold $c$. Instead,
we choose  $c$ based on the Bayesian Information Criterion
\citep[BIC;][]{schwarz1978estimating}. The steps
of this procedure are summarised in Algorithm~\ref{alg:bic} below, where the notation is simplified for the ease of exposition.

\begin{algorithm}
\caption{Hard-thresholding for model selection based on $L^p$ rotation}\label{alg:bic}
\textbf{Input:} A sequence of candidate thresholds $\mathcal C$, observed data, and the rotated loading matrix $\hat{\boldsymbol\Lambda} = (\hat \lambda_{jk})_{J\times K}$ given by the $L^p$ CLF criterion.

\medskip
For each value of $c \in \mathcal C$, we perform the following two steps: \begin{enumerate}
    \item[] \textbf{Step 1:} Obtain the corresponding selected loading structure $\hat{\boldsymbol\Gamma}_c = \left( \mbox{sgn}(\hat\lambda_{jk})\times 1_{\{|\hat \lambda_{jk}|>c\}}\right)_{J\times K}$.

\item[] \textbf{Step 2:} Fit a Confirmatory Factor Analysis (CFA) model based on $\hat{\boldsymbol\Gamma}_c$ using the maximum likelihood estimator, in which the $(i,j)$th loading parameter satisfies the sign constraint implied by the corresponding entry of $\hat{\boldsymbol\Gamma}_c$.   Calculate the BIC value for this CFA model, denoted by $\mbox{BIC}_c$.
\end{enumerate}

Obtain
$\hat c = \operatornamewithlimits{arg\,min}_{c\in \mathcal C} \mbox{BIC}_c.$

\medskip
\textbf{Output:} The selected sign pattern $\hat{\boldsymbol\Gamma}_{\hat c}$.
\end{algorithm}

When the candidate values of $c$ are chosen properly (i.e., $\mathcal C$ includes values that are below $c_0$), then Theorem~\ref{thm:selection} implies that with probability tending to one, the true model will be in the candidate models. Together with the consistency of BIC for parametric models \citep{shao1997asymptotic,vrieze2012model}, the true non-zero pattern can be consistently recovered.

\subsection{Confidence Intervals}

Often, we are not only interested in the point estimate of the underlying sparse loading matrix, but also in quantifying its uncertainty. Uncertainty quantification is typically achieved by constructing confidence intervals for the loadings of the rotated solution. Traditionally, this is achieved by establishing the asympotic normality of the rotated loading matrix using the delta method, which involves calculating the partial
derivatives of a rotation algorithm using implicit differentiation \citep{jennrich1973standard}. Unfortunately, this procedure is no longer suitable, if the true loading matrix is sparse and the loss function is not differentiable with respect to the zero loadings.

Motivated by a simple while well-performing post-selection inference procedure in regression analysis \citep{zhao2021defense}, we propose a procedure for constructing confidence intervals for individual loading parameters of the rotated solution. More precisely, this procedure runs a loop over all the manifest variables, $j=1, ..., J$. Each time, the procedure obtains the confidence intervals for the loading parameters of manifest variable $j$, by fitting a CFA model whose the loading structure is determined by the selected sign pattern  of the rest $J-1$ manifest variables. More precisely, the loading parameters of the CFA model satisfy the sign constraints imposed by  the selected sign pattern $\hat{\boldsymbol\Gamma}_{\hat c}$ from Algorithm~\ref{alg:bic}, for all the items except for $j$. No constraint is imposed on the loading parameters of item $j$. Confidence intervals for the loading parameters of item $j$ are obtained based on the asymptotic normality of the estimator for this CFA model.
We summarise this procedure in Algorithm~\ref{alg:CI} below.

\begin{algorithm}
\caption{Post-selection confidence intervals}\label{alg:CI}
\textbf{Input:}  The selected sign pattern $\hat{\boldsymbol\Gamma} = (\hat \gamma_{jk})_{J\times K}$, observed data, and significance level $\alpha \in (0,1)$.

\medskip

For each manifest variable $s = 1, ..., J$, we perform the following two steps.
\begin{enumerate}
    \item[] \textbf{Step 1:} Obtain a CFA model whose loadings $\lambda_{jk}$ satisfy the constraints that
    $\mbox{sgn}(\lambda_{jk}) = \hat \gamma_{jk}$ for all $j \neq s$.

\item[] \textbf{Step 2:}  Fit the CFA model and obtain the $(1-\alpha)$-confidence intervals for parameters $\lambda_{s1}$, ..., $\lambda_{sK}$ using a standard inference procedure for CFA (e.g., based on the maximum likelihood estimator). We denote these confidence intervals by $(l_{sk}, u_{sk})$. If the CFA model in \textbf{Step 1} is not identifiable, we let the confidence intervals to be $(-\infty, \infty)$.
\end{enumerate}
\textbf{Output:} Confidence intervals $(l_{sk}, u_{sk})$, $s = 1, ..., J, k=1, ..., K$.
\end{algorithm}

In what follows, we establish the consistency of confidence intervals given by Algorithm~\ref{alg:CI}. To emphasize that the statistics in Algorithm~\ref{alg:CI} depend on the sample size $N$, we attach $N$ as a subscript or superscript when describing this consistency result. We require the following conditions.

\begin{enumerate}
    \item[C5.] The selected sign pattern $\hat{\boldsymbol\Gamma}_N$ is consistent. That is, there exist  $\mathbf D_{N} \in \mathcal D_1$ and $\tilde{\mathbf D}_{N} \in \mathcal D_2$, such that the probability
$P(\hat{\boldsymbol\Gamma}_{N,p}\mathbf D_{N}\tilde{\mathbf D}_{N} = \boldsymbol\Gamma^*)$
converges to 1 as the sample size $N$ goes to infinity.
\end{enumerate}
Thanks to the consistency of BIC selection and when the candidate thresholds are chosen properly, condition C5 holds if $\hat{\boldsymbol\Gamma}_N$ is obtained by Algorithm \ref{alg:bic}.

\begin{enumerate}
    \item[C6.]  For each manifest variable $s = 1, ..., J$, the CFA model whose loading parameters satisfy
    $\mbox{sgn}(\lambda_{jk}) =  \mbox{sgn}(\lambda_{jk}^*)$ for all $j \neq s$ is identifiable, and using the same procedure in Step 2 of Algorithm~\ref{alg:CI} leads to consistent confidence intervals for $\lambda_{s1}$, ..., $\lambda_{sK}$.  That is, let $(l_{sk}^{*(N)}, u_{sk}^{*(N)})$ be the resulting confidence interval for $\lambda_{sk}$, then
    $P(\lambda_{sk}^* \in (l_{sk}^{*(N)}, u_{sk}^{*(N)}))$ converges to $1-\alpha$, as the sample size $N$ goes to infinity.
\end{enumerate}
Note that C6 is a condition imposed on the sign pattern of the true loading matrix. It essentially requires that the factors can be identified by the sign pattern of any $(J-1)$-subset of the manifest variables. Given an identified CFA model, the consistent confidence intervals can be easily constructed based on the asymptotic normality of any reasonable estimator of the CFA model, e.g., the maximum likelihood estimator.
Under conditions C5 and C6, the following theorem holds.

\begin{theorem}\label{thm:ci}
Suppose that conditions C5 and C6 hold for the selected sign pattern $\hat{\boldsymbol\Gamma}_N$ and the true model, where  $\mathbf D_{N} \in \mathcal D_1$ and
$\tilde{\mathbf D}_{N} \in \mathcal D_2$ are from condition C5. Suppose we input $\hat{\boldsymbol\Gamma}_N$, observed data from the true model, and significance level $\alpha$ into the true model, and obtain output
$(l_{sk}^{(N)}, u_{sk}^{(N)})$, $s = 1, ..., J, k=1, ..., K$.
Then we have
$P(\lambda_{sk}^{*(N)} \in (l_{sk}^{(N)}, u_{sk}^{(N)}))$
converges to $1-\alpha$, for all  $s = 1, ..., J, k=1, ..., K$,
where $\lambda_{sk}^{*(N)}$ are entries of $\boldsymbol\Lambda^{*(N)} = \boldsymbol\Lambda^* \tilde{\mathbf D}_{N}^{-1}\mathbf D_{N}^{-1}$. Note that
$\boldsymbol\Lambda^{*(N)}$ is equivalent to $\boldsymbol\Lambda^{*}$ up to column permutation and sign flips.

We remark that under the conditions of Theorem  \ref{thm:ci}, all the CFA models fitted in Step 2 of Algorithm \ref{alg:CI} should be identifiable for sufficiently large $N$. However, in practice, it may happen that some CFA models are not identifiable, either due to that the sample size is not large enough or that the regularity conditions C5 or C6 does not hold. In that case, we set the corresponding confidence intervals  to be $(-\infty, \infty)$ as a conservative choice.

\end{theorem}

\section{Computation}\label{sec:comp}

\subsection{Proposed IRGP Algorithm}
We now discuss the computation for the proposed rotation.
Recall that we aim to solve the optimisation problem
$$\mathbf{\hat{T}} \in \operatornamewithlimits{arg\,min}_{ \mathbf{T} \in \mathcal{M}} Q_p(\hat{\mathbf{A}} \mathbf{T'}^{-1}),$$
where $Q_p$ is the $L^p$ CLF defined in \eqref{eq:CLC_Lp}. Note that this objective function is not differentiable when $\hat{\mathbf{A}} \mathbf{T'}^{-1}$ has zero elements, as the $L^p$ function is not smooth at zero. Consequently, standard numerical solvers fail, especially when the true solution is approximately sparse. To solve this optimisation problem, we develop an Iteratively Reweighted Gradient Projection (IRGP) algorithm that combines the iteratively reweighted least square algorithm \citep{ba2013convergence,daubechies2010iteratively} and the gradient projection algorithm \citep{jennrich2002simple}.

Similar to \citet{jennrich2006rotation}, the IRGP algorithm also solves a smooth approximation to the objective function $Q_p(\hat{\mathbf{A}} \mathbf{T'}^{-1})$. That is, we introduce a sufficiently small constant $\epsilon > 0$, and approximate the objective function by $Q_{p,\epsilon}(\hat{\mathbf{A}} \mathbf{T'}^{-1})$,
where
$$Q_{p,\epsilon}(\boldsymbol\Lambda) = \sum_{j=1}^J\sum_{k=1}^K (\epsilon^2 + \lambda_{jk}^2)^{\frac{p}{2}}.$$
As discussed in the sequel, the $\epsilon$ is introduced to robustify the computation. The IRGP algorithm alternates between two steps -- (1) function approximation step and (2) Projected Gradient Descent (PGD) step. More precisely, let
$T_t$ be the parameter value at the $t$th iteration.

The function approximation step involves approximating the objective function by
\begin{equation}\label{eq:appr}
G_t(\mathbf T) = \sum_{j=1}^J\sum_{k=1}^K w_{jk}^{(t)} \left((\hat{\mathbf{A}} \mathbf{T'}^{-1})_{jk}\right)^2,
\end{equation}
where the weights $w_{jk}^{(t)}$ are given by
$$w_{jk}^{(t)} = \frac{1}{((\hat{\mathbf{A}} (\mathbf{T}_t')^{-1})_{jk}^2 + \epsilon^2)^{\frac{p}{2}}}.$$
Here $\epsilon > 0$ is a pre-specified parameter that is chosen sufficiently small. We provide some remarks about  this approximation. First, the small tuning parameter is chosen to stabilise the algorithm when certain  $\hat{\mathbf{A}} (\mathbf{T}_t')^{-1})_{jk}$s are close to zero. Without $\epsilon$, the weight
$w_{jk}^{(t)}$ can become vary large, resulting in an unstable PGD step. Second, supposing that $(\hat{\mathbf{A}} (\mathbf{T}_t')^{-1})_{jk} \neq 0$ for all $j$ and $k$, then $G_t(\mathbf{T}_t) \approx Q_p(\hat{\mathbf{A}} (\mathbf{T}'_t)^{-1})$ when $\epsilon$ is sufficiently small, i.e., the function approximation and the objective function value are close to each other at the current
parameter value. Lastly, this approximation is similar to the E-step of the Expectation-Maximisation algorithm \citep{dempster1977maximum}; see \citet{ba2013convergence} for this correspondence.

The PGD step involves updating the parameter value based on the $G_t(\mathbf T)$ via projected gradient descent.
This step is similar to the update in each iteration of the gradient projection algorithm for oblique rotations \citep{jennrich2002simple}.
PGD can be performed to $G_t(\mathbf T)$, as this function approximation is smooth in $\mathbf T$. More precisely, we define a projection operator as
\begin{equation}\label{eq:proj}
    \mbox{Proj}(\mathbf T) = \mathbf T (\mbox{diag}(\mathbf T' \mathbf T))^{-\frac{1}{2}},
\end{equation}
where $(\mbox{diag}(\mathbf T' \mathbf T))^{-\frac{1}{2}}$ is a diagonal matrix whose $i$th diagonal entry is given by
$1/\sqrt{(\mathbf T' \mathbf T)_{ii}}$. This operator projects any invertible matrix into the space of oblique matrices $\mathcal M$.  The PGD update is given by
\begin{equation}\label{eq:PGD}
\mathbf T_{t+1} = \mbox{Proj}(\mathbf T_t - \alpha \grad G_t(\mathbf T)),
\end{equation}
where $\alpha > 0$ is a step size chosen by line search
and $\grad G_t(\mathbf T)$ is a $K\times K$ matrix whose $(i,j)$th entry is   the partial derivative of $G_t(\mathbf T)$ with respect to $(i,j)$th entry of $\mathbf T$. We summarise the IRGP algorithm below.

\begin{algorithm}
\caption{IRGP algorithm for $L^p$ rotation}\label{alg:cap}
\textbf{Input:} The initial loading matrix estimate $\hat{\mathbf{A}}$, parameter $\epsilon >0$,
and an initial value $\mathbf T_0$.

\medskip

For iterations $t = 0, 1, 2, ...$, we iterate between the following two steps:

\begin{enumerate}
\item[] \textbf{Step 1:} Construct $G_t(\mathbf T)$ using equation \eqref{eq:appr}.
\item[] \textbf{Step 2:} Obtain $\mathbf T_{t+1}$  using equation \eqref{eq:PGD}, where the step size $\alpha$ is chosen by line search.
\end{enumerate}

Stop until the convergence criterion is met. Let $t_{max}$ be the final iteration number.

\medskip
\textbf{Output:} $\mathbf{T}_{t_{max}}$.
\end{algorithm}

Under reasonable regularity conditions \citep{ba2013convergence}, every limit point of $\{\mathbf T_t\}_{t=1}^\infty$ will be a stationary point of the approximated objective function
$Q_{p,\epsilon}(\hat{\mathbf{A}} \mathbf{T'}^{-1})$.
In addition, the algorithm has local linear convergence
when $p = 1$ and super-linear convergence when $0<p<1$.

We remark on the choice of initial value $\mathbf T_0$ when $0< p <1$. As discussed previously in Section~\ref{subsec:propose}, when $0< p <1$, the objective function $Q_p(\hat{\mathbf{A}} \mathbf{T'}^{-1})$ is highly nonconvex, and thus, may contain many stationary points. To avoid the algorithm getting stuck at a local optimum, the choice of $\mathbf T_0$ is important. We recommend trying multiple values of $p$, starting with $p=1$. For each value of $p$, we can use
the solution from the previous larger value of $p$ as
the initial value $\mathbf T_0$.

\subsection{Comparison with Regularised Estimation}

To compare the computation of the proposed rotation method and that of regularised estimation, we also describe a proximal gradient algorithm for the $L_1$ regularised estimator.
The proximal algorithm is a state-of-the-art algorithm for solving non-smooth optimisation problems \citep{parikh2014proximal}. It can be viewed as a generalisation of projected gradient descent. As will be discussed below, each iteration of the algorithm can be computed easily. In principle, the proximal algorithm can also be applied to the $L_p$ regularised estimator, for $0 < p <1$. However, it is computationally much more costly than the case when $p=1$, and thus, will not be discussed here.

The $L_1$ regularised estimator, also referred to as the Lasso estimator, solves the following optimisation problem
$$\min_{\boldsymbol \theta}~~ L(\boldsymbol{\Sigma}(\boldsymbol\theta)) + \gamma  \sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|.$$
To apply the proximal gradient algorithm, we reparameterise the covariance matrix $\boldsymbol\Phi$ by
$\mathbf T' \mathbf T$, and reparameterise the diagonal entries of the diagonal matrix $\boldsymbol\Omega$ by $\mathbf v = (v_{1}, ..., v_{J})$, where $v_i = \log(\omega_{ii})$. With slight abuse of notation, we can write the optimisation problem as
$$\min_{\boldsymbol\Lambda, \mathbf T, \mathbf v}~~ L(\boldsymbol{\Sigma}(\boldsymbol\Lambda, \mathbf T, \mathbf v)) + \gamma  \sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|.$$

We define a proximal operator for the loading matrix as
\begin{equation}\label{eq:prox}
\mbox{Prox}_{\alpha, \gamma}(\tilde{\boldsymbol \Lambda}_t) = \operatornamewithlimits{arg\,min}_{\boldsymbol\Lambda} ~~\frac{1}{2} \sum_{j=1}^J\sum_{k=1}^K (\lambda_{jk} - \tilde\lambda_{jk}^{(t)})^2 + \alpha\gamma \sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|,
\end{equation}
where $\alpha > 0$ will be a step size  and $\tilde{\boldsymbol \Lambda}_t = (\tilde\lambda_{jk}^{(t)})_{J\times K}$
will be the value of $\boldsymbol \Lambda$ from the previous step in the proximal gradient algorithm. Note that  \eqref{eq:prox} has a closed-form solution given by soft-thresholding \citep{parikh2014proximal} that can be easily computed. We summarise the proximal gradient algorithm in Algorithm
\ref{alg:prox} below.

\begin{algorithm}
\caption{Proximal gradient algorithm for $L^1$ regularised estimation.}\label{alg:prox}
\textbf{Input:} The initial values $\boldsymbol \Lambda_0$, $\mathbf T_0$, and $\mathbf v_0$.

\medskip

For iterations $t = 0, 1, 2, ...$, we iterate between the following two steps:

\begin{enumerate}
\item[] \textbf{Step 1:}  Calculate the gradients of $L(\boldsymbol{\Sigma}(\boldsymbol\Lambda, \mathbf T, \mathbf v))$  with respect to $\boldsymbol\Lambda$, $\mathbf T$, and $\mathbf v$, respectively, at $(\boldsymbol \Lambda_t, \mathbf T_t, \mathbf v_t)$. Denote these gradients by $\grad L_{t,\boldsymbol\Lambda}$
$\grad L_{t,\mathbf T}$, and $\grad L_{t,\mathbf v}$.

\item[] \textbf{Step 2:} Update the parameters by
$$\boldsymbol \Lambda_{t+1} = \mbox{Prox}_{\alpha, \gamma}(\boldsymbol \Lambda_t - \alpha \grad L_{t,\boldsymbol\Lambda}),$$
$$\mathbf T_{t+1} = \mbox{Proj}(\mathbf T_t - \alpha \grad L_{t,\mathbf T}),$$
and
$$\mathbf v_{t+1} = \mathbf v_t - \alpha \grad L_{t,\mathbf v}.$$

Recall that the operator $\mbox{Proj}(\cdot)$ is defined in \eqref{eq:proj}, and $\alpha$ is a step size chosen by line search.

\end{enumerate}

Stop until the convergence criterion is met. Let $t_{max}$ be the final iteration number.

\medskip
\textbf{Output:} $(\boldsymbol \Lambda_{t_{max}}, \mathbf T_{t_{max}}, \mathbf v_{t_{max}})$.
\end{algorithm}
Under suitable conditions, this proximal gradient algorithm converges to stationary points of the objective function, and has local linear convergence rate \citep{karimi2016linear}.
We notice that when $p=1$, Algorithms~\ref{alg:cap} and \ref{alg:prox} have similar convergence properties.  However, their per-iteration computational complexities
are different. In particular,  Algorithm \ref{alg:prox}
involve parameters $\boldsymbol\Lambda$ and $\mathbf v$, which substantially increases its computational complexity. If fact, the per-iteration complexity for Algorithm~\ref{alg:cap} is $O(K^3+K^2J)$, while that for Algorithm~\ref{alg:prox} is $O(J^3+J^2K+K^2J)$. The difference can be substantial, when $J$ is much larger than $K$.

\section{Simulation Study}\label{sec:sim}

\subsection{Study I}

In this study, we evaluate the performance of $L^{0.5}$ and $L^1$ rotations, and compare them  with some traditional rotation methods and $L^1$ regularised estimation. Several traditional oblique rotation methods are considered, including the oblimin, quartmin, simplimax, geomin and promax methods. These methods have been considered in the simulation studies in \citet{jennrich2006rotation}.
They are implemented using the \texttt{GPArotation} package \citep{bernaards2005gradient} in R.

\paragraph{Settings.}
Two simulation settings are considered, one with $J=15$ manifest variables and $K=3$ factors, and the other with $J=30$ and $K=5$. The first setting has nine manifest variables that each loads on a single factor (three variables for each factor), and six manifest variables that each loads on two factors. The second setting has  15 manifest variables that each loads on a single factor (three variables for each factor), 10 manifest variables that each loads on two factors, and 5 manifest variables that each loads on three factors. The true model parameters are given in the supplementary material.
By numerical evaluations, the true loading matrices satisfies condition C3 for both $L^{0.5}$ and $L^1$ criteria. Under each setting, three sample sizes are considered, including $N = 400, 800$, and 1600.
For each setting and each sample size, we run $B = 500$ independent replications.

\paragraph{Evaluation criteria.} We evaluate the proposed method from three aspects. First, we compare all estimators in terms of accuracy of point estimation. Second, we compare the proposed method and the $L^1$ regularised estimator in terms of their model selection accuracy. Finally, we examine the coverage rate of the proposed method for constructing confidence intervals.

When evaluating the performance of different estimators,
we take into account the indeterminacy due to column permutations and sign flips. Let $\tilde{\mathbf{\Lambda}}^{(b)}$ be the loading matrix estimate given by a rotation or regularised estimation method in the $b$th replication. We then find
$$\hat{\mathbf{\Lambda}}^{(b)} = \operatornamewithlimits{arg\,min}_{\mathbf{\Lambda}}\{\Vert\mathbf{\Lambda} - \mathbf{\Lambda}^*\Vert^2: \mathbf{\Lambda} =\tilde{\mathbf{\Lambda}}^{(b)}\mathbf D \tilde{\mathbf D}, \mathbf D \in \mathcal D_1,\tilde{\mathbf D} \in \mathcal D_2 \},$$
which is the one closest to the true loading matrix $\mathbf{\Lambda}^*$ among all the loading matrices that are equivalent to $\tilde{\mathbf{\Lambda}}^{(b)}$. Our evaluation criteria are constructed based on
$\hat{\mathbf{\Lambda}}^{(b)}$:

\begin{enumerate}

\item The accuracy of point estimation is estimated by the mean squared error (MSE):
$$
\text{MSE} =   \frac{||\hat{\mathbf{\Lambda}}^{(b)}  - \mathbf{\Lambda}^*||_F^2}{JK},
$$
where $\hat{\mathbf{\Lambda}}^{(b)}$
is obtained by a certain rotation or regularisation method in the $b$-th replication.

\item The model selection accuracy is assessed by the area under the curve (AUC) from the corresponding receiver operating characteristic (ROC) curve. For each threshold $c$, we compute the average true positive rate ($\widebar{\text{TPR}}_c$), which is the proportion of successfully identified non-zero elements in the true loading matrix:
\begin{equation}\label{eq:TPR}
 \widebar{\text{TPR}}_c = \frac{1}{B} \sum_{b=1}^B\frac{\sum_{j,k}\mathbbm{1}_{ \{\hat{\lambda}^{(b,c)}_{jk} \neq 0, \lambda_{jk}^* \neq 0 \}} }{\sum_{j,k} \mathbbm{1}_{\{\lambda_{jk}^* \neq 0\}}},
\end{equation}
where
where
$ \{ \hat{\lambda}^{(b, c)}_{jk} \}_{J \times K}=
 \hat{\mathbf{\Lambda}}^{(b,c)}
$
is the estimated loading matrix in the $b$-th replication from a CFA model based on $\hat{\boldsymbol\Gamma}_c$ using the maximum likelihood estimator.  Similarly, we calculate the average true negative rate ($\widebar{\text{TNR}}_c$), which is the success rate of identifying zero elements:
\begin{equation}\label{eq:TNR}
 \widebar{\text{TNR}}_c = \frac{1}{B} \sum_{b=1}^B \frac{\sum_{j,k}\mathbbm{1}_{ \{\hat{\lambda}^{(b,c)}_{jk} = 0, \lambda_{jk}^* = 0 \}} }{\sum_{j,k} \mathbbm{1}_{\{\lambda_{jk}^* = 0\}}}.
\end{equation}
The AUC is consequently calculated by plotting $\widebar{\text{TPR}}_c$ against $1 - \widebar{\text{TNR}}_c$ by varying the threshold $c$. We also use the overall selection accuracy, i.e., the true selection rate ($\text{TR}$), to evaluate the model selection procedure described in Algorithm~\ref{alg:bic}. The TR is calculated as
$$
\text{TR} = \frac{1}{B} \sum_{b=1}^B\frac{\sum_{j,k}\mathbbm{1}_{ \{\hat{\lambda}^{(b,\hat{c})}_{jk} \neq 0, \lambda_{jk}^* \neq 0 \}} + \sum_{j,k}\mathbbm{1}_{ \{\hat{\lambda}^{(b,\hat{c})}_{jk} = 0, \lambda_{jk}^* = 0 \}}}{JK},
$$
where $\hat{c}$ is the BIC selected threshold value from Algorithm~\ref{alg:bic}. Correspondingly, we calculate the $\text{TPR}$ and $\text{TNR}$ of the selected model as
$$\text{TPR} = \widebar{\text{TPR}}_{\hat c} ~\mbox{and}~ \text{TNR} = \widebar{\text{TNR}}_{\hat c}.$$
\item The entry-wise 95\% confidence interval coverage rate (ECIC) is calculated to evaluate the performance of our post-selection confidence interval procedure in Algorithm~\ref{alg:CI}. For each entry of the loading matrix, the empirical probability of the true loading falling within the estimated confidence interval is calculated as
$$
\text{ECIC}_{jk}=\frac{\sum_{b=1}^B \mathbbm{1}_{ \{ \lambda_{jk}^{*(N)} \in (l_{jk}^{(N)}, u_{jk}^{(N)})\} }}{B}.
$$
\end{enumerate}

\paragraph{Results on point estimation.}

In Table \ref{tab:MSE}, the MSE of the estimated loading matrix is presented, for both simulation settings   and  $N \in \{400, 800, 1600\}$. The first five rows show the results based on traditional oblique rotation criteria, followed by the results of the proposed $L^p$ loss function for two choices of $p$, and finally those of the Lasso estimator for five choices of $\gamma$. For both settings and all sample sizes, geomin performed the best among the traditional rotation methods.
The geomin results were very similar to those of $L^p$ rotation and the Lasso estimator with sufficiently small tuning parameter $\gamma$. For the Lasso estimator, the MSE increased as $\gamma$ increased. For $L^p$ rotation, only very small differences were observed between $p = 0.5$ and $p = 1$. In addition, their
MSEs were close to those of the Lasso estimator with $\gamma = 0.01$ and $\gamma = 0.05$.

\begin{table}[h]
\centering
\caption{MSE obtained by using different rotation criteria under various settings, Study I.}
\label{tab:MSE}
\begin{tabular}{ccccccccccc}
\hline
&\multicolumn{3}{c}{$15 \times 3$}                                                    & \multicolumn{3}{c}{$30 \times 5$} \\
\hline

                               & $N=400$ & $N=800$ & $N=1600$  & $N=400$  & $N=800$  & $N=1600$ \\ \hline
\multicolumn{1}{l|}{Oblimin}   & 0.012 & 0.007 & \multicolumn{1}{c|}{0.004} & 0.012  & 0.008  & 0.006  \\
\multicolumn{1}{l|}{Quartimin} & 0.012 & 0.007 & \multicolumn{1}{c|}{0.004} & 0.012  & 0.008  & 0.006  \\
\multicolumn{1}{l|}{Simplimax} & 0.019 & 0.017 & \multicolumn{1}{c|}{0.019} & 0.021  & 0.019  & 0.017  \\
\multicolumn{1}{l|}{Geomin}   & 0.010 & 0.005 & \multicolumn{1}{c|}{0.002} & 0.010  & 0.005  & 0.002  \\
\multicolumn{1}{l|}{Promax}    & 0.013 & 0.007 & \multicolumn{1}{c|}{0.005} & 0.014  & 0.009  & 0.007  \\
\multicolumn{1}{l|}{$L^{0.5}$ rotation}   & 0.011 & 0.005 & \multicolumn{1}{c|}{0.003} & 0.010  & 0.005  & 0.002  \\
\multicolumn{1}{l|}{$L^1$ rotation}     & 0.010  & 0.005 & \multicolumn{1}{c|}{0.003} & 0.009  & 0.004  & 0.002  \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.01$} & 0.009 & 0.004 & \multicolumn{1}{c|}{0.002} & 0.008  & 0.003  & 0.002  \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.05$} & 0.009 & 0.006 & \multicolumn{1}{c|}{0.005} & 0.007  & 0.005  & 0.004  \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.1$} & 0.017 & 0.015 & \multicolumn{1}{c|}{0.014} & 0.012  & 0.011  & 0.010  \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.2$} & 0.079 & 0.076 & \multicolumn{1}{c|}{0.074} & 0.037  & 0.034  & 0.032  \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.5$} & 0.244 & 0.244 & \multicolumn{1}{c|}{0.244} & 0.146  & 0.149  & 0.150  \\ \hline
\end{tabular}

\end{table}

\paragraph{Results on model selection.}
In Table \ref{tab:modacc}, the AUC, TR, TPR and TNR are presented for the $L^p$ rotations and the Lasso estimator with different tuning parameters.
For both scenarios and all sample sizes, the AUC and TR were very similar for the rotation estimator with $p = 0.5$ and $p = 1$. The AUC of the Lasso estimator with a small tuning parameter is similar to that of the $L^1$ rotation method.
We noted that the model selection performance was poor for the Lasso estimator when $\gamma$ became large. This is due to the presence of many false negative selections (i.e., nonzero loading parameters selected as zeros), as a result of over-regularisation.

\begin{table}[h]
\centering
\caption{The AUC, TR, TPR and TNR for the $L^p$-based rotation estimator and the regularised estimator, Study I.}
\label{tab:modacc}
\begin{tabular}{lcccccccc}
\hline
                               & \multicolumn{4}{c|}{$15 \times 3$}                         & \multicolumn{4}{c}{$30 \times 5$}     \\
\hline
                               & AUC   & TR    & TPR   & \multicolumn{1}{l|}{TNR}   & AUC   & TR    & TPR   & TNR   \\ \hline
                               & \multicolumn{7}{c}{$N = 400$}                                          \\ \hline
\multicolumn{1}{l|}{$L^{0.5}$ rotation}   & 0.996 & 0.979 & 0.979 & \multicolumn{1}{l|}{0.979}  & 0.989 & 0.965 & 0.93  & 0.982 \\
\multicolumn{1}{l|}{$L^1$ rotation}     & 0.997 & 0.979 & 0.979 & \multicolumn{1}{l|}{0.979}  & 0.988 & 0.964 & 0.928 & 0.982 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.01$} & 0.997 & 0.981 & 0.979 & \multicolumn{1}{l|}{0.983}  & 0.989 & 0.967 & 0.933 & 0.984 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.05$}  & 0.997 & 0.984 & 0.978 & \multicolumn{1}{l|}{0.989}  & 0.992 & 0.971 & 0.937 & 0.989 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.1$}  & 0.992 & 0.982 & 0.973 & \multicolumn{1}{l|}{0.991}  & 0.987 & 0.969 & 0.929 & 0.989 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.2$}  & 0.870 & 0.848 & 0.780 & \multicolumn{1}{l|}{0.907}  & 0.904 & 0.917 & 0.798 & 0.977 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.5$}  & 0.500 & 0.533 & 0.000 & \multicolumn{1}{l|}{1.000}  & 0.514 & 0.675 & 0.053 & 0.985 \\ \hline
                               & \multicolumn{7}{c}{$N = 800$}                                                                         \\ \hline
\multicolumn{1}{l|}{$L^{0.5}$ rotation}   & 1.000     & 0.993 & 0.992 & \multicolumn{1}{l|}{0.993}  & 0.998 & 0.987 & 0.978 & 0.992 \\
\multicolumn{1}{l|}{$L^1$ rotation}      & 1.000     & 0.992 & 0.993 & \multicolumn{1}{l|}{0.992}  & 0.998 & 0.987 & 0.978 & 0.991 \\

\multicolumn{1}{l|}{Lasso, $\gamma = 0.01$}  & 1.000     & 0.992 & 0.992 & \multicolumn{1}{l|}{0.993}  & 0.998 & 0.989 & 0.981 & 0.993 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.05$}  & 1.000     & 0.993 & 0.992 & \multicolumn{1}{l|}{0.993}  & 0.999 & 0.991 & 0.982 & 0.995 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.1$}   & 0.996 & 0.992 & 0.988 & \multicolumn{1}{l|}{0.995}  & 0.995 & 0.989 & 0.977 & 0.995 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.2$}   & 0.88  & 0.861 & 0.815 & \multicolumn{1}{l|}{0.902}  & 0.919 & 0.933 & 0.824 & 0.987 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.5$}   & 0.500   & 0.533 & 0.000     & \multicolumn{1}{l|}{1.000}      & 0.504 & 0.669 & 0.019 & 0.994 \\ \hline
                               & \multicolumn{7}{c}{$N = 1600$}                                                                        \\ \hline
\multicolumn{1}{l|}{$L^{0.5}$ rotation}    & 1.000 & 0.997 & 0.999 & \multicolumn{1}{l|}{0.996}  & 1.000 & 0.997 & 0.996 & 0.997 \\
\multicolumn{1}{l|}{$L^1$ rotation}      & 1.000 & 0.997 & 0.999 & \multicolumn{1}{l|}{0.995}  & 1.000 & 0.997 & 0.996 & 0.997 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.01$}  & 1.000 & 0.997 & 1.000 & \multicolumn{1}{l|}{0.995}  & 1.000 & 0.997 & 0.997 & 0.997 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.05$}  & 1.000 & 0.997 & 0.999 & \multicolumn{1}{l|}{0.996}  & 1.000 & 0.997 & 0.997 & 0.997 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.1$}   & 0.998 & 0.997 & 0.995 & \multicolumn{1}{l|}{0.999}  & 0.998 & 0.996 & 0.993 & 0.998 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.2$}   & 0.886 & 0.870 & 0.831 & \multicolumn{1}{l|}{0.904}  & 0.929 & 0.941 & 0.838 & 0.993 \\
\multicolumn{1}{l|}{Lasso, $\gamma = 0.5$}   & 0.500 & 0.533 & 0.000 & \multicolumn{1}{l|}{1.000}  & 0.501 & 0.667 & 0.003 & 0.999 \\ \hline
\end{tabular}
\end{table}

\paragraph{Results on confidence intervals.} In Figure \ref{fig:ECIC}, boxplots of the ECIC are shown for the $L^p$ rotations, for $p = 0.5$ and $p = 1$ and  $N \in \{ 400, 800, 1600 \}$. For both $p = 0.5$ and $p = 1$, the
$\mbox{ECIC}_{jk}$s are close to the 95\% nominal level, supporting the consistency of the proposed procedure for constructing confidence intervals.

\paragraph{Some remarks.} Under the current simulation settings, condition C3 is satisfied by both the $L^{0.5}$  and  $L^1$ criteria, in which cases the two criteria tend to perform similarly.  As will be shown in Section~\ref{subsec:study2} below, the performance of the two criteria can be substantially different when C3 holds for one criterion but not the other.
In addition, we see that the Lasso estimator with a small tuning parameter performed similarly as the $L^1$ rotation method. This is expected, since the
$L^1$ rotation solution can be viewed as the limiting case of the Lasso estimator when the tuning parameter goes to zero. The Lasso estimator performed poorly for large tuning parameters, due to the bias brought by the regularisation.
Finally, it is not clear to us why the geomin rotation performs similarly as the $L^{0.5}$ and  $L^1$ criteria, which is worth future investigation.

\begin{figure}[h]
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{15_3.pdf}
\caption{Boxplots of $\mbox{ECIC}_{jk}$ for the $15 \times 3$ setting}
\end{subfigure}
\newline
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{30_5.pdf}
\caption{Boxplots of $\mbox{ECIC}_{jk}$ for the $30 \times 5$ setting}
\end{subfigure}
       \caption{Boxplots of $\mbox{ECIC}_{jk}$. The label 0 means that $\lambda_{jk}^*=0$ and the label 1 means that $\lambda_{jk}^*\neq 0$.}
        \label{fig:ECIC}
\end{figure}

\begin{comment}
\begin{figure}[h]
\includegraphics[width=\textwidth]{510_30_5_400.png}
\end{figure}

\begin{figure}[h]
\includegraphics[width=\textwidth]{510_30_5_800.png}
\end{figure}

\begin{figure}[h]
\includegraphics[width=\textwidth]{510_30_5_1600.png}
\end{figure}
\end{comment}

\subsection{Study II}\label{subsec:study2}

This study compares the $L^{0.5}$ and $L^1$ rotations,
under a setting where condition C3 holds for the $L^{0.5}$ rotation but not the $L^1$ rotation. The setting is chosen to somewhat exaggerate the differences, in order to show  the consequence of
misspecifying $p$.

\paragraph{Setting and evaluation criteria.}
The true loading matrix is of dimension $J = 18$ and $K = 3$.
Each
item is set to load on two factors, so that no item has a perfect simple structure.
Given the loading structure, the model is identifiable as a confirmatory factor analysis model. The true model parameters can be found in the supplementary material. By grid search, it is checked that C3 holds for the $L^{0.5}$ criterion but not the $L^{1}$ criterion. The sample size is chosen to be $N = 3000$. Similar to Study I, we compare the two rotation criteria using the MSE, AUC, TR, TPR and TNR by running $B = 500 $ independent replications.

\paragraph{Results.}
The results are presented in Table \ref{tab:counterex}.
The $L^{0.5}$ criterion performed better in terms of both point estimation and model selection, as its MSE was lower and the AUC, TR, TPR and TNR were higher.
In particular, we noted that the $L^{0.5}$ rotation achieved a much higher TNR than the $L^1$ rotation, meaning that the
$L^1$ rotation tended to make many false positive selections (i.e., zero loading parameters selected as non-zeros), as a consequence of violating condition C3.

\begin{table}[t]
\centering
\caption{The MSE, AUC, TR, TPR and TNR for the $L^{p}$-based rotation estimator, Study II.}
\label{tab:counterex}
\resizebox{0.5\textwidth}{!}{%
\begin{tabular}{lccccc}
\hline
                             & MSE   & AUC   & TR & TPR &TNR \\ \hline
\multicolumn{1}{l|}{$L^{0.5}$ rotation} & 0.003 & 0.984 & 0.951  & 0.944   & 0.967   \\
\multicolumn{1}{l|}{$L^1$ rotation}   & 0.025 & 0.954 & 0.864  & 0.940   & 0.712   \\
 \hline
\end{tabular}%
}
\end{table}

\section{An Application to Big-Five Personality Test} \label{sec:real}

We illustrate the proposed method through an application to the big-five personality test. We consider the Big-Five Factor Markers  from the International Personality Item Pool \citep[][]{goldberg1992development}, which contains 50 items that were designed to measure five personality factors, including  Extraversion (E), Emotional Stability (ES), Agreeableness (A), Conscientiousness (C), and Intellect/Imagination (I). Each item is a statement describing a personality pattern like ``\textit{I am the life of the party}" and ``\textit{I get stressed out easily}", designed to primarily measure one personality factor.
The 50 items can be divided into five equal-sized groups, with each group primarily measure one personality factor.
Responses to the items are on  a five-level Likert scale, which are treated as continuous variables in the current analysis.

Although the big-five personality test was designed to have a perfect simple structure, cross loadings are often found in empirical data \citep[e.g.,][]{gow2005goldberg}. To better understand the loading structure of this widely used personality test, we applied the proposed $L^{0.5}$  and $L^1$ rotations to a dataset\footnote{The dataset is  downloaded from: \url{http://personality-testing.info/_rawdata/}.} on this test. To avoid possible complexities brought by measurement non-invariance, we selected the subset of male respondents from the United Kingdom, which has a sample size  $N = 609$. In the analysis, the number of factors is set to be $K=5$.

After applying the proposed rotations, we further adjusted the estimates by column permutation and sign flip transformations, so that the resulting factors correspond to the E, ES, A, C, I factors, respectively.
Our results are given in Tables~\ref{tab:empcovmat} through \ref{tab:empCI0.5_3}. Table \ref{tab:empcovmat} shows the estimated covariance matrices from the two rotations. The estimated correlation matrices from the two criteria are similar to each other. In particular, all the signs of the correlations are consistent, except for the correlation between A and I, in which case both correlations are close to zero.  In addition,
for each pair of factors, the correlations obtained by the two criteria are close. The sign pattern of
the correlations between the big-five factors is largely consistent with those found in the literature \citep[e.g.,][]{booth2014exploratory,gow2005goldberg}.

Tables \ref{tab:empCI0.5_1} through \ref{tab:empCI0.5_3} show the estimated loading parameters and the corresponding 95\% confidence intervals obtained from the $L^{0.5}$ rotation.
The loadings that are significantly different from zero according to the 95\% confidence intervals are indicated by asterisks.
The results of the
$L^{1}$ rotation are similar, and thus, are given in the supplementary material. In Tables \ref{tab:empCI0.5_1}-\ref{tab:empCI0.5_3}, the items are labelled based on the personality factor that they are designed to measure, and their scoring keys\footnote{Positively scored items are indicated by ``$(+)$" and negatively scored items are indicated by ``$(-)$".}. The estimated loading matrix is largely consistent with the true loading structure, where all the items have relatively strong loadings on the factors that they are designed to measure, and the signs of the loadings are consistent with the scoring keys.  The confidence intervals shed additional light on the uncertainty of each loading. Specifically, we notice that many loadings are statistically insignificantly different from zero, suggesting that the true loading structure is sparse.  There are also items with fairly strong cross loadings.
For example, item C4 ``I make a mess of things" loads strongly on not only factor C, but also factor ES. In fact, its estimated loading on ES is -0.607, with a  confidence interval  (-0.677, -0.504). This seems to sensible, as item C4 may remind people about stressful situations that are covered by the ES factor. For another example, item ES4 ``I seldom feel blue" also loads positively on factor E, with an estimated loading 0.237 and a confidence interval $(0.115, 0.310)$. It also makes sense, as this item seems to be closely related to how energetic a person is.

\begin{table}[h]
\centering
\caption{Estimated correlation matrices based on $L^{0.5}$ and $L^1$ rotations,big-five personality test}
\label{tab:empcovmat}
\begin{tabular}{lccccc|ccccc}
\hline
\multicolumn{1}{c}{}    & \multicolumn{5}{c|}{$p=0.5$}         & \multicolumn{5}{c}{$p=1$}           \\ \hline
\multicolumn{1}{c}{}    & E     & ES     & A      & C      & I & E     & ES     & A     & C      & I \\ \hline
\multicolumn{1}{l|}{E}  & 1     &        &        &        &   & 1     &        &       &        &   \\
\multicolumn{1}{l|}{ES} & 0.117 & 1      &        &        &   & 0.186 & 1      &       &        &   \\
\multicolumn{1}{l|}{A}  & 0.194 & -0.015 & 1      &        &   & 0.194 & -0.002 & 1     &        &   \\
\multicolumn{1}{l|}{C}  & 0.005 & 0.106  & 0.035  & 1      &   & 0.019 & 0.144  & 0.038 & 1      &   \\
\multicolumn{1}{l|}{I}  & 0.035 & 0.020   & -0.005 & -0.028 & 1 & 0.161 & 0.043  & 0.020  & -0.017 & 1 \\ \hline
\end{tabular}
\end{table}

\begin{table}
\centering
\caption{Part I: Point estimates and confidence intervals constructed by $L^{0.5}$, big-five personality test. The loadings that are significantly different from zero according to the 95\% confidence intervals are indicated by asterisks.}
\label{tab:empCI0.5_1}
\begin{tabular}{lccccc}
\hline
 &
  E &
  ES &
  A &
  C &
  I \\ \hline
\multicolumn{1}{l|}{E1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.882*\\      (  0.793,  0.980)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.055\\      (-0.161,  0.003)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.068*\\      (-0.183,-0.017)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.015\\      (-0.078,  0.100)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.087*\\      (  0.005,  0.170)\end{tabular} \\
\multicolumn{1}{l|}{E2$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.845*\\      (-0.962,-0.760)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.117*\\      (  0.048,  0.228)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.057,  0.126)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.031\\      (-0.061,  0.132)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.020\\      (-0.104,  0.079)\end{tabular} \\
\multicolumn{1}{l|}{E3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.786*\\      (  0.702,  0.876)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.286*\\      (  0.192,  0.347)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.203*\\      (  0.120,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123*\\      (  0.050,  0.213)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088*\\      (-0.175,-0.020)\end{tabular} \\
\multicolumn{1}{l|}{E4$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.913*\\      (-1.019,-0.842)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.072\\      (-0.140,  0.011)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.022\\      (-0.075,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.099,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.084*\\      (  0.033,  0.184)\end{tabular} \\
\multicolumn{1}{l|}{E5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.895*\\      (  0.812,  0.987)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.012\\      (-0.118,  0.032)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.155*\\      (  0.059,  0.209)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.104*\\      (  0.023,  0.183)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085*\\      (  0.021,  0.172)\end{tabular} \\
\multicolumn{1}{l|}{E6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.751*\\      (-0.849,-0.660)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.011\\      (-0.069,  0.103)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088\\      (-0.163,  0.010)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062\\      (-0.161,  0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.141*\\      (-0.226,-0.052)\end{tabular} \\
\multicolumn{1}{l|}{E7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}1.114*\\      (  1.020,  1.221)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062*\\      (-0.189,-0.018)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.083*\\      (  0.001,  0.172)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.105*\\      (  0.020,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.117,  0.052)\end{tabular} \\
\multicolumn{1}{l|}{E8$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.722*\\      (-0.823,-0.630)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.099*\\      (-0.176,  0.000)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.118*\\      (  0.024,  0.210)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.069\\      (-0.157,  0.022)\end{tabular} \\
\multicolumn{1}{l|}{E9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.858*\\      (  0.748,  0.953)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.054,  0.131)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.111,  0.075)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.013\\      (-0.110,  0.086)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.246*\\      (  0.145,  0.332)\end{tabular} \\
\multicolumn{1}{l|}{E10$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.829*\\      (-0.930,-0.741)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.128*\\      (-0.193,-0.025)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.048\\      (-0.123,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.121*\\      (-0.215,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.078,  0.092)\end{tabular} \\
\multicolumn{1}{l|}{ES1$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.152*\\      (-0.210,-0.028)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.975*\\      (-1.073,-0.875)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.008\\      (-0.121,  0.078)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.081*\\      (  0.016,  0.200)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.104*\\      (-0.193,-0.017)\end{tabular} \\
\multicolumn{1}{l|}{ES2$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.161*\\      (  0.039,  0.215)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.675*\\      (  0.589,  0.771)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.064,  0.115)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.084*\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085*\\      (  0.012,  0.185)\end{tabular} \\
\multicolumn{1}{l|}{ES3$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.203*\\      (-0.271,-0.093)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.786*\\      (-0.881,-0.696)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.232*\\      (  0.124,  0.304)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.122*\\      (  0.031,  0.208)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.042\\      (-0.033,  0.139)\end{tabular} \\
\multicolumn{1}{l|}{ES4$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.237*\\      (  0.115,  0.310)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.568*\\      (  0.474,  0.670)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.001\\      (-0.069,  0.125)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.066\\      (-0.049,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.079,  0.113)\end{tabular} \\
\multicolumn{1}{l|}{ES5$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.073,  0.138)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.471*\\      (-0.567,-0.357)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.163,  0.046)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.110\\      (-0.208,  0.006)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.225*\\      (-0.345,-0.135)\end{tabular} \\
\multicolumn{1}{l|}{ES6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.147*\\      (-0.202,-0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.808*\\      (-0.905,-0.717)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.259*\\      (  0.145,  0.327)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.025\\      (-0.106,  0.074)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.132*\\      (-0.227,-0.053)\end{tabular} \\
\multicolumn{1}{l|}{ES7$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.048,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.962*\\      (-1.064,-0.883)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111*\\      (-0.226,-0.055)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.044\\      (-0.102,  0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.089,  0.071)\end{tabular} \\
\multicolumn{1}{l|}{ES8$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.026\\      (-0.080,  0.106)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-1.130*\\      (-1.240,-1.046)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.133*\\      (-0.261,-0.080)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.140,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.103,  0.065)\end{tabular} \\
\multicolumn{1}{l|}{ES9$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.131,  0.048)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.866*\\      (-0.956,-0.768)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.292*\\      (-0.399,-0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.171*\\      (  0.092,  0.267)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.018\\      (-0.104,  0.068)\end{tabular} \\
\multicolumn{1}{l|}{ES10$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.364*\\      (-0.435,-0.256)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.842*\\      (-0.940,-0.752)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.029,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.109*\\      (-0.181,-0.004)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.096*\\      (  0.016,  0.187)\end{tabular} \\ \hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Part II: Point estimates and confidence intervals constructed by $L^{0.5}$, big-five personality test.}
\label{tab:empCI0.5_2}
\begin{tabular}{lccccc}
\hline
 &
  E &
  ES &
  A &
  C &
  I \\ \hline
\multicolumn{1}{l|}{A1$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.112,  0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.127*\\      (-0.205,-0.013)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.779*\\      (-0.876,-0.670)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.030\\      (-0.071,  0.127)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.044\\      (-0.060,  0.134)\end{tabular} \\
\multicolumn{1}{l|}{A2$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.437*\\      (  0.359,  0.523)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.099,  0.053)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.557*\\      (  0.464,  0.626)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.043\\      (-0.124,  0.034)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.026,  0.128)\end{tabular} \\
\multicolumn{1}{l|}{A3$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.178*\\      (  0.080,  0.286)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571*\\      (-0.674,-0.467)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.565*\\      (-0.692,-0.480)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.077\\      (-0.177,  0.030)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.138*\\      (  0.029,  0.231)\end{tabular} \\
\multicolumn{1}{l|}{A4$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.013\\      (-0.034,  0.142)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.100,  0.039)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.980*\\      (  0.895,  1.050)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.098,  0.032)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.060,  0.071)\end{tabular} \\
\multicolumn{1}{l|}{A5$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.156*\\      (-0.248,-0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.105,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.815*\\      (-0.894,-0.725)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.078,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.086*\\      (  0.011,  0.163)\end{tabular} \\
\multicolumn{1}{l|}{A6$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.157,  0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184*\\      (-0.270,-0.102)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.717*\\      (  0.630,  0.812)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.007\\      (-0.100,  0.073)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.014\\      (-0.073,  0.096)\end{tabular} \\
\multicolumn{1}{l|}{A7$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.367*\\      (-0.452,-0.297)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.095*\\      (-0.161,-0.015)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.733*\\      (-0.800,-0.640)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.057\\      (-0.016,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.026\\      (-0.047,  0.098)\end{tabular} \\
\multicolumn{1}{l|}{A8$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.110*\\      (  0.039,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.038\\      (-0.128,  0.011)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.692*\\      (  0.616,  0.770)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.079*\\      (  0.007,  0.149)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.034,  0.106)\end{tabular} \\
\multicolumn{1}{l|}{A9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.119*\\      (  0.040,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111*\\      (-0.206,-0.055)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.752*\\      (  0.667,  0.835)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.015,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114*\\      (  0.042,  0.194)\end{tabular} \\
\multicolumn{1}{l|}{A10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.440*\\      (  0.352,  0.515)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.075\\      (-0.008,  0.145)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.321*\\      (  0.244,  0.401)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.125*\\      (  0.033,  0.193)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.056\\      (-0.019,  0.136)\end{tabular} \\
\multicolumn{1}{l|}{C1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.111*\\      (  0.002,  0.180)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.101\\      (-0.003,  0.179)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.037\\      (-0.108,  0.079)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.686*\\      (  0.583,  0.771)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.131*\\      (  0.064,  0.244)\end{tabular} \\
\multicolumn{1}{l|}{C2$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.015,  0.191)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.190*\\      (-0.285,-0.078)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.108\\      (-0.020,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.658*\\      (-0.793,-0.581)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.145*\\      (  0.012,  0.215)\end{tabular} \\
\multicolumn{1}{l|}{C3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.078,  0.083)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.094,  0.066)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114*\\      (  0.046,  0.206)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.401*\\      (  0.300,  0.465)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.280*\\      (  0.217,  0.379)\end{tabular} \\
\multicolumn{1}{l|}{C4$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.140*\\      (-0.201,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.607*\\      (-0.677,-0.504)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.049\\      (-0.052,  0.116)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.496*\\      (-0.576,-0.407)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041*\\      (-0.170,-0.006)\end{tabular} \\
\multicolumn{1}{l|}{C5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.081*\\      (  0.001,  0.192)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.047,  0.161)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.053,  0.151)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.779*\\      (  0.691,  0.887)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.126,  0.066)\end{tabular} \\
\multicolumn{1}{l|}{C6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.088,  0.124)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184*\\      (-0.277,-0.062)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.078,  0.139)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.705*\\      (-0.841,-0.621)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.089\\      (-0.024,  0.188)\end{tabular} \\
\multicolumn{1}{l|}{C7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.129*\\      (-0.221,-0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.139*\\      (-0.235,-0.057)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.110*\\      (  0.035,  0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.528*\\      (  0.434,  0.618)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.038\\      (-0.021,  0.157)\end{tabular} \\
\multicolumn{1}{l|}{C8$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.010\\      (-0.087,  0.094)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.275*\\      (-0.355,-0.173)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.241*\\      (-0.356,-0.174)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.524*\\      (-0.608,-0.421)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.122,  0.059)\end{tabular} \\
\multicolumn{1}{l|}{C9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.058,  0.130)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.014\\      (-0.134,  0.067)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123*\\      (  0.052,  0.241)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.723*\\      (  0.626,  0.821)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.149,  0.040)\end{tabular} \\
\multicolumn{1}{l|}{C10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.107,  0.059)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.016\\      (-0.117,  0.051)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.128*\\      (  0.065,  0.231)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.524*\\      (  0.416,  0.587)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.234*\\      (  0.172,  0.337)\end{tabular} \\ \hline
\end{tabular}
\end{table}

\begin{table}
\centering
\caption{Part III: Point estimates and confidence intervals constructed by $L^{0.5}$, big-five personality test.}
\label{tab:empCI0.5_3}
\begin{tabular}{lccccc}
\hline
 &
  E &
  ES &
  A &
  C &
  I \\ \hline
\multicolumn{1}{l|}{I1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.079\\      (-0.021,  0.145)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.109,  0.057)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.045\\      (-0.146,  0.014)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.108,  0.056)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.623*\\      (  0.544,  0.711)\end{tabular} \\
\multicolumn{1}{l|}{I2$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.051,  0.121)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.220*\\      (-0.283,-0.114)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.087*\\      (-0.183,-0.015)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.009\\      (-0.055,  0.117)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.581*\\      (-0.675,-0.502)\end{tabular} \\
\multicolumn{1}{l|}{I3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.026,  0.135)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.153*\\      (-0.251,-0.092)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.025\\      (-0.061,  0.096)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.098,  0.063)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.589*\\      (  0.502,  0.665)\end{tabular} \\
\multicolumn{1}{l|}{I4$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.027,  0.144)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.202*\\      (-0.267,-0.098)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.154*\\      (-0.230,-0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.022,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571*\\      (-0.660,-0.487)\end{tabular} \\
\multicolumn{1}{l|}{I5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.237*\\      (  0.133,  0.271)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.002,  0.137)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.058\\      (-0.132,  0.002)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.176*\\      (  0.069,  0.205)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.581*\\      (  0.509,  0.651)\end{tabular} \\
\multicolumn{1}{l|}{I6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.211*\\      (-0.272,-0.102)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.003\\      (-0.060,  0.108)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.047\\      (-0.121,  0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.510*\\      (-0.605,-0.434)\end{tabular} \\
\multicolumn{1}{l|}{I7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.076\\      (-0.017,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.166*\\      (  0.084,  0.221)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.035\\      (-0.100,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097*\\      (  0.012,  0.153)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.450*\\      (  0.382,  0.524)\end{tabular} \\
\multicolumn{1}{l|}{I8$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.024\\      (-0.163,  0.021)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.165*\\      (-0.270,-0.090)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.107*\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.123,  0.060)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.656*\\      (  0.577,  0.764)\end{tabular} \\
\multicolumn{1}{l|}{I9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.153,  0.008)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.217*\\      (-0.309,-0.151)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.240*\\      (  0.156,  0.315)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.107*\\      (  0.017,  0.179)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.258*\\      (  0.183,  0.343)\end{tabular} \\
\multicolumn{1}{l|}{I10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.240*\\      (  0.132,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.004\\      (-0.113,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.072,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097*\\      (  0.012,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.685*\\      (  0.613,  0.762)\end{tabular} \\ \hline
\end{tabular}
\end{table}

\begin{comment}
\begin{table}[h]
\centering
\caption{Point estimate and confidence interval constructed by $L^{0.5}$}
\label{tab:empCI0.5_1}
\begin{tabular}{lccccc}
\hline
 &
  E &
  N &
  A &
  C &
  O \\ \hline
\multicolumn{1}{l|}{E1$$$(+)$$$} &
  \begin{tabular}[c]{@{}c@{}}0.882\\      (  0.793,  0.980)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.055\\      (-0.161,  0.003)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.068\\      (-0.183,-0.017)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.015\\      (-0.078,  0.100)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.087\\      (  0.005,  0.170)\end{tabular} \\
\multicolumn{1}{l|}{E2$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.845\\      (-0.962,-0.760)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.117\\      (-0.228,-0.048)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.126,  0.057)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.031\\      (-0.061,  0.132)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.020\\      (-0.079,  0.104)\end{tabular} \\
\multicolumn{1}{l|}{E3$$$(+)$$$} &
  \begin{tabular}[c]{@{}c@{}}0.786\\      (  0.702,  0.876)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.286\\      (-0.347,-0.192)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.203\\      (  0.120,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123\\      (  0.050,  0.213)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088\\      (-0.175,-0.020)\end{tabular} \\
\multicolumn{1}{l|}{E4$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.913\\      (-1.019,-0.842)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.072\\      (-0.011,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.022\\      (-0.075,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.099,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.084\\      (-0.184,-0.033)\end{tabular} \\
\multicolumn{1}{l|}{E5$$$(+)$$$} &
  \begin{tabular}[c]{@{}c@{}}0.895\\      (  0.812,  0.987)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.012\\      (-0.032,  0.118)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.155\\      (  0.059,  0.209)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.104\\      (  0.023,  0.183)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085\\      (  0.021,  0.172)\end{tabular} \\
\multicolumn{1}{l|}{E6$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.751\\      (-0.849,-0.660)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.011\\      (-0.103,  0.069)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088\\      (-0.163,  0.010)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062\\      (-0.161,  0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.141\\      (  0.052,  0.226)\end{tabular} \\
\multicolumn{1}{l|}{E7$$$(+)$$$} &
  \begin{tabular}[c]{@{}c@{}}1.114\\      (  1.020,  1.221)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062\\      (  0.018,  0.189)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.083\\      (  0.001,  0.172)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.105\\      (  0.020,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.052,  0.117)\end{tabular} \\
\multicolumn{1}{l|}{E8$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.722\\      (-0.823,-0.630)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.099\\      (  0.000,  0.176)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.118\\      (  0.024,  0.210)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.069\\      (-0.022,  0.157)\end{tabular} \\
\multicolumn{1}{l|}{E9$$$(+)$$$} &
  \begin{tabular}[c]{@{}c@{}}0.858\\      (  0.748,  0.953)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.131,  0.054)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.111,  0.075)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.013\\      (-0.110,  0.086)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.246\\      (-0.332,-0.145)\end{tabular} \\
\multicolumn{1}{l|}{E10$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.829\\      (-0.930,-0.741)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.128\\      (  0.025,  0.193)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.048\\      (-0.123,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.121\\      (-0.215,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.092,  0.078)\end{tabular} \\
\multicolumn{1}{l|}{N1$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.152\\      (-0.210,-0.028)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.975\\      (  0.875,  1.073)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.008\\      (-0.121,  0.078)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.081\\      (  0.016,  0.200)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.104\\      (  0.017,  0.193)\end{tabular} \\
\multicolumn{1}{l|}{N2$$$(+)$$$} &
  \begin{tabular}[c]{@{}c@{}}0.161\\      (  0.039,  0.215)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.675\\      (-0.771,-0.589)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.064,  0.115)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.084\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085\\      (-0.185,-0.012)\end{tabular} \\
\multicolumn{1}{l|}{N3$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.203\\      (-0.271,-0.093)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.786\\      (  0.696,  0.881)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.232\\      (  0.124,  0.304)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.122\\      (  0.031,  0.208)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.042\\      (-0.139,  0.033)\end{tabular} \\
\multicolumn{1}{l|}{N4$$$(+)$$$} &
  \begin{tabular}[c]{@{}c@{}}0.237\\      (  0.115,  0.310)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.568\\      (-0.670,-0.474)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.001\\      (-0.069,  0.125)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.066\\      (-0.049,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.113,  0.079)\end{tabular} \\
\multicolumn{1}{l|}{N5$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.073,  0.138)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.471\\      (  0.357,  0.567)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.163,  0.046)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.110\\      (-0.208,  0.006)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.225\\      (  0.135,  0.345)\end{tabular} \\
\multicolumn{1}{l|}{N6$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.147\\      (-0.202,-0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.808\\      (  0.717,  0.905)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.259\\      (  0.145,  0.327)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.025\\      (-0.106,  0.074)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.132\\      (  0.053,  0.227)\end{tabular} \\
\multicolumn{1}{l|}{N7$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.048,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.962\\      (  0.883,  1.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111\\      (-0.226,-0.055)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.044\\      (-0.102,  0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.071,  0.089)\end{tabular} \\
\multicolumn{1}{l|}{N8$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.026\\      (-0.080,  0.106)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-1.130\\      (  1.046,  1.240)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.133\\      (-0.261,-0.080)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.140,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.065,  0.103)\end{tabular} \\
\hline
\end{tabular}%
\end{table}
\end{comment}
\begin{comment}
\begin{table}[h]
\centering
\caption{Point estimate and confidence interval constructed by $L^{0.5}$}
\label{tab:empCI0.5_2}
\begin{tabular}{lccccc}
\hline
 &
  E &
  N &
  A &
  C &
  O \\ \hline

\multicolumn{1}{l|}{N9$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.131,  0.048)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.866\\      (  0.768,  0.956)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.292\\      (-0.399,-0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.171\\      (  0.092,  0.267)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.018\\      (-0.068,  0.104)\end{tabular} \\
\multicolumn{1}{l|}{N10$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.364\\      (-0.435,-0.256)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.842\\      (  0.752,  0.940)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.029,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.109\\      (-0.181,-0.004)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.096\\      (-0.187,-0.016)\end{tabular} \\
\multicolumn{1}{l|}{A1$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.112,  0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.127\\      (  0.013,  0.205)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.779\\      (-0.876,-0.670)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.030\\      (-0.071,  0.127)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.044\\      (-0.134,  0.060)\end{tabular} \\
\multicolumn{1}{l|}{A2$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.437\\      (  0.359,  0.523)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.053,  0.099)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.557\\      (  0.464,  0.626)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.043\\      (-0.124,  0.034)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.128,  0.026)\end{tabular} \\
\multicolumn{1}{l|}{A3$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}0.178\\      (  0.080,  0.286)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571\\      (  0.467,  0.674)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.565\\      (-0.692,-0.480)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.077\\      (-0.177,  0.030)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.138\\      (-0.231,-0.029)\end{tabular} \\
\multicolumn{1}{l|}{A4$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.013\\      (-0.034,  0.142)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.039,  0.100)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.980\\      (  0.895,  1.050)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.098,  0.032)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.071,  0.060)\end{tabular} \\
\multicolumn{1}{l|}{A5$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.156\\      (-0.248,-0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.047,  0.105)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.815\\      (-0.894,-0.725)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.078,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.086\\      (-0.163,-0.011)\end{tabular} \\
\multicolumn{1}{l|}{A6$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.157,  0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184\\      (  0.102,  0.270)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.717\\      (  0.630,  0.812)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.007\\      (-0.100,  0.073)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.014\\      (-0.096,  0.073)\end{tabular} \\
\multicolumn{1}{l|}{A7$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}-0.367\\      (-0.452,-0.297)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.095\\      (  0.015,  0.161)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.733\\      (-0.800,-0.640)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.057\\      (-0.016,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.026\\      (-0.098,  0.047)\end{tabular} \\
\multicolumn{1}{l|}{A8$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.110\\      (  0.039,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.038\\      (-0.011,  0.128)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.692\\      (  0.616,  0.770)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.079\\      (  0.007,  0.149)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.106,  0.034)\end{tabular} \\
\multicolumn{1}{l|}{A9$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.119\\      (  0.040,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111\\      (  0.055,  0.206)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.752\\      (  0.667,  0.835)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.015,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114\\      (-0.194,-0.042)\end{tabular} \\
\multicolumn{1}{l|}{A10$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.440\\      (  0.352,  0.515)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.075\\      (-0.145,  0.008)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.321\\      (  0.244,  0.401)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.125\\      (  0.033,  0.193)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.056\\      (-0.136,  0.019)\end{tabular} \\
\multicolumn{1}{l|}{C1$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.111\\      (  0.002,  0.180)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.101\\      (-0.179,  0.003)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.037\\      (-0.108,  0.079)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.686\\      (  0.583,  0.771)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.131\\      (-0.244,-0.064)\end{tabular} \\
\multicolumn{1}{l|}{C2$$(-)$$} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.015,  0.191)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.190\\      (  0.078,  0.285)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.108\\      (-0.020,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.658\\      (-0.793,-0.581)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.145\\      (-0.215,-0.012)\end{tabular} \\
\multicolumn{1}{l|}{C3$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.078,  0.083)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.066,  0.094)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114\\      (  0.046,  0.206)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.401\\      (  0.300,  0.465)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.280\\      (-0.379,-0.217)\end{tabular} \\
\multicolumn{1}{l|}{C4$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.140\\      (-0.201,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.607\\      (  0.504,  0.677)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.049\\      (-0.052,  0.116)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.496\\      (-0.576,-0.407)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (  0.006,  0.170)\end{tabular} \\
\multicolumn{1}{l|}{C5$$(+)$$} &
  \begin{tabular}[c]{@{}c@{}}0.081\\      (  0.001,  0.192)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.161,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.053,  0.151)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.779\\      (  0.691,  0.887)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.066,  0.126)\end{tabular} \\
\multicolumn{1}{l|}{C6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.088,  0.124)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184\\      (  0.062,  0.277)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.078,  0.139)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.705\\      (-0.841,-0.621)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.089\\      (-0.188,  0.024)\end{tabular} \\
\hline
\end{tabular}%
\end{table}

\begin{table}[h]
\centering
\caption{Point estimate and confidence interval constructed by $L^{0.5}$}
\label{tab:empCI0.5_3}
\begin{tabular}{lccccc}
\hline
 &
  E &
  N &
  A &
  C &
  O \\ \hline

\multicolumn{1}{l|}{C7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.129\\      (-0.221,-0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.139\\      (  0.057,  0.235)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.110\\      (  0.035,  0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.528\\      (  0.434,  0.618)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.038\\      (-0.157,  0.021)\end{tabular} \\
\multicolumn{1}{l|}{C8$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.010\\      (-0.087,  0.094)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.275\\      (  0.173,  0.355)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.241\\      (-0.356,-0.174)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.524\\      (-0.608,-0.421)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.059,  0.122)\end{tabular} \\
\multicolumn{1}{l|}{C9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.058,  0.130)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.014\\      (-0.067,  0.134)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123\\      (  0.052,  0.241)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.723\\      (  0.626,  0.821)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.040,  0.149)\end{tabular} \\
\multicolumn{1}{l|}{C10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.107,  0.059)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.016\\      (-0.051,  0.117)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.128\\      (  0.065,  0.231)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.524\\      (  0.416,  0.587)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.234\\      (-0.337,-0.172)\end{tabular} \\
\multicolumn{1}{l|}{O1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.079\\      (-0.021,  0.145)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.057,  0.109)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.045\\      (-0.146,  0.014)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.108,  0.056)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.623\\      (-0.711,-0.544)\end{tabular} \\
\multicolumn{1}{l|}{O2$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.051,  0.121)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.220\\      (  0.114,  0.283)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.087\\      (-0.183,-0.015)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.009\\      (-0.055,  0.117)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.581\\      (  0.502,  0.675)\end{tabular} \\
\multicolumn{1}{l|}{O3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.026,  0.135)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.153\\      (  0.092,  0.251)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.025\\      (-0.061,  0.096)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.098,  0.063)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.589\\      (-0.665,-0.502)\end{tabular} \\
\multicolumn{1}{l|}{O4$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.027,  0.144)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.202\\      (  0.098,  0.267)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.154\\      (-0.230,-0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.022,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571\\      (  0.487,  0.660)\end{tabular} \\
\multicolumn{1}{l|}{O5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.237\\      (  0.133,  0.271)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.137,  0.002)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.058\\      (-0.132,  0.002)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.176\\      (  0.069,  0.205)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.581\\      (-0.651,-0.509)\end{tabular} \\
\multicolumn{1}{l|}{O6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.211\\      (-0.272,-0.102)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.003\\      (-0.108,  0.060)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.047\\      (-0.121,  0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.510\\      (  0.434,  0.605)\end{tabular} \\
\multicolumn{1}{l|}{O7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.076\\      (-0.017,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.166\\      (-0.221,-0.084)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.035\\      (-0.100,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097\\      (  0.012,  0.153)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.450\\      (-0.524,-0.382)\end{tabular} \\
\multicolumn{1}{l|}{O8$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.024\\      (-0.163,  0.021)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.165\\      (  0.090,  0.270)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.107\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.123,  0.060)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.656\\      (-0.764,-0.577)\end{tabular} \\
\multicolumn{1}{l|}{O9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.153,  0.008)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.217\\      (  0.151,  0.309)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.240\\      (  0.156,  0.315)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.107\\      (  0.017,  0.179)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.258\\      (-0.343,-0.183)\end{tabular} \\
\multicolumn{1}{l|}{O10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.240\\      (  0.132,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.004\\      (-0.036,  0.113)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.072,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097\\      (  0.012,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.685\\      (-0.762,-0.613)\end{tabular} \\ \hline
\end{tabular}%
\end{table}
\end{comment}

\begin{comment}
\begin{longtable}[c]{lccccc}
\caption{Point estimate and confidence interval constructed by $L^{0.5}$}
\label{tab:empCI1}\\
\hline
 &
  E &
  N &
  A &
  C &
  O \\ \hline
\endhead
\hline
\endfoot
\endlastfoot
\multicolumn{1}{l|}{E1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.882\\      (  0.793,  0.980)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.055\\      (-0.161,  0.003)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.068\\      (-0.183,-0.017)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.015\\      (-0.078,  0.100)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.087\\      (  0.005,  0.170)\end{tabular} \\
\multicolumn{1}{l|}{E2$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.845\\      (-0.962,-0.760)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.117\\      (-0.228,-0.048)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.126,  0.057)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.031\\      (-0.061,  0.132)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.020\\      (-0.079,  0.104)\end{tabular} \\
\multicolumn{1}{l|}{E3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.786\\      (  0.702,  0.876)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.286\\      (-0.347,-0.192)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.203\\      (  0.120,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123\\      (  0.050,  0.213)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088\\      (-0.175,-0.020)\end{tabular} \\
\multicolumn{1}{l|}{E4$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.913\\      (-1.019,-0.842)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.072\\      (-0.011,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.022\\      (-0.075,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.099,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.084\\      (-0.184,-0.033)\end{tabular} \\
\multicolumn{1}{l|}{E5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.895\\      (  0.812,  0.987)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.012\\      (-0.032,  0.118)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.155\\      (  0.059,  0.209)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.104\\      (  0.023,  0.183)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085\\      (  0.021,  0.172)\end{tabular} \\
\multicolumn{1}{l|}{E6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.751\\      (-0.849,-0.660)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.011\\      (-0.103,  0.069)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088\\      (-0.163,  0.010)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062\\      (-0.161,  0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.141\\      (  0.052,  0.226)\end{tabular} \\
\multicolumn{1}{l|}{E7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}1.114\\      (  1.020,  1.221)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062\\      (  0.018,  0.189)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.083\\      (  0.001,  0.172)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.105\\      (  0.020,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.052,  0.117)\end{tabular} \\
\multicolumn{1}{l|}{E8$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.722\\      (-0.823,-0.630)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.099\\      (  0.000,  0.176)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.118\\      (  0.024,  0.210)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.069\\      (-0.022,  0.157)\end{tabular} \\
\multicolumn{1}{l|}{E9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.858\\      (  0.748,  0.953)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.131,  0.054)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.111,  0.075)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.013\\      (-0.110,  0.086)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.246\\      (-0.332,-0.145)\end{tabular} \\
\multicolumn{1}{l|}{E10$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.829\\      (-0.930,-0.741)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.128\\      (  0.025,  0.193)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.048\\      (-0.123,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.121\\      (-0.215,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.092,  0.078)\end{tabular} \\
\multicolumn{1}{l|}{N1$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.152\\      (-0.210,-0.028)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.975\\      (  0.875,  1.073)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.008\\      (-0.121,  0.078)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.081\\      (  0.016,  0.200)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.104\\      (  0.017,  0.193)\end{tabular} \\
\multicolumn{1}{l|}{N2$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.161\\      (  0.039,  0.215)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.675\\      (-0.771,-0.589)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.064,  0.115)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.084\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085\\      (-0.185,-0.012)\end{tabular} \\
\multicolumn{1}{l|}{N3$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.203\\      (-0.271,-0.093)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.786\\      (  0.696,  0.881)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.232\\      (  0.124,  0.304)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.122\\      (  0.031,  0.208)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.042\\      (-0.139,  0.033)\end{tabular} \\
\multicolumn{1}{l|}{N4$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.237\\      (  0.115,  0.310)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.568\\      (-0.670,-0.474)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.001\\      (-0.069,  0.125)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.066\\      (-0.049,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.113,  0.079)\end{tabular} \\
\multicolumn{1}{l|}{N5$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.073,  0.138)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.471\\      (  0.357,  0.567)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.163,  0.046)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.110\\      (-0.208,  0.006)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.225\\      (  0.135,  0.345)\end{tabular} \\
\multicolumn{1}{l|}{N6$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.147\\      (-0.202,-0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.808\\      (  0.717,  0.905)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.259\\      (  0.145,  0.327)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.025\\      (-0.106,  0.074)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.132\\      (  0.053,  0.227)\end{tabular} \\
\multicolumn{1}{l|}{N7$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.048,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.962\\      (  0.883,  1.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111\\      (-0.226,-0.055)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.044\\      (-0.102,  0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.071,  0.089)\end{tabular} \\
\multicolumn{1}{l|}{N8$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.026\\      (-0.080,  0.106)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-1.130\\      (  1.046,  1.240)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.133\\      (-0.261,-0.080)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.140,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.065,  0.103)\end{tabular} \\
\multicolumn{1}{l|}{N9$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.131,  0.048)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.866\\      (  0.768,  0.956)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.292\\      (-0.399,-0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.171\\      (  0.092,  0.267)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.018\\      (-0.068,  0.104)\end{tabular} \\
\multicolumn{1}{l|}{N10$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.364\\      (-0.435,-0.256)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.842\\      (  0.752,  0.940)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.029,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.109\\      (-0.181,-0.004)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.096\\      (-0.187,-0.016)\end{tabular} \\
\multicolumn{1}{l|}{A1$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.112,  0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.127\\      (  0.013,  0.205)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.779\\      (-0.876,-0.670)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.030\\      (-0.071,  0.127)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.044\\      (-0.134,  0.060)\end{tabular} \\
\multicolumn{1}{l|}{A2$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.437\\      (  0.359,  0.523)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.053,  0.099)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.557\\      (  0.464,  0.626)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.043\\      (-0.124,  0.034)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.128,  0.026)\end{tabular} \\
\multicolumn{1}{l|}{A3$(-)$} &
  \begin{tabular}[c]{@{}c@{}}0.178\\      (  0.080,  0.286)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571\\      (  0.467,  0.674)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.565\\      (-0.692,-0.480)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.077\\      (-0.177,  0.030)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.138\\      (-0.231,-0.029)\end{tabular} \\
\multicolumn{1}{l|}{A4$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.013\\      (-0.034,  0.142)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.039,  0.100)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.980\\      (  0.895,  1.050)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.098,  0.032)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.071,  0.060)\end{tabular} \\
\multicolumn{1}{l|}{A5$(-)$} &
  \begin{tabular}[c]{@{}c@{}}-0.156\\      (-0.248,-0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.047,  0.105)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.815\\      (-0.894,-0.725)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.078,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.086\\      (-0.163,-0.011)\end{tabular} \\
\multicolumn{1}{l|}{A6$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.157,  0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184\\      (  0.102,  0.270)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.717\\      (  0.630,  0.812)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.007\\      (-0.100,  0.073)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.014\\      (-0.096,  0.073)\end{tabular} \\
\multicolumn{1}{l|}{A7(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.367\\      (-0.452,-0.297)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.095\\      (  0.015,  0.161)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.733\\      (-0.800,-0.640)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.057\\      (-0.016,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.026\\      (-0.098,  0.047)\end{tabular} \\
\multicolumn{1}{l|}{A8$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.110\\      (  0.039,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.038\\      (-0.011,  0.128)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.692\\      (  0.616,  0.770)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.079\\      (  0.007,  0.149)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.106,  0.034)\end{tabular} \\
\multicolumn{1}{l|}{A9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.119\\      (  0.040,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111\\      (  0.055,  0.206)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.752\\      (  0.667,  0.835)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.015,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114\\      (-0.194,-0.042)\end{tabular} \\
\multicolumn{1}{l|}{A10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.440\\      (  0.352,  0.515)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.075\\      (-0.145,  0.008)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.321\\      (  0.244,  0.401)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.125\\      (  0.033,  0.193)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.056\\      (-0.136,  0.019)\end{tabular} \\
\multicolumn{1}{l|}{C1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.111\\      (  0.002,  0.180)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.101\\      (-0.179,  0.003)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.037\\      (-0.108,  0.079)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.686\\      (  0.583,  0.771)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.131\\      (-0.244,-0.064)\end{tabular} \\
\multicolumn{1}{l|}{C2(-)} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.015,  0.191)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.190\\      (  0.078,  0.285)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.108\\      (-0.020,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.658\\      (-0.793,-0.581)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.145\\      (-0.215,-0.012)\end{tabular} \\
\multicolumn{1}{l|}{C3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.078,  0.083)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.066,  0.094)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114\\      (  0.046,  0.206)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.401\\      (  0.300,  0.465)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.280\\      (-0.379,-0.217)\end{tabular} \\
\multicolumn{1}{l|}{C4(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.140\\      (-0.201,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.607\\      (  0.504,  0.677)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.049\\      (-0.052,  0.116)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.496\\      (-0.576,-0.407)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (  0.006,  0.170)\end{tabular} \\
\multicolumn{1}{l|}{C5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.081\\      (  0.001,  0.192)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.161,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.053,  0.151)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.779\\      (  0.691,  0.887)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.066,  0.126)\end{tabular} \\
\multicolumn{1}{l|}{C6(-)} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.088,  0.124)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184\\      (  0.062,  0.277)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.078,  0.139)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.705\\      (-0.841,-0.621)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.089\\      (-0.188,  0.024)\end{tabular} \\
\multicolumn{1}{l|}{C7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.129\\      (-0.221,-0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.139\\      (  0.057,  0.235)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.110\\      (  0.035,  0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.528\\      (  0.434,  0.618)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.038\\      (-0.157,  0.021)\end{tabular} \\
\multicolumn{1}{l|}{C8(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.010\\      (-0.087,  0.094)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.275\\      (  0.173,  0.355)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.241\\      (-0.356,-0.174)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.524\\      (-0.608,-0.421)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.059,  0.122)\end{tabular} \\
\multicolumn{1}{l|}{C9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.058,  0.130)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.014\\      (-0.067,  0.134)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123\\      (  0.052,  0.241)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.723\\      (  0.626,  0.821)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.040,  0.149)\end{tabular} \\
\multicolumn{1}{l|}{C10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.107,  0.059)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.016\\      (-0.051,  0.117)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.128\\      (  0.065,  0.231)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.524\\      (  0.416,  0.587)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.234\\      (-0.337,-0.172)\end{tabular} \\
\multicolumn{1}{l|}{O1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.079\\      (-0.021,  0.145)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.057,  0.109)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.045\\      (-0.146,  0.014)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.108,  0.056)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.623\\      (-0.711,-0.544)\end{tabular} \\
\multicolumn{1}{l|}{O2(-)} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.051,  0.121)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.220\\      (  0.114,  0.283)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.087\\      (-0.183,-0.015)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.009\\      (-0.055,  0.117)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.581\\      (  0.502,  0.675)\end{tabular} \\
\multicolumn{1}{l|}{O3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.026,  0.135)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.153\\      (  0.092,  0.251)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.025\\      (-0.061,  0.096)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.098,  0.063)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.589\\      (-0.665,-0.502)\end{tabular} \\
\multicolumn{1}{l|}{O4(-)} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.027,  0.144)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.202\\      (  0.098,  0.267)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.154\\      (-0.230,-0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.022,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571\\      (  0.487,  0.660)\end{tabular} \\
\multicolumn{1}{l|}{O5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.237\\      (  0.133,  0.271)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.137,  0.002)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.058\\      (-0.132,  0.002)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.176\\      (  0.069,  0.205)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.581\\      (-0.651,-0.509)\end{tabular} \\
\multicolumn{1}{l|}{O6(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.211\\      (-0.272,-0.102)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.003\\      (-0.108,  0.060)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.047\\      (-0.121,  0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.510\\      (  0.434,  0.605)\end{tabular} \\
\multicolumn{1}{l|}{O7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.076\\      (-0.017,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.166\\      (-0.221,-0.084)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.035\\      (-0.100,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097\\      (  0.012,  0.153)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.450\\      (-0.524,-0.382)\end{tabular} \\
\multicolumn{1}{l|}{O8$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.024\\      (-0.163,  0.021)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.165\\      (  0.090,  0.270)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.107\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.123,  0.060)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.656\\      (-0.764,-0.577)\end{tabular} \\
\multicolumn{1}{l|}{O9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.153,  0.008)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.217\\      (  0.151,  0.309)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.240\\      (  0.156,  0.315)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.107\\      (  0.017,  0.179)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.258\\      (-0.343,-0.183)\end{tabular} \\
\multicolumn{1}{l|}{O10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.240\\      (  0.132,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.004\\      (-0.036,  0.113)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.072,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097\\      (  0.012,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.685\\      (-0.762,-0.613)\end{tabular} \\ \hline
\end{longtable}
\end{comment}

\begin{comment}
\begin{table}[h]
{
\begin{longtable}[c]{llllll}
\caption{Point estimate and confidence interval constructed by $L^{0.5}$ }
\label{tab:empCI} \\

\hline

\endfirsthead
\endhead
\hline
\endfoot
\endlastfoot
                            & E                     & N                     & A                     & C                     & O                     \\ \hline
\multicolumn{1}{l|}{$E1_{$(+)$} $}  & $ ~~0.882_{( 0.793, 0.980)}  $ & $ -0.055_{(-0.161, 0.003)} $ & $ -0.068_{(-0.183,-0.017)} $ & $ ~~0.015_{(-0.078, 0.100)}  $ & $ ~~0.087_{( 0.005, 0.170)}  $ \\
\multicolumn{1}{l|}{$E2_{(-)} $}  & $ -0.845_{(-0.962,-0.760)} $ & $ ~~0.117_{(-0.228,-0.048)}  $ & $ ~~0.002_{(-0.126, 0.057)}  $ & $ ~~0.031_{(-0.061, 0.132)}  $ & $ -0.020_{(-0.079, 0.104)} $ \\
\multicolumn{1}{l|}{$E3_{$(+)$} $}  & $ ~~0.786_{( 0.702, 0.876)}  $ & $ ~~0.286_{(-0.347,-0.192)}  $ & $ ~~0.203_{( 0.120, 0.275)}  $ & $ ~~0.123_{( 0.050, 0.213)}  $ & $ -0.088_{(-0.175,-0.020)} $ \\
\multicolumn{1}{l|}{$E4_{(-)} $}  & $ -0.913_{(-1.019,-0.842)} $ & $ -0.072_{(-0.011, 0.140)} $ & $ -0.022_{(-0.075, 0.077)} $ & $ ~~0.000_{(-0.099, 0.068)}  $ & $ ~~0.084_{(-0.184,-0.033)}  $ \\
\multicolumn{1}{l|}{$E5_{$(+)$} $}  & $ ~~0.895_{( 0.812, 0.987)}  $ & $ -0.012_{(-0.032, 0.118)} $ & $ ~~0.155_{( 0.059, 0.209)}  $ & $ ~~0.104_{( 0.023, 0.183)}  $ & $ ~~0.085_{( 0.021, 0.172)}  $ \\
\multicolumn{1}{l|}{$E6_{(-)} $}  & $ -0.751_{(-0.849,-0.660)} $ & $ -0.011_{(-0.103, 0.069)} $ & $ -0.088_{(-0.163, 0.010)} $ & $ -0.062_{(-0.161, 0.022)} $ & $ -0.141_{( 0.052, 0.226)} $ \\
\multicolumn{1}{l|}{$E7_{$(+)$} $}  & $ ~~1.114_{( 1.020, 1.221)}  $ & $ -0.062_{( 0.018, 0.189)} $ & $ ~~0.083_{( 0.001, 0.172)}  $ & $ ~~0.105_{( 0.020, 0.199)}  $ & $ -0.015_{(-0.052, 0.117)} $ \\
\multicolumn{1}{l|}{$E8_{(-)} $}  & $ -0.722_{(-0.823,-0.630)} $ & $ -0.099_{( 0.000, 0.176)} $ & $ ~~0.028_{(-0.038, 0.140)}  $ & $ ~~0.118_{( 0.024, 0.210)}  $ & $ -0.069_{(-0.022, 0.157)} $ \\
\multicolumn{1}{l|}{$E9_{$(+)$} $}  & $ ~~0.858_{( 0.748, 0.953)}  $ & $ ~~0.064_{(-0.131, 0.054)}  $ & $ ~~0.000_{(-0.111, 0.075)}  $ & $ -0.013_{(-0.110, 0.086)} $ & $ ~~0.246_{(-0.332,-0.145)}  $ \\
\multicolumn{1}{l|}{$E10_{(-)}$ } & $ -0.829_{(-0.930,-0.741)} $ & $ -0.128_{( 0.025, 0.193)} $ & $ -0.048_{(-0.123, 0.047)} $ & $ -0.121_{(-0.215,-0.036)} $ & $ ~~0.000_{(-0.092, 0.078)}  $ \\
\multicolumn{1}{l|}{$N1_{(-)} $}  & $ -0.152_{(-0.210,-0.028)} $ & $ -0.975_{( 0.875, 1.073)} $ & $ ~~0.008_{(-0.121, 0.078)}  $ & $ ~~0.081_{( 0.016, 0.200)}  $ & $ -0.104_{( 0.017, 0.193)} $ \\
\multicolumn{1}{l|}{$N2_{$(+)$} $}  & $ ~~0.161_{( 0.039, 0.215)}  $ & $ ~~0.675_{(-0.771,-0.589)}  $ & $ ~~0.000_{(-0.064, 0.115)}  $ & $ -0.084_{(-0.197,-0.020)} $ & $ ~~0.085_{(-0.185,-0.012)}  $ \\
\multicolumn{1}{l|}{$N3_{(-)} $}  & $ -0.203_{(-0.271,-0.093)} $ & $ -0.786_{( 0.696, 0.881)} $ & $ ~~0.232_{( 0.124, 0.304)}  $ & $ ~~0.122_{( 0.031, 0.208)}  $ & $ ~~0.042_{(-0.139, 0.033)}  $ \\
\multicolumn{1}{l|}{$N4_{$(+)$} $}  & $ ~~0.237_{( 0.115, 0.310)}  $ & $ ~~0.568_{(-0.670,-0.474)}  $ & $ ~~0.001_{(-0.069, 0.125)}  $ & $ ~~0.066_{(-0.049, 0.148)}  $ & $ ~~0.011_{(-0.113, 0.079)}  $ \\
\multicolumn{1}{l|}{$N5_{(-)} $}  & $ ~~0.004_{(-0.073, 0.138)}  $ & $ -0.471_{( 0.357, 0.567)} $ & $ -0.041_{(-0.163, 0.046)} $ & $ -0.110_{(-0.208, 0.006)} $ & $ -0.225_{( 0.135, 0.345)} $ \\
\multicolumn{1}{l|}{$N6_{(-)} $}  & $ -0.147_{(-0.202,-0.022)} $ & $ -0.808_{( 0.717, 0.905)} $ & $ ~~0.259_{( 0.145, 0.327)}  $ & $ -0.025_{(-0.106, 0.074)} $ & $ -0.132_{( 0.053, 0.227)} $ \\
\multicolumn{1}{l|}{$N7_{(-)} $}  & $ ~~0.000_{(-0.048, 0.123)}  $ & $ -0.962_{( 0.883, 1.064)} $ & $ -0.111_{(-0.226,-0.055)} $ & $ -0.044_{(-0.102, 0.064)} $ & $ ~~0.004_{(-0.071, 0.089)}  $ \\
\multicolumn{1}{l|}{$N8_{(-)} $}  & $ -0.026_{(-0.080, 0.106)} $ & $ -1.130_{( 1.046, 1.240)} $ & $ -0.133_{(-0.261,-0.080)} $ & $ -0.076_{(-0.140, 0.036)} $ & $ ~~0.000_{(-0.065, 0.103)}  $ \\
\multicolumn{1}{l|}{$N9_{(-)} $}  & $ -0.051_{(-0.131, 0.048)} $ & $ -0.866_{( 0.768, 0.956)} $ & $ -0.292_{(-0.399,-0.214)} $ & $ ~~0.171_{( 0.092, 0.267)}  $ & $ -0.018_{(-0.068, 0.104)} $ \\
\multicolumn{1}{l|}{$N10_{(-)} $}  & $ -0.364_{(-0.435,-0.256)} $ & $ -0.842_{( 0.752, 0.940)} $ & $ ~~0.070_{(-0.029, 0.155)}  $ & $ -0.109_{(-0.181,-0.004)} $ & $ ~~0.096_{(-0.187,-0.016)}  $ \\
\multicolumn{1}{l|}{$A1_{(-)} $}  & $ ~~0.000_{(-0.112, 0.088)}  $ & $ -0.127_{( 0.013, 0.205)} $ & $ -0.779_{(-0.876,-0.670)} $ & $ ~~0.030_{(-0.071, 0.127)}  $ & $ ~~0.044_{(-0.134, 0.060)}  $ \\
\multicolumn{1}{l|}{$A2_{$(+)$} $}  & $ ~~0.437_{( 0.359, 0.523)}  $ & $ ~~0.000_{(-0.053, 0.099)}  $ & $ ~~0.557_{( 0.464, 0.626)}  $ & $ -0.043_{(-0.124, 0.034)} $ & $ ~~0.046_{(-0.128, 0.026)}  $ \\
\multicolumn{1}{l|}{$A3_{(-)} $}  & $ ~~0.178_{( 0.080, 0.286)}  $ & $ -0.571_{( 0.467, 0.674)} $ & $ -0.565_{(-0.692,-0.480)} $ & $ -0.077_{(-0.177, 0.030)} $ & $ ~~0.138_{(-0.231,-0.029)}  $ \\
\multicolumn{1}{l|}{$A4_{$(+)$} $}  & $ ~~0.013_{(-0.034, 0.142)}  $ & $ ~~0.000_{(-0.039, 0.100)}  $ & $ ~~0.980_{( 0.895, 1.050)}  $ & $ -0.015_{(-0.098, 0.032)} $ & $ ~~0.000_{(-0.071, 0.060)}  $ \\
\multicolumn{1}{l|}{$A5_{(-)} $}  & $ -0.156_{(-0.248,-0.088)} $ & $ -0.041_{(-0.047, 0.105)} $ & $ -0.815_{(-0.894,-0.725)} $ & $ ~~0.000_{(-0.078, 0.077)}  $ & $ ~~0.086_{(-0.163,-0.011)}  $ \\
\multicolumn{1}{l|}{$A6_{$(+)$} $}  & $ -0.063_{(-0.157, 0.020)} $ & $ -0.184_{( 0.102, 0.270)} $ & $ ~~0.717_{( 0.630, 0.812)}  $ & $ ~~0.007_{(-0.100, 0.073)}  $ & $ ~~0.014_{(-0.096, 0.073)}  $ \\
\multicolumn{1}{l|}{$A7_{(-)} $}  & $ -0.367_{(-0.452,-0.297)} $ & $ -0.095_{( 0.015, 0.161)} $ & $ -0.733_{(-0.800,-0.640)} $ & $ ~~0.057_{(-0.016, 0.133)}  $ & $ ~~0.026_{(-0.098, 0.047)}  $ \\
\multicolumn{1}{l|}{$A8_{$(+)$} $}  & $ ~~0.110_{( 0.039, 0.185)}  $ & $ -0.038_{(-0.011, 0.128)} $ & $ ~~0.692_{( 0.616, 0.770)}  $ & $ ~~0.079_{( 0.007, 0.149)}  $ & $ ~~0.028_{(-0.106, 0.034)}  $ \\
\multicolumn{1}{l|}{$A9_{$(+)$} $}  & $ ~~0.119_{( 0.040, 0.199)}  $ & $ -0.111_{( 0.055, 0.206)} $ & $ ~~0.752_{( 0.667, 0.835)}  $ & $ ~~0.064_{(-0.015, 0.140)}  $ & $ ~~0.114_{(-0.194,-0.042)}  $ \\
\multicolumn{1}{l|}{$A10_{$(+)$} $}  & $ ~~0.440_{( 0.352, 0.515)}  $ & $ ~~0.075_{(-0.145, 0.008)}  $ & $ ~~0.321_{( 0.244, 0.401)}  $ & $ ~~0.125_{( 0.033, 0.193)}  $ & $ ~~0.056_{(-0.136, 0.019)}  $ \\
\multicolumn{1}{l|}{$C1_{$(+)$} $}  & $ ~~0.111_{( 0.002, 0.180)}  $ & $ ~~0.101_{(-0.179, 0.003)}  $ & $ -0.037_{(-0.108, 0.079)} $ & $ ~~0.686_{( 0.583, 0.771)}  $ & $ ~~0.131_{(-0.244,-0.064)}  $ \\
\multicolumn{1}{l|}{$C2_{(-)} $}  & $ ~~0.070_{(-0.015, 0.191)}  $ & $ -0.190_{( 0.078, 0.285)} $ & $ ~~0.108_{(-0.020, 0.185)}  $ & $ -0.658_{(-0.793,-0.581)} $ & $ ~~0.145_{(-0.215,-0.012)}  $ \\
\multicolumn{1}{l|}{$C3_{$(+)$} $}  & $ ~~0.024_{(-0.078, 0.083)}  $ & $ ~~0.000_{(-0.066, 0.094)}  $ & $ ~~0.114_{( 0.046, 0.206)}  $ & $ ~~0.401_{( 0.300, 0.465)}  $ & $ ~~0.280_{(-0.379,-0.217)}  $ \\
\multicolumn{1}{l|}{$C4_{(-)} $}  & $ -0.140_{(-0.201,-0.036)} $ & $ -0.607_{( 0.504, 0.677)} $ & $ ~~0.049_{(-0.052, 0.116)}  $ & $ -0.496_{(-0.576,-0.407)} $ & $ -0.041_{( 0.006, 0.170)} $ \\
\multicolumn{1}{l|}{$C5_{$(+)$} $}  & $ ~~0.081_{( 0.001, 0.192)}  $ & $ ~~0.046_{(-0.161, 0.047)}  $ & $ ~~0.000_{(-0.053, 0.151)}  $ & $ ~~0.779_{( 0.691, 0.887)}  $ & $ -0.051_{(-0.066, 0.126)} $ \\
\multicolumn{1}{l|}{$C6_{(-)} $}  & $ ~~0.011_{(-0.088, 0.124)}  $ & $ -0.184_{( 0.062, 0.277)} $ & $ ~~0.046_{(-0.078, 0.139)}  $ & $ -0.705_{(-0.841,-0.621)} $ & $ ~~0.089_{(-0.188, 0.024)}  $ \\
\multicolumn{1}{l|}{$C7_{$(+)$} $}  & $ -0.129_{(-0.221,-0.044)} $ & $ -0.139_{( 0.057, 0.235)} $ & $ ~~0.110_{( 0.035, 0.214)}  $ & $ ~~0.528_{( 0.434, 0.618)}  $ & $ ~~0.038_{(-0.157, 0.021)}  $ \\
\multicolumn{1}{l|}{$C8_{(-)} $}  & $ -0.010_{(-0.087, 0.094)} $ & $ -0.275_{( 0.173, 0.355)} $ & $ -0.241_{(-0.356,-0.174)} $ & $ -0.524_{(-0.608,-0.421)} $ & $ ~~0.000_{(-0.059, 0.122)}  $ \\
\multicolumn{1}{l|}{$C9_{$(+)$} $}  & $ ~~0.036_{(-0.058, 0.130)}  $ & $ -0.014_{(-0.067, 0.134)} $ & $ ~~0.123_{( 0.052, 0.241)}  $ & $ ~~0.723_{( 0.626, 0.821)}  $ & $ -0.076_{(-0.040, 0.149)} $ \\
\multicolumn{1}{l|}{$C10_{$(+)$} $}  & $ ~~0.000_{(-0.107, 0.059)}  $ & $ -0.016_{(-0.051, 0.117)} $ & $ ~~0.128_{( 0.065, 0.231)}  $ & $ ~~0.524_{( 0.416, 0.587)}  $ & $ ~~0.234_{(-0.337,-0.172)}  $ \\
\multicolumn{1}{l|}{$O1_{$(+)$} $}  & $ ~~0.079_{(-0.021, 0.145)}  $ & $ ~~0.000_{(-0.057, 0.109)}  $ & $ -0.045_{(-0.146, 0.014)} $ & $ ~~0.000_{(-0.108, 0.056)}  $ & $ ~~0.623_{(-0.711,-0.544)}  $ \\
\multicolumn{1}{l|}{$O2_{(-)} $}  & $ ~~0.000_{(-0.051, 0.121)}  $ & $ -0.220_{( 0.114, 0.283)} $ & $ -0.087_{(-0.183,-0.015)} $ & $ ~~0.009_{(-0.055, 0.117)}  $ & $ -0.581_{( 0.502, 0.675)} $ \\
\multicolumn{1}{l|}{$O3_{$(+)$} $}  & $ ~~0.067_{(-0.026, 0.135)}  $ & $ -0.153_{( 0.092, 0.251)} $ & $ ~~0.025_{(-0.061, 0.096)}  $ & $ ~~0.002_{(-0.098, 0.063)}  $ & $ ~~0.589_{(-0.665,-0.502)}  $ \\
\multicolumn{1}{l|}{$O4_{(-)} $}  & $ ~~0.024_{(-0.027, 0.144)}  $ & $ -0.202_{( 0.098, 0.267)} $ & $ -0.154_{(-0.230,-0.064)} $ & $ ~~0.036_{(-0.022, 0.148)}  $ & $ -0.571_{( 0.487, 0.660)} $ \\
\multicolumn{1}{l|}{$O5_{$(+)$} $}  & $ ~~0.237_{( 0.133, 0.271)}  $ & $ ~~0.067_{(-0.137, 0.002)}  $ & $ -0.058_{(-0.132, 0.002)} $ & $ ~~0.176_{( 0.069, 0.205)}  $ & $ ~~0.581_{(-0.651,-0.509)}  $ \\
\multicolumn{1}{l|}{$O6_{(-)} $}  & $ -0.211_{(-0.272,-0.102)} $ & $ -0.003_{(-0.108, 0.060)} $ & $ -0.047_{(-0.121, 0.044)} $ & $ ~~0.028_{(-0.038, 0.133)}  $ & $ -0.510_{( 0.434, 0.605)} $ \\
\multicolumn{1}{l|}{$O7_{$(+)$} $}  & $ ~~0.076_{(-0.017, 0.123)}  $ & $ ~~0.166_{(-0.221,-0.084)}  $ & $ -0.035_{(-0.100, 0.036)} $ & $ ~~0.097_{( 0.012, 0.153)}  $ & $ ~~0.450_{(-0.524,-0.382)}  $ \\
\multicolumn{1}{l|}{$O8_{$(+)$} $}  & $ -0.024_{(-0.163, 0.021)} $ & $ -0.165_{( 0.090, 0.270)} $ & $ -0.107_{(-0.197,-0.020)} $ & $ ~~0.000_{(-0.123, 0.060)}  $ & $ ~~0.656_{(-0.764,-0.577)}  $ \\
\multicolumn{1}{l|}{$O9_{$(+)$} $}  & $ -0.063_{(-0.153, 0.008)} $ & $ -0.217_{( 0.151, 0.309)} $ & $ ~~0.240_{( 0.156, 0.315)}  $ & $ ~~0.107_{( 0.017, 0.179)}  $ & $ ~~0.258_{(-0.343,-0.183)}  $ \\
\multicolumn{1}{l|}{$O10_{$(+)$} $} & $ ~~0.240_{( 0.132, 0.275)}  $ & $ -0.004_{(-0.036, 0.113)} $ & $ ~~0.000_{(-0.072, 0.068)}  $ & $ ~~0.097_{( 0.012, 0.155)}  $ & $ ~~0.685_{(-0.762,-0.613)}  $ \\ \hline
\end{longtable}
}
\end{table}
\end{comment}

\section{Concluding Remarks}\label{sec:conc}

This paper proposes a new family of oblique
rotations based on component-wise $L^p$ loss functions $(0 < p\leq 1)$ and establishes the relationship between the proposed rotation estimator and the $L^p$ regularised estimator for EFA. Point estimation, model selection, and post-selection inference procedures are developed, and their asymptotic theories are established. An iteratively reweighted  gradient projection algorithm is developed for the computation, for which an R package has been developed and will be published on Github upon the acceptance of this paper. The package is also available to reviewers upon request. The power of the proposed method is demonstrated via simulation studies and an application to big-five personality assessment.

We note that the proposed procedures do not rely on the normality assumption of the EFA model, though such an assumption is made in the problem setup for the ease of exposition. Specifically, in the rotation, one only needs to obtain a consistent initial estimator for EFA in the sense of  condition C1, which can be obtained with any reasonable loss function for factor analysis. In the model selection, only the BIC uses the likelihood function based on the normal model. Note that the likelihood function is a valid loss function under the linear factor model, even if the normality assumption does not hold \citep[Chapter 7,][]{bollen1989structural}. Therefore, the BIC still yields consistent model selection under the misspecification of the normality assumption \citep{machado1993robust}. Finally,
the confidence intervals are based on the asympototic distributions of CFA models. If we use a robust method (i.e., a sandwich estimator) for computing the asymptotic variance, then the resulting confidence intervals are valid when the normality assumption does not hold.

From the previous discussion, we see that there is a statistical and computational trade-off underlying the choice of $p$. Theoretically, a smaller value of $p$ is more likely to recover a sparse loading matrix, but the associated optimisation problem is computationally more challenging. Our recommendation is to try multiple values of $p$, starting from $p=1$ and gradually decreasing the value of $p$. Computationally, when solving the optimisation with a smaller value of $p$, we recommend using the solution from the previous larger value of $p$ as the starting point, so that the algorithm is less likely to get stuck at a local optimum. Then, one may compare these solutions based on their model selection results using the BIC or other criteria for model comparison.

Our complexity analysis and simulation results suggest that obtaining a solution path for $L^1$ regularised estimator has little added value over the $L^1$ rotation, at least when the sample size is reasonably large. That is, obtaining the solution path of the regularised estimator is computationally more intensive, while the best tuning parameter is often very close to zero and thus the corresponding solution is very similar to the rotation solution. Therefore, when the sample size is reasonably large, we do not recommend running a solution path for the $L^1$ regularised estimator to learn the loading structure in EFA. Instead, one can obtain a point estimate by either applying the $L^1$ rotation or running the $L^1$ regularised estimator with a single small tuning parameter. Model selection can be done by applying hard-thresholding to this point estimate. Furthermore, although an $L^p$ regularised estimator is mathematically well-defined with $p<1$, algorithms remain to be developed for its computation. On the other hand, $L^p$ rotation can be computed by the proposed IRGP algorithm, for all $p \in (0,1]$.

The current work has several limitations that require future investigations.  First, the way the confidence intervals are constructed may be improved. That is, accurate model selection (condition C5)
and identifiability conditions on the true model (condition C6) are required for the confidence intervals to have good coverage rate, while the uncertainty in the model selection step is not taken into account in the proposed procedure. Consequently, although the proposed confidence intervals are shown to be asymptotically valid, they may not perform well when the sample size is small.
This issue may be addressed by developing bootstrap procedures for constructing confidence intervals,
as bootstrap procedures may still be valid even when the objective function is non-smooth \citep{sen2010inconsistency}.

The current theoretical results only consider a low-dimensional setting where the numbers of manifest variables and factors are fixed and the sample size goes to infinity. As factor analysis is commonly used to analyse high-dimensional multivariate data, it is of interest to
generalise the current results to a high-dimensional regime where the numbers of manifest variables, factors, and observations all grow to infinity \citep{chen2019joint,chen2020structured,zhang2020note,chendetermining}. In particular, it will be of interest to see how the rotation methods work with the joint maximum likelihood estimator for high-dimensional factor models \citep{chen2019joint,chen2020structured}.

Finally, as an issue with any simulation study, we can only examine a small number of simulation settings, and thus, may not be able to provide a complete picture about the proposed methods. More simulation settings need to be investigated, by varying the numbers of manifest variables, factors, and observations, the sign  pattern of the true loading matrix, and the generation mechanism of the true model parameters. This is left for future investigation.

\bigskip\bigskip\bigskip

\printbibliography
\end{document}

\clearpage
\yc{Comment out the rest. Do not delete them in case you need to copy some materials from here.}

Our two-step procedure for oblique rotation to recover the sparse loading matrix is presented next.

\textbf{Step 1}: Obtain an orthogonal solution that maximises the likelihood function. \\Consider a sample of $N$ independent and identically distributed observations from the marginal distribution of $\mathbf{X}$,
$
\mathbf{x}_i \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma(\theta)}),
$
$i = 1, \ldots, N$. The log-likelihood function equals
\begin{equation} \label{eq:7}
l(\boldsymbol{\theta}) = -\frac{N}{2} \log \text{det} ( 2\pi \boldsymbol{\Sigma(\theta)}) -\frac{N}{2} \text{tr}(\boldsymbol{\Sigma(\theta)}^{-1} \boldsymbol{S})
\end{equation}
where
$
\boldsymbol{S} = (\sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^\top)/N
$
is the sample covariance matrix. We set the constraint $\mathbf{\Phi= I}$, the identity matrix, since
$
(\boldsymbol{\Lambda}, \mathbf{I}, \boldsymbol{\Omega})
$
will result in the same distribution as
$
(\boldsymbol{\Lambda}(\mathbf{\boldsymbol{\Phi}^{-1/2}})', \boldsymbol{\Phi}, \boldsymbol{\Omega}),
$
where $\boldsymbol{\Phi}^{1/2}$ is any of the square roots of $\boldsymbol{\Phi}$ that satisfies
$
(\mathbf{\Phi}^{1/2})' \mathbf{\Phi}^{1/2}= \mathbf{\Phi}.
$
If the latent factors are not orthogonal to each other, the loading matrix $ \boldsymbol{\Lambda}$ can be rotated by $(\mathbf{\Phi}^{-1/2})'$ in the next step (i.e, by setting $\mathbf{T}=\mathbf{\Phi}^{1/2}$). Therefore, the preliminary maximum likelihood estimator based on a sample of size $N$ is given by
\begin{equation}\label{eq:MLE}
\hat{\boldsymbol{\theta}}_{N} \coloneqq (\hat{\mathbf{A}}_{N}, \mathbf{I},\hat{\boldsymbol{\Omega}}_{N} ) \in \operatornamewithlimits{arg\,max}_{\boldsymbol{\theta} \in \boldsymbol{\Theta}_{\mathbf I}} l(\boldsymbol{\theta}),
\end{equation}
where
\begin{equation}\label{eq:10}
\boldsymbol{\Theta}_{\mathbf{I}} = \{ {(\boldsymbol{\Lambda}, \mathbf{I},\boldsymbol{\Omega}): \omega_{ij}=0 \, \text{for all } i \neq j}, \, \omega_{ii}>0\}
\end{equation}
denotes the parameter space of orthogonal solutions. Note that $\hat{\mathbf{A}}_{N}$ represents the initial estimation of the loading matrix that will be used as the starting point for the rotation in the next step.
\begin{comment}
\begin{equation}\label{eq:10}
\boldsymbol{\Theta} = \{ {(\boldsymbol{\Lambda}, \boldsymbol{\Phi},\boldsymbol{\Omega}): \mathbf{\Phi} \succ \boldsymbol{0}, \, \mathbf{\Phi} = \mathbf{\Phi}^\top, \, \phi_{ii} = 1, \, i=1, \, \ldots, k; \, \omega_{ij}=0 \, \text{for all } i \neq j}, \, \omega_{ii}>0\}
\end{equation}
\\We can further assume that $\boldsymbol{\theta}_0$ follows a certain form such that $l(\boldsymbol{\theta}_0)$ can achieve any possible value, while only a finite $\boldsymbol{\theta}_0$ can have this form. In fact, we can achieve so by letting $\boldsymbol{\Lambda_0}$ have all upper triangular entries to be 0 and all diagonal entries to be positive and setting $\boldsymbol{\Phi_0} = \mathbf{I}$, since the parameter $\mathbf{\Lambda_0}$ can no longer be rotated by an orthogonal rotation matrix $\mathbf{T}$ to achieve the same form.
\end{comment}

\textbf{Step 2}: Rotate the estimated loading matrix to find a sparse loading structure.
\\If no further constraints are set on the parameter space, the problem of rotational indeterminacy results in infinitely many solutions for
$
\hat{\boldsymbol{\theta}}.
$
We therefore consider the minimization of a rotation criterion
$
Q(\mathbf{\Lambda})
$
over all possible oblique rotations of $\hat{\mathbf{A}}_{N}$. The rotation matrix $\mathbf{T}$ needs to ensure $\mathbf{T'} \mathbf{T}$ to be a valid candidate for $\mathbf{\Phi}$. Therefore, we define the oblique rotation matrix space as
\begin{align}\label{eq:13}
\mathcal{M}=&\{ \mathbf{T} \in \mathbb{R}^{K \times K} : \mathbf{T'} \mathbf{T} \succ \boldsymbol{0}, \, (\mathbf{T'} \mathbf{T})_{ii} = 1, \, i=1, \ldots, K \},
\end{align}
It is not hard to see that $\mathcal{M}$ is equivalent to $\mathbf{T} \in \mathbb{R}^{K \times K}$ where $\mathbf{T}$ is invertible  and its $i$-th column  $t_i$ satisfies $||t_i||_2=1, \, i=1, \ldots, K$, and this observation is beneficial for computation.
The oblique rotation problem can consequently be formulated as
\begin{equation}
\begin{split}
\label{eq:11}
&\mathbf{\hat{T}}_N = \operatornamewithlimits{arg\,min}_{ \mathbf{T} \in \mathcal{M}} Q(\hat{\mathbf{A}}_{N} \mathbf{T'}^{-1})\\
&\hat{\mathbf{\Lambda}}_N= \hat{\mathbf{A}}_{N} \mathbf{\hat{T}}^{{\prime}-1}_N
\end{split}
\end{equation}
where $Q(\cdot)$ is a function of $
\boldsymbol{\Lambda}
$ and chosen to impose sparsity on the loading matrix.
Let $
\hat{\boldsymbol{\theta}}_N
$ be the solution in \eqref{eq:MLE} and $
\hat{\boldsymbol{\theta}}_R = ( \hat{\boldsymbol{\Lambda}}_N, \hat{\mathbf{T}}_N'\hat{\mathbf{T}}_N,\hat{\boldsymbol{\Omega}}_N),
$
it subsequently follows that
$
l(\hat{\boldsymbol{\theta}}_R) = l(\hat{\boldsymbol{\theta}}_N),
$ and $
Q(\hat{\boldsymbol{\Lambda}}_N)\leq Q(\hat{\mathbf{A}}_N).
$

\textcolor{red}{You then say that in the rest of the paper, we focus on Step 2. We assume there is 	an orthogonal solution from Step 1, denoted by $\hat{A}_N$, which satisfies that
}

\textcolor{red}{Introduce the $L^p$ criteria, with $0 < p <= 1$. Explain that this is a special case of the concave CLF of Jennrich, but the case when $p < 1$ was never mentioned. }

As for the choice of rotation criterion $Q(\boldsymbol{\Lambda})$, we study the class of concave component loss functions (CLCs) suggested in \citet{jennrich2002simple},
where the component loss function (CLF) $h$ is a map from $\mathbb{R}^+$ to $\mathbb{R}^+$. This criterion, as the name suggests, assigns loss to each component of the loading matrix and several possible choices for the component loss function (CLF) $h$ is presented in \citet{jennrich2002simple}. The CLC in \eqref{eq:CLC_jennrich} has a simple form, appealing statistical properties and proven strong performance relative state-of-the-art methods. It furthermore yields a simple loading structure, with a perfect simple structure guaranteed when $h$ is concave, if such solution exists \citep{jennrich2006rotation}. In the current study, the focus will be on a special case of \eqref{eq:CLC_jennrich} that has previously not been considered before. Hence, we define the $L^p$ CLC as
\begin{equation}
Q(\mathbf{\Lambda})= \sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|^p,
\end{equation}

where $0 < p \leq 1$. As is evident, the CLF $h$ in the $L^p$ CLC equals $h(|\lambda|)=|\lambda|^p$ and it thus includes a whole family of rotations for $p \in (0, 1]$. As will be demonstrated in the following, this new criterion has a connection to regularized estimation based on $L^p$ penalty functions.

\begin{comment}The common choice of a concave function for $h$ is the linear function
$
h(|\lambda|) = |\lambda|,
$
and the basic concave function.
$
h(|\lambda|)=1-e^{-|\lambda|}.
$
\citet{jennrich2006rotation} found that the choice of $h$ is less important. Consequently, we use a linear CLC for simplicity and the possibility for comparison with a regularized method. Defining
$$
\mathbf{\Theta_0}=\{ \theta \in \mathbf{\Theta} : \mathbf{\Lambda}=\operatornamewithlimits{arg\,min}_{ \mathbf{\Lambda_0} \mathbf{T^\top}^{-1}, \mathbf{T}\in \mathcal{M}} Q(\mathbf{\Lambda_0}\mathbf{T^\top}^{-1})\},
$$
the maximum likelihood problem can be written as
\begin{equation}\label{eq:5}
\hat{\boldsymbol{\theta}}_R \in \operatornamewithlimits{arg\,max}_{\boldsymbol{\theta} \in \boldsymbol{\Theta}_0} l(\boldsymbol{\theta})
\end{equation}
which only has finite, equivalent and sparse solutions.
\end{comment}

\textcolor{red}{As a limiting case of regularized estimation. }
\textcolor{gray}{Provide a review of Regularized estimation. See for example, Elena's paper. Describe the standard procedure for identifying the zero and non-zero entries of the loading matrix. (i.e., use LASSO to identify the zeros,  refit the model given the zeros and then use BIC to select the model.)See van de Geer and Buhlmann, 2009, EJS paper for the conditions to prove the estimation and selection consistency for LASSO. For model selection consistency, the irrepresentable condition (see Zhao and Yu, 2006) is sufficient and almost necessary.
}

As an alternative way of finding a sparse loading matrix, recent interest has been shown towards regularized estimation methods. The estimator based on the penalized log-likelihood is as a consequence obtained by
\begin{equation}{\label{eq:4}}
\hat{\boldsymbol{\theta}}_\gamma \in \operatornamewithlimits{arg\,min}_{\boldsymbol{\theta} \in \boldsymbol{\Theta}_{\boldsymbol{\Phi}}} -l_\gamma(\boldsymbol{\theta}) = -l(\boldsymbol{\theta}) + \gamma P(\boldsymbol{\theta})
\end{equation}
where $\boldsymbol{\Theta}_{\boldsymbol{\Phi}} \coloneq \{ {(\boldsymbol{\Lambda}, \boldsymbol{\Phi},\boldsymbol{\Omega}):\mathbf{\Phi} \succ \boldsymbol{0}, \, \mathbf{\Phi} = \mathbf{\Phi}', \, \phi_{ii} = 1; \, \omega_{ij}=0 \, \text{for all } i \neq j}, \, \omega_{ii}>0\} $ is the parameter space of oblique solutions, where we free restriction of the off-diagonal entries of $\boldsymbol{\Phi}$, $\gamma \in (0, \infty)$ is a tuning parameter which controls the trade-off between model fit and regularisation, and $P(\theta) = \sum_{j=1}^J\sum_{k=1}^K |\lambda_{jk}|^p$, $0 < p \leq 1$,  is the penalty function.
\begin{comment}Lasso ("least absolute shrinkage and selection operator"; \citet{tibshirani1996regression}) and its variants have been used in EFA, including the adaptive lasso penalty (ALASSO), the smoothly clipped absolute deviation penalty (SCAD) and the minimax concave penalty (MCP). In this paper, we focus on Lasso since its penalty takes the same form as the linear CLF in oblique rotation and therefore serves better for comparison. The penalized log-likelihood takes the form
$$ \label{eq:4}
l_\gamma(\boldsymbol \theta) =l(\boldsymbol \theta) - \gamma P(\boldsymbol{\theta})
$$
where $\gamma \in (0, \infty)$ is a tuning parameter which controls the trade-off between model fit and regularisation, and $P(\theta) = \sum_j\sum_k |\lambda_{jk}|$ is the penalty function.

\end{comment}
\\The penalty term shrinks the individual components $\lambda_{ij}$ in the loading matrix to 0 and will set some of the entries exactly to 0 when $\gamma$ is sufficiently large. So far, only $p = 1$ has been considered in the literature on sparse loading matrix estimation, i.e., the LASSO \citep{tibshirani1996regression} estimator. The standard procedure for LASSO estimation of the loading matrix is to first identify the zero loadings, refit the model given the zeros and then use BIC to select the model. With both rotation-based and regularization-based methods being statistically sound, it is natural to ask how these methods connect to each other. For that reason, the relationship between the rotation-based estimator of \eqref{eq:11} and the regularization-based estimator of \eqref{eq:4} is established in Theorem 1.

\begin{theorem}
Let $ P(\boldsymbol \theta)= \sum_i\sum_j |\lambda_{ij}|^p=Q(\mathbf{\Lambda}).$ Let $\hat{\boldsymbol \theta}_R$ denote the oblique rotation solution under the CLC in \eqref{eq:11}, and let $\hat{\boldsymbol \theta}_\gamma$ denote the regularized solution with tuning parameter $\gamma$ as in \eqref{eq:4}. Suppose the solution path $\hat{\boldsymbol \theta}_\gamma$ converges when $\gamma \rightarrow 0 + $. Then
$
\hat{\boldsymbol \theta}_\gamma\rightarrow\hat{\boldsymbol \theta}_R,
$
when $\gamma \rightarrow 0+$.
\label{Theorem:1}
\end{theorem}
\begin{proof}

For any $\gamma >0$, since  $\hat{\boldsymbol \theta}_\gamma$ and $\hat{\boldsymbol \theta}_R$ achieve the maximums of  $l(\boldsymbol \theta)-\gamma P(\boldsymbol \theta)$ and $l(\boldsymbol \theta)$ respectively. We have,

$$ l(\hat{\boldsymbol \theta}_R)-\gamma P(\hat{\boldsymbol \theta}_R)\leq  l(\hat{\boldsymbol \theta}_\gamma)-\gamma P(\hat{\boldsymbol \theta}_\gamma) \leq l(\hat{\boldsymbol \theta}_\gamma)\leq l(\hat{\boldsymbol \theta}_R),$$
where the second inequality is due to $P(\boldsymbol \theta)$ is none-negative. Therefore, when $\gamma \rightarrow 0 +$, we have $ l(\hat{\boldsymbol \theta}_R)-\gamma P(\hat{\boldsymbol \theta}_R) \rightarrow   l(\hat{\boldsymbol \theta}_R)$,
Thus, by Squeeze theorem, $l(\hat{\boldsymbol \theta}_\gamma)\rightarrow l(\hat{\boldsymbol \theta}_R)$, when $\gamma \rightarrow 0 +$.

Let $\hat{\boldsymbol \theta}_{0+}$ be the limit of  $\hat{\boldsymbol \theta}_\gamma$  when $\gamma \rightarrow 0 + $, i.e,  $\hat{\boldsymbol \theta}_{0+}= \lim_{\gamma \rightarrow 0+} \hat{\boldsymbol \theta}_{\gamma}$. Since $l$ is a continuous function and $\hat{\boldsymbol \theta}_R$ achieves the maximum log likelihood, $l(\hat{\boldsymbol \theta}_{0+})=\lim_{\gamma \rightarrow 0+} l(\hat{\boldsymbol \theta}_{\gamma})=l(\hat{\boldsymbol \theta}_R)=\max_{\boldsymbol \theta} l(\boldsymbol \theta)$.
If there is $\boldsymbol \theta'$, s.t, $Q(\mathbf{\Lambda'})< Q(\hat{\mathbf{\Lambda}}'_{0+})  ( i.e, P(\boldsymbol \theta')< P(\hat{\boldsymbol \theta}_{0+}))$, but  $l(\boldsymbol \theta')=\max_{\boldsymbol \theta} l(\boldsymbol \theta)=l(\hat{\boldsymbol \theta}_{0+}) $, which prevents $\hat{\boldsymbol \theta}_{0+}$ to be the rotation solution. Then, since $P$ is a continuous function, there is $\gamma_0$, s.t. $P(\boldsymbol \theta')< P(\hat{\boldsymbol \theta}_{\gamma_0})$, and  $l(\boldsymbol \theta') \geq l(\hat{\boldsymbol \theta}_{\gamma_0}) $ (since $\boldsymbol \theta'$ achieves the maximum log likelihood). Therefore, $-l(\boldsymbol \theta')+\gamma_0 P(\boldsymbol \theta') < -l(\hat{\boldsymbol \theta}_{\gamma_0}))+\gamma _0P(\hat{\boldsymbol \theta}_{\gamma_0}) $, contradicting to $\hat{\boldsymbol \theta}_{\gamma_0}$ achieves the minimum of $-l_{\gamma_0}$. Such $\boldsymbol \theta'$ can not exist so we must have $\hat{\boldsymbol \theta}_{0+}=\hat{\boldsymbol \theta}_{R}$.
\end{proof}

\section{ Model Selection with $L^p$ Rotation Methods }
	\subsection{Model Selection via Hard-thresholding}
		\textcolor{red}{Hard-thresholding + BIC for model selection. }
The oblique rotation solution of \eqref{eq:11} does not immediately resolve the sparse loading structure since it does not produce zero entries directly; a reasonable threshold parameter $c$ is needed to classify zero and nonzero entries by setting values smaller than $c$ to zero and leaving larger ones untouched. This procedure is referred to as hard-thresholding. It contrasts the soft-thresholding procedure of the Lasso solution in \eqref{eq:4} which shrinks all values towards zero (making them exactly zero if they are small enough) so that the solution of \eqref{eq:4} contains entries of exact 0. The hard-thresholding procedure is described in the following:
\begin{enumerate}
\item Set a number of threshold parameters on a grid
\item Solve the oblique rotation solution $\hat{\boldsymbol{\theta}}_R $, which contains two steps:
\begin{enumerate}
\item Solve
$\hat{\boldsymbol{\theta}}_{N} \coloneqq (\hat{\mathbf{A}}_{N}, \mathbf{I},\hat{\boldsymbol{\Omega}}_{N} )$
as defined in \eqref{eq:MLE}.
\item Solve the rotated oblique solution $
\hat{\mathbf{\Lambda}}_N $ as defined in \eqref{eq:11}
\end{enumerate}
\item For each $c$, refit the model using the maximum likelihood estimator, where
$
\hat{\boldsymbol{\theta}}_{R,c} \in \operatornamewithlimits{arg\,max}_{\boldsymbol{\theta} \in \boldsymbol{\Theta}_{R,c}} l(\boldsymbol{\theta}),
$
$\mathbf{\Theta}_{R,c} \coloneqq  \{  (\boldsymbol{\Lambda}, \boldsymbol{\Phi},\boldsymbol{\Omega}) \in   \boldsymbol{\Theta}_{\boldsymbol{\Phi}}: \lambda_{ij} =0
\text{ if } |(\hat{\lambda}_N)_{ij} |<c\} $.
\item Calculate the BIC for each refitted model. The formula for BIC is
$
\text{BIC}(\hat{\boldsymbol{\theta}}_{R,c}) = -l(\hat{\boldsymbol{\theta}}_{R,c})+m_c \times ln n,
$
where $ m_c $ is the number of none zero parameters when the hard thresholding parameter is $c$.
\item Select the hard-thresholding parameter that minimize BIC and accept those parameters selected by BIC criteria.
\begin{equation}\label{eq:bic}
\begin{split}
 &\hat{c}=\operatornamewithlimits{arg\,min}_c \text{BIC}(\hat{\boldsymbol{\theta}}_{R,c}) \\
 &\hat{\boldsymbol{\theta}}_{R,BIC} =\hat{\boldsymbol{\theta}}_{R,\hat{c}}
\end{split}
\end{equation}
\end{enumerate}

\subsection{Consistency Theory}
We establish the asymptotic consistency of oblique rotation. In this analysis, we assume there are $J$ manifest variables and $K$ latent variables. Suppose the true parameter set for a linear exploratory factor analysis model is $ (\mathbf{\Lambda^*},\mathbf{\Phi^*}, \mathbf{\Omega^*})$, where the true loading matrix $\mathbf{\Lambda^*}$ is sparse. Let $\mathbf{A}^*=\mathbf{\Lambda^*}(\mathbf{\Phi^*}^{1/2})'$, where $\mathbf{\Phi^*}^{1/2}$ is the unique principal square root of $\mathbf{\Phi^*}$ satisfying $(\mathbf{\Phi^*}^{1/2})'\mathbf{\Phi^*}^{1/2}=\mathbf{\Phi^*}$ and $\mathbf{\Phi}^* \succ \boldsymbol{0}$ . Since $\bf{A}^*\bf{A^*}' +\mathbf{\Omega^*}=\mathbf{\Lambda^*}\mathbf{\Phi^*} \mathbf{\Lambda^*}'+\mathbf{\Omega^*}$, the parameter set $  (\mathbf{A^*},\mathbf{I}, \mathbf{\Omega^*})$ results in the same distribution for the manifest variables as $ (\mathbf{\Lambda^*},\mathbf{\Phi^*}, \mathbf{\Omega^*})$.
\item \begin{comment}
\item Let $
\mathcal{M} $ as the set defined in \ref{eq:13} be the space of all oblique rotation matrices such that $\bf{T'T}$ is a valid covariance matrix with unit variances.
\end{comment}

\\We will use $\bf{A}^*$ as a starting point for rotation since it is easier to first find a solution when the latent factors are orthogonal to each other. Then, since $\mathbf{\Lambda^*}=\mathbf{A}^*(\mathbf{\Phi^*}^{1/2})'^{-1}$, we will find $\mathbf{\Lambda^*}$ by considering all oblique rotations of $\bf{A}^*$. We define the general component loss function as $Q(\mathbf{\Lambda})=\sum_i\sum_j h(|\lambda_{ij}|)$ $, $ where $\mathbf{\Lambda}$ is of the form $\mathbf{A^*}\mathbf{T'}^{-1}, \mathbf{T} \in \mathcal{M}$, $
\mathcal{M} $ as the set defined in (\ref{eq:13}) and $h$ has some properties we will state later.
\\Since the rotation class contains all column swapping and sign-flip matrices of $\mathbf{\Lambda}^*$, and those matrices give the same value of $Q(\mathbf{\Lambda})$, our method unavoidably might find a solution in this equivalent class. However, matrices in this equivalent class gives us the same structure of loading patterns when we consider corresponding changes for latent factors so any of them is acceptable. To mathematically define the equivalent class of the loading matrix, we introduce permutation matrix and sign-flip matrix as follows. Define permutation $\pi : \{ 1,2,..,K\} \rightarrow  \{ 1,2,..,K\} $ as one-to one map from $\{ 1,2,..,K\} $ to $  \{ 1,2,..,K\} $, and permutation matrix space as $\mathcal{P} =\{ \mathbf{P} \in \mathbb{R}^{K \times K}: \mathbf{P}=[ \mathbf{e}_{\pi(1)},\mathbf{e}_{\pi(2)}, ...., \mathbf{e}_{\pi(K)}] \}$,where $\mathbf{e}_{i}$ is a column vector, with 1 in the i-th position and 0 in every other positions. Define the sign-filp matrix space  $\mathcal{S} =\{ \mathbf{S} \in \mathbb{R}^{K \times K}: \mathbf{S}= diag(a_1,a_2,...,a_K), a_i \in \{-1,1\} \}$. Then, the equivalent class of $\mathbf{\Lambda}^*$ can be defined as as $\mathcal{H}^*=\{ \mathbf{\Lambda}: \mathbf{\Lambda}=\mathbf{\Lambda}^*\mathbf{PS}, \mathbf{P} \in \mathcal{P}, \mathbf{S} \in \mathcal{S}\}$. If the solution set of this minimisation problem only contains those equivalent matrices of $\mathbf{\Lambda}^*$, i.e, $\mathcal{H}^*=\operatornamewithlimits{arg\,min}_{\mathbf{A}^*\mathbf{T'^{-1}} ,\mathbf{T}\in \mathcal{M}  }Q(\mathbf{A}^*\mathbf{T'^{-1}})$. Then ,when the samples size N goes to infinity, we can estimate $\mathbf{\Lambda^*}$ consistently up to some sign flips and column swapping.

Let $||\cdot||_{max}$ denotes the matrix norm induced by vector 1-norm, $ ||\mathbf{A}||_{max}=\max_{i,j} |A_{ij}|$. Let $||\cdot||_{2}$ denotes matrix norm induced by vector 2-norm, $ ||\mathbf{A}||_2=\max_{||x_2||=1} ||\mathbf{A} x||_2= \sqrt{\max eig(\mathbf{A'A})}=d_{max}(\mathbf{A})$.  We introduce the following notations:
\begin{enumerate}
    \item $\mathbf{\Lambda^*}=(\lambda^*_{ij})_{J \times K}$ denotes the true loading matrix, which is an oblique solution with sparse structure.
\item  $\mathbf{A^*}=(a^*_{ij})_{J \times K}=\mathbf{\Lambda^*}(\mathbf{\Phi^*}^{1/2})'$ , which is an orthogonal rotation that produces the same distribution as $\mathbf{\Lambda^*}$.
    \item $\mathbf{\hat{A}}_N=( (\hat{a}_N)_{ij})_{J \times K}$  denotes the MLE of $n$ samples solving $l(\mathbf{\theta})$ defined in \eqref{eq:7} over parameter space $\mathbf{\Theta}$.
    \item  $g:\mathbb{R}^{J \times K} \times \mathcal{M}\rightarrow \mathbb{R}, g(\mathbf{A},\mathbf{T})= Q(\mathbf{AT'^{-1}})=\sum_i\sum_j h(|(\mathbf{A}\mathbf{T'}^{-1})_{ij}|)$ is a general concave CLF.
     \item$\mathcal{T}^*=\operatornamewithlimits{arg\,min}_\mathbf{T\in \mathcal{M}} g(\mathbf{A}^*,\mathbf{T})$, which denotes the true rotation matrix set and can contain multiple minimum points.
     \item $\hat{\mathbf{T}}_N \in \operatornamewithlimits{arg\,min}_\mathbf{T\in \mathcal{M}} g(\mathbf{\hat{A}}_N,\mathbf{T})$ is the estimated rotation matrix.
\item $ \mathbf{\hat{\Lambda}}_N = \mathbf{\hat{A}}_N\mathbf{\hat{T}'}_N^{-1} $ is the estimated loading matrix.
       \item $\mathcal{H}^*=\{ \mathbf{\Lambda}: \mathbf{\Lambda}=\mathbf{\Lambda}^*\mathbf{PS},\mathbf{P} \in \mathcal{P}, \mathbf{S} \in \mathcal{S}\}$ be the set of equivalent matrices to the true (sparse) loading matrix.
\end{enumerate}
Note that we use $\mathbf{A^*}$ to denote a oblique rotated version of $\mathbf{\Lambda^*}$ which is needless to be sparse.  $\mathbf{A}^*$ can be any of the square root of $\mathbf{\Lambda^*\Phi ^*}\mathbf{\Lambda^* }'$, where they only differ by an orthogonal rotation, which can be proved to not effect the rotation solution.
We can prove by the following two lemmas that the starting orthogonal solution for loading matrix can only differ by a orthogonal matrix and that doesn't affect us to find $\bf \Lambda^*$ and $\mathbf{\hat{\Lambda}}_N$.
\begin{lemma}
For $J \times J$ matrix $\bf \Sigma$, with $ rank(\mathbf \Sigma)=K$, If there exists $J \times K $ matrix, $\bf A_1,\bf A_2$, satisfies $\bf A_1 \bf A_1'=\bf \Sigma=\bf A_2\bf A_2'$.Then there is an orthogonal matrix $\mbox{$\mathbf O$}, (\mbox{$\mathbf O$} \mbox{$\mathbf O$}'=I), s.t. \bf A_1= \bf A_2 \mbox{$\mathbf O$}$, \label{lemma:2}
\end{lemma}
\begin{lemma}
If  $J \times K $ matrix, $\bf A_1,\bf A_2$, $s.t. \bf A_2= \bf A_1 \mbox{$\mathbf O$}$, for some orthogonal transformation matrix $\bf \mbox{$\mathbf O$}$. Then,  $\operatornamewithlimits{arg\,min}_{\mathbf {A_1 T'^{-1} , T} \in \mathcal{M}} g(\mathbf A_1,\mathbf T)$= $\operatornamewithlimits{arg\,min}_{\mathbf {A_2 T'^{-1} , T} \in \mathcal{M}} g(\mathbf A_2,\mathbf T)$\label{lemma:1}
\end{lemma}

In addition, we require the following conditions to hold:
\begin{enumerate}[\bf C1]
\item   $\mathbf{\hat{A}}_N  \mathbf{\hat{A}}_N' \overset{p}{\to} \mathbf{\Lambda^*\Phi ^*}\mathbf{\Lambda^* }'$
\item  $\mathcal{H}^*=\operatornamewithlimits{arg\,min}_{\mathbf{A^*T'^{-1}} ,\mathbf{T}\in \mathcal{M}  }Q(\mathbf{A^*T'^{-1}})=\{\mathbf{A^*T'^{-1}} :  \mathbf{T} \in \mathcal{T}^*\}$
\item  The k-th largest singular value of $\mathbf{A^*}$ is positive, i.e, $d_{K}(\mathbf{A^*})>0$ \label{cond:3} \yc{This is not sufficient! See my condition C2.}
\item  $h(0)=0$, $h$ is concave, non-decreasing and $h$ is continuous \yc{Not needed, as we prove for the $L^p$}
\item  when $x \rightarrow +\infty, h(x) \rightarrow +\infty$ \yc{Not needed, as we prove for the $L^p$}
\end{enumerate}
Note that $\bf C1$ is a mild condition that generally holds for a MLE estimator.
$\bf C4$  is the condition in Jennrich's paper required for component loss function in order to reproduce perfect simple structure. $\bf C2$ is proved to be held in Jennrichs paper when $\bf\Lambda^*$ is of perfect simple structure and $h$ is strictly concave, which suits for all $L^p$ CLF when $0<p<1$. $\bf C3 $ is also a mild condition which holds when $\bf\Lambda^*$ is of full column rank, i.e, we do not assume unnecessary common factors. Any matrix satisfies the two-indicator rule will satisfy this condition.  $\bf C5$ holds for most concave CLF, including all $L^p$ CLF when $0<p\leq 1$, it does not hold when $ h(x)=1-e^{-|x|}$.

\begin{theorem}
Under condition $\mathbf{C 1-5}$, $\mathbf{\hat{T}}_N \overset{p}{\to} \mathcal{T}^*$ and  $ \mathbf{\hat{\Lambda}}_N = \mathbf{\hat{A}}_N\mathbf{\hat{T}'}_N^{-1}  \overset{p}{\to}  \mathcal{H}^*, N \rightarrow \infty$
\end{theorem}

\section{Statistical Inference}

\subsection{Post-selection confidence intervals. }

\textcolor{red}{Point out that due to the non-smoothness of such objective functions, one cannot construct confidence intervals for the loadings using the delta method. We propose a new method here, for the post-selection inference. }
 Due to the non-smoothness of such objective functions, one cannot construct confidence intervals for the loadings using the delta method which requires the information of gradient. We propose a new method here, for the post-selection inference. This method requires us to calculate the confidence interval row by row using the information of the active set of the BIC selected model. The detailed implementation works as follows:
 \begin{enumerate}

\item Let $ \hat{\bf U}$ denotes the active set of $\hat{\mathbf{\Lambda}}_{R,BIC} $ , where the latter is obtained as in \eqref{eq:bic} by the model selection procedure. So we have $\hat{u}_{ij}= \mathbbm{1}{ \{ \hat{(\mathbf{\lambda}}_{R,BIC} )_{ij}\neq 0\}}$ and we use the notation of $\hat{\bf u}_i$ for the i-th row of $ \hat{\bf U}$.
\item For each row $m, m=1,2,...,p$, we construct the Q-matrix for confirmatory factor analysis $\mathbf{U}^{(m)}$ as follows, \[
   \mathbf{u}^{(m)}_{i}=
\begin{cases}
   \hat{\bf u}_i ,& \text{if } i \neq m\\
    [1,1,..,1]_{1 \times K},              & \text{if } i=m
\end{cases}
\]where $ \mathbf{u}^{(m)}_{i}$ denotes the i-th of $\mathbf{U}^{(m)}$. Then we use this Q-matrix to fit a confirmatory factor analysis model, and use the estimated confidence interval of this cfa model for the m-th row of the true loading parameters.
\end{enumerate}

\subsection{Asymptotic Theory}
\textcolor{red}{Give conditions and a theorem that your confidence interval is asymptotically valid. }

\section{Computation }
\textcolor{red}{Give the IRLS for a general Lp objective function.
  Do not give the details for the proximal algorithm for L1. You only mention that the complexity of the state-of-the-art proximal algorithm is … and compare it with the complexity of the IRLS. And explain the reason why the IRLS algorithm is computationally less intensive (due to the two-step estimation).}
In order to use $L^p$ CLC to find a sparse loading matrix, the following optimization problem needs to be solved:
$$
\min_\mathbf{T\in \mathcal{M}} Q(\hat{\mathbf{A}}_{N}\mathbf{T'}^{-1}),
$$
where
$
Q(\mathbf{\Lambda})
$ is defined in \eqref{eq:CLC_Lp}
and
$
\mathcal{M}
$ is defined in \eqref{eq:13}.
However,  due to the non-smoothness of the objective function $Q$, there is no suitable method for solving the optimisation problem.  In particular, in Jennrich's paper, a smooth approximation to the objective function is minimized where he suggests to approximate $|\lambda|$ by a parabola on a small interval $[-\epsilon,\epsilon]$, instead of the original problem. Therefore, the problem is not solved exactly. In this paper, we propose an algorithm based on iterated reweighted least squares (IRLS) and gradient projection (GP) to solve the issue of nonsmoothness. The main idea of this algorithm is to solve a weighted least squares problem which is smooth in order to gradually approximate the true objective function. In each step $t=1,2,3,...$, the function $Q(\mathbf{\Lambda})$ is approximated by a differentiable function
$$
Q^{(t)}(\mathbf{\Lambda})=\sum_i\sum_j w_{ij}^{(t)} \lambda_{ij}^2,
$$
where
$
w_{ij}^{(t)}=|\lambda_{ij}^{(t-1)}    |^{p-2},
$
such that
$
Q^{(t)}(\mathbf{\Lambda}^{(t-1)}) = Q(\mathbf{\Lambda}^{(t-1)}).
$
Since $Q^{(t)}$ needs to be minimized over the space $\mathcal{A} =   \big\{\mathbf{\Lambda}_0 \mathbf{T'}^{-1} : \mathbf{T} \in \mathcal{M}\}$ , a gradient projection algorithm is used to project $\mathbf{T}$ back to the space $\mathcal{M}$ after each gradient descent step. Since the goal is to decrease $Q^{(t)}$ rather than minimize it (since $Q^{(t)}$ is different from the objective function $Q$), the maximum number of iterations can be restricted to increase the computational speed. Both the IRLS algorithm and the GP algorithm are given below.
The rotation solution is generally solved faster than LASSO solution in that the rotation criterion is only a function of $\mathbf{\Lambda}$ and updates less number of parameters than LASSO. As a comparison, the computational complexities per iteration for our method and the state-of-the-art proximal algorithm for LASSO are listed below,
\begin{table}[h]
\centering
\resizebox{0.7\textwidth}{!}{%
\begin{tabular}{lll}
 \hline
      &Method &Computational Complexity per Iteration \\
       \hline
IRLS     &IRLS-GP            & o($K^3+K^2J$)                          \\
LASSO   & proximal   gradient descent     & o($J^3+J^2K+K^2J$)                     \\  \hline
\end{tabular}
}
\end{table}

 \begin{algorithm}
\caption{IRLS Algorithm for linear CLF oblique rotation}\label{alg:cap}
\textbf{Input} Set $\hat{\mathbf{A}}_{N}=( (\hat{a}_N)_{ij})_{J \times K} $ to be an orthogonal MLE as defined in \eqref{eq:MLE},
$t \gets 1$, maximum iteration $t_{max}\gets 10000, w_{ij}^{(0)}\gets {(|\hat{a}_N)_{ij} |+mean(|\hat{a}_N)_{ij} |) * \epsilon_t)}^{p-2}$ , $\mathbf{T}^{(0)} \gets\mathbf{I}_{K \times K} $ ,
\Repeat
\State $\mathbf{T}^{(t)}= \operatornamewithlimits{arg\,min}_{\mathbf{T} \in \mathcal{M}} \sum_i\sum_j w^{(t)}_{ij}(\hat{\mathbf{A}}_{N}  \mathbf{T'}^{-1})_{ij}^2 \coloneqq Q^{(t)}(\mathbf{\Lambda}(\mathbf{T}))$. $\mathbf{T}^{(t)}$ is solved by gradient projection algorithm.
\State $\mathbf{\Lambda^{(t)}} \gets \hat{\mathbf{A}}_{N}{\mathbf{T^{(t)}}'}^{-1}$
\State $ w^{(t+1)}_{ij} \gets {\max(|\lambda^{(t)}_{ij} |, \epsilon_t)}^{p-2}$where $\lambda_{ij}^{(t)}$ is the $(i,j)$ th entry of $\mathbf{\Lambda^{(t)}}$.
\State
$ \epsilon_{t+1}  \gets 0.001$
\Comment{$\epsilon_t$ is used to maintain numerical stability for very small $\lambda_{ij}$}
\Until {\textcolor{red}{stopping criterion is met}}
\end{algorithm}

\begin{algorithm}
\caption{Gradient Projection algorithm for solving $\mathbf{T}^{(t)}$}\label{alg:cap}
\textbf{Input}{ $k \gets 1$, maximum iteration $k_{max} \gets 5, \mathbf{T_0} \gets \mathbf{T}^{(t-1)}$, $w_{ij} \gets w_{ij}^{(t)}$, fitted value $ft \gets Q^{(t)}(\mathbf{\Lambda}(\mathbf{T}_0))$, step size $\alpha \gets 1$
\For {$k = 1$ to $k_{max}$};
\State $ \mathbf{\Lambda}_{k-1} \gets \hat{\mathbf{A}}_{N}\mathbf{T_{k-1}'}^{-1}$
\State $\bf G_k \coloneqq \frac{\partial Q^{(t)}(\mathbf{\Lambda}(\mathbf{T}))}{ \partial\mathbf{T}} (\mathbf{T}_{k-1}) \gets -({\mathbf{\Lambda}_{k-1}}' (2\mathbf{W} \cdot \mathbf{\Lambda}_{k-1} )\mathbf{T_{k-1}}^{-1})'$; \Comment{$\cdot$ means elementwise matrix multiplication}
\State $\alpha\gets 2 \alpha$; \Comment{ avoid to have too small $\alpha $ and stuck at local minimum}
\For {m = 0 to 20};  \Comment{line search of $\alpha$}

\State $\mathbf{T_k}\gets \textbf{proj}_\mathcal{M} (\mathbf{T_{k-1}}–  \alpha \bf G_k)$ \Comment{ $\textbf{proj}_\mathcal{M}(\mathbf{T}) = \mathbf{T}* diag(\mathbf{T}' \mathbf{T})^{-1/2} $.}
\If {$ Q^{(t)}(\mathbf{\Lambda}(\mathbf{T}_k)) <ft$}
\then $ft \gets Q^{(t)}(\mathbf{\Lambda}(\mathbf{T}_k))$,
 break ;
\Else $ \alpha\gets \alpha/2$;
\EndIf;
\EndFor
\EndFor;
\text{Output} $\mathbf{T}_{k_{max}}$
\end{algorithm}

\section{Simulation and Real Data in the outline. }
\textcolor{red}{Please add that to the paper. @Liu114,X (pgr) Please run some simulations with p=0.5, to see the performance (and compare with the case when p=1). Note that you will need to modify the IRLS algorithm (which should be quite straightforward).}

\subsection{Simulation sett-up}
 \subsubsection{Simulation}

 In order to calculate the confidence interval, we need the loading matrix to be identifiable when calculating the confidence interval for each item. That means, our matrix need to satisfy the 2-indicator rule for multi-latent variables when deleting any of its row. We therefore consider true sparse loading matrices, with 3 indicators for each latent variable. Two settings where the size of the true loading matrix differs are considered to compare the performance of the rotation-based estimator and the penalization-based estimator. In each setting, we generate $B = 500$ data sets, each containing $N =400$ or $800$ samples generated using the same true parameters $\theta^*$.

In the smaller setting, we consider $J=15$ items and $K=3$ latent variables, each latent variable have 2 indicators and the last 6 items load on 2 latent variables.

In the larger setting, we consider $J=30$ items and $K=5$ latent variables.  The first 15 items loads only on 1 latent variable with 3 indicators for each latent variable, and the following 10 items load on 2 latent variables and the last 5 items load on 3 latent variables with as many combinations of latent variables as possible.The parameters are listed as follows. The diagonal entries of each square sub-matrix are generated uniformly from 0.5 to 1, and the remained cross-loadings are generated uniformly from 0.2 to 0.5. The Item Unique Variance for Item 1-15 is the same in both settings.
\begin{comment}
Since the accuracy of hard-thresholding procedure also depends on the scale of cross-loading, i.e those entries that is very small may not be selected by the hard-thresholding procedure. We also consider the scale of cross-loading. In the small cross-loading set, each matrix has 5 entries that is smaller than 0.3. For the large cross loading setting, we will modify every none-zero entry that is smaller than 1 by making it plus 1. We believe in the large cross loading setting, we will have almost perfect variable selection accuracy, so the post-selection confidence interval coverage will near 95\%.

\begin{figure}[h]
\includegraphics[width=\textwidth]{Parameters.png}
\end{figure}
\end{comment}

\subsubsection{Case Study}
The Big 5 personality test data set is a classical psychometric application containing the responses to p=50 items from an interactive online personality test and very large sample size. In order to get a smaller sample size with a relatively more homogeneous group, we choose to analyse the sub-population of British male and that results in $n=609$ samples. The items were rated on a five point scale where 1=Disagree, 3=Neutral, 5=Agree. In this analysis, we treat them as continuous and run factor analysis. We do the point estimation and the confidence intervals of the loading matrix. All parameters in IRLS and post-selection procedure are the same as the section above.

\subsection{Methods compared and evaluation criteria}
In this study, we compare the following 2 methods:
\begin{enumerate}
\end{enumerate}
\begin{enumerate}
\item Oblique rotation with linear CLF rotation criteria with $p=1$ and $p=0.5$. We use IRLS-GP algorithm to solve it. The maximum iterations for IRLS algorithms(outer loop) is set to 10000 and for gradient projection algorithms (inner loop) is set to 5 since we only expect gradient projection algorithm to decrease the objective function rather than find the exact solution for the quadratic approximation function.
\begin{comment}
\item Lasso path solution. We use proximal gradient descent algorithm to solve it where the maximum iteration is set to 4000. The tuning parameter increases from 0 to 2. To ensure $\mathbf{\Omega}$ to have positive diagonal entries, we re-parametrize it with a vector $\mathbf{\Psi} \in \mathbb{R}^K$, where the i-th diagonal entry of $\bf \Omega$ is expressed as $exp(\psi_i)$ . In order to avoid  $\mathbf{\Phi}$ to collapse, (i.e, the correlation coefficient of two latent variables to be 1), we use the true value for  $\mathbf{\Phi}$ rather than optimize it. This certainly give our Lasso an advantage in computation time and accuracy.
\end{comment}
\item Lasso solution with tuning parameter set to 0.01,0.05,0.1,0.2,0.5. We use the proximal gradient descent algorithm to solve it except we keep updating $\mathbf{\Phi}$ since it won't collapse when $\lambda$ is small. In order to ensure $\mathbf{\Phi}$ to be symmetric and positive definite, we re-parametrize $\mathbf{\Phi}$ as $\mathbf{T' T}$, where $\mathbf{T}$ is an upper-triangular matrix with unit columns. To ensure $\mathbf{\Omega}$ to have positive diagonal entries, we re-parametrize it with a vector $\mathbf{\Psi} \in \mathbb{R}^K$, where the i-th diagonal entry of $\bf \Omega$ is expressed as $exp(\psi_i)$ .
\end{enumerate}

The stopping criteria for all algorithms is the maximum elementwise error to be smaller than $10^{-5}$ between two IRLS iterations or Proximal gradient descent steps.
$$ \max|\lambda^{(t+1)}_{ij}-\lambda^{(t)}_{ij}| \leq 10^{-5}$$

\begin{comment}
\subsection{results}
\textcolor{red}{Design a study to compare LASSO and our method. Simulation settings? Evaluation criteria? Think of the design and we discuss in our next meeting.

I think that for a relatively fairer comparison, you should use the same starting point (i.e., the one from CFA)  for IRLS and the LASSO with the smallest tuning parameter value.

In the comparison, you may also include LASSO with a very small tuning parameter value. You want to show that this solution behaves very similarly to the solution from the IRLS, because we know that the solution of IRLS is the limiting case of the LASSO when the tuning parameter goes to zero. You may also want to compare the computation speed. It might be possible that the LASSO solution can be obtained in a shorter time, while the accuracy is similar. This is still good, as our key message is that computing a solution path seems not very useful here.
}

\end{comment}

\subsection{Data Generating process and model selection process}
We then calculate $ \bf \Sigma^*= \bf{\Lambda^* \Phi^* {\Lambda^*}' +\Omega^*}$, and we do a Cholesky decomposition to get $(\bf\Sigma^*)^{1/2}$. We then generate $N=500$ i.i.d samples $z_i \sim \mathcal{N}(0,\mathbf I_K)$, and let $x_i=\mathbf {(\Sigma^*)^{1/2}} z_i$. Since the only information needed to calculate log-likelihood is the sample covariance and the number of samples, we then calculate $\boldsymbol{S} = (\sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^\top)/N$ and pass that to the cfa function to solve the initial point of the rotation problem. Since the optimisation problem of LASSO is nonconvex which has many local minimums, we use a warm start for all methods. The starting point is set to be the varimax rotation of the lower triangular loading matrix. It is solved by the cfa function in the lavaan R package, where the lower triangular loading matrix and the assumption that $\mathbf{\Phi}=\mathbf{I}$ is used to ensure its identifiablity.

After that, we will do the hard-thresholding and BIC selection procedure according to section 3.1 for model selection, where the hard-thresholding parameter is increased from 0.02 to 0.4 by 0.02. Then, we calculate the post-selection confidence intervals by the procedure in section 4.1. If the confidence interval can not be calculated due to identifiablity issue, we will use $-\infty$ as the lower bound, and $\infty$ as upper bound for the 95\% confidence interval.

\subsubsection{Results}
In order to compare the performance of CLF with $p=1$ and $p=0.5$ and lasso, we compare their mean square error(mse), AUC and overall selection accuracy and element-wise 95\% confidence interval coverage rate.

\begin{comment}
\begin{table}[h]
\begin{tabular}{l|lll|lll}
                     & \multicolumn{3}{c|}{p=10, k=3}    & \multicolumn{3}{c}{p=30, k=5}     \\ \cline{2-7}
                     & average\_iter & time /min & AUC   & average\_iter & time /min & AUC   \\ \hline
IRLS                 & 4473.9        & 3.477     & 0.956 & 10272.2       & 9.11717   & 0.96  \\
LASSO                & 1209          & 1.789     & 0.891 & 2727.8        & 17.4773   & 0.904 \\
LASSO\_ap(0.01,0.02) & 390.2         & 0.569833  & 0.948 & 1065.5        & 5.0685    & 0.961
\end{tabular}
\end{table}
\end{comment}

For big 5 data set, here is the BIC selected model by using $L^{0.5}$ and $L^1$ rotation criteria, and the hypothesis testing results for testing zero entries.
\begin{comment}
\begin{figure}[h]
\includegraphics[width=\textwidth]{big5_res1.png}
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{big5_res2.png}
\end{figure}
\end{comment}

The estimated confidence interval are shown below in Table \ref{tab:ci1} and Table \ref{tab:ci0.5}.

{\footnotesize
\begin{longtable}[c]{llllll}
\caption{Point estimate and confidence interval constructed by $L^{0.5}$ }
\label{tab:my-table}\\

\hline
\multicolumn{6}{c}{p=0.5}                                                                                \\
\endfirsthead
\endhead
\hline
\endfoot
\endlastfoot
                            & E                     & N                     & A                     & C                     & O                     \\ \hline
\multicolumn{1}{l|}{E1$(+)$}  & 0.882( 0.793, 0.980)  & -0.055(-0.161, 0.003) & -0.068(-0.183,-0.017) & 0.015(-0.078, 0.100)  & 0.087( 0.005, 0.170)  \\
\multicolumn{1}{l|}{E2(-)}  & -0.845(-0.962,-0.760) & 0.117(-0.228,-0.048)  & 0.002(-0.126, 0.057)  & 0.031(-0.061, 0.132)  & -0.020(-0.079, 0.104) \\
\multicolumn{1}{l|}{E3$(+)$}  & 0.786( 0.702, 0.876)  & 0.286(-0.347,-0.192)  & 0.203( 0.120, 0.275)  & 0.123( 0.050, 0.213)  & -0.088(-0.175,-0.020) \\
\multicolumn{1}{l|}{E4(-)}  & -0.913(-1.019,-0.842) & -0.072(-0.011, 0.140) & -0.022(-0.075, 0.077) & 0.000(-0.099, 0.068)  & 0.084(-0.184,-0.033)  \\
\multicolumn{1}{l|}{E5$(+)$}  & 0.895( 0.812, 0.987)  & -0.012(-0.032, 0.118) & 0.155( 0.059, 0.209)  & 0.104( 0.023, 0.183)  & 0.085( 0.021, 0.172)  \\
\multicolumn{1}{l|}{E6(-)}  & -0.751(-0.849,-0.660) & -0.011(-0.103, 0.069) & -0.088(-0.163, 0.010) & -0.062(-0.161, 0.022) & -0.141( 0.052, 0.226) \\
\multicolumn{1}{l|}{E7$(+)$}  & 1.114( 1.020, 1.221)  & -0.062( 0.018, 0.189) & 0.083( 0.001, 0.172)  & 0.105( 0.020, 0.199)  & -0.015(-0.052, 0.117) \\
\multicolumn{1}{l|}{E8(-)}  & -0.722(-0.823,-0.630) & -0.099( 0.000, 0.176) & 0.028(-0.038, 0.140)  & 0.118( 0.024, 0.210)  & -0.069(-0.022, 0.157) \\
\multicolumn{1}{l|}{E9$(+)$}  & 0.858( 0.748, 0.953)  & 0.064(-0.131, 0.054)  & 0.000(-0.111, 0.075)  & -0.013(-0.110, 0.086) & 0.246(-0.332,-0.145)  \\
\multicolumn{1}{l|}{E10(-)} & -0.829(-0.930,-0.741) & -0.128( 0.025, 0.193) & -0.048(-0.123, 0.047) & -0.121(-0.215,-0.036) & 0.000(-0.092, 0.078)  \\
\multicolumn{1}{l|}{N1(-)}  & -0.152(-0.210,-0.028) & -0.975( 0.875, 1.073) & 0.008(-0.121, 0.078)  & 0.081( 0.016, 0.200)  & -0.104( 0.017, 0.193) \\
\multicolumn{1}{l|}{N2$(+)$}  & 0.161( 0.039, 0.215)  & 0.675(-0.771,-0.589)  & 0.000(-0.064, 0.115)  & -0.084(-0.197,-0.020) & 0.085(-0.185,-0.012)  \\
\multicolumn{1}{l|}{N3(-)}  & -0.203(-0.271,-0.093) & -0.786( 0.696, 0.881) & 0.232( 0.124, 0.304)  & 0.122( 0.031, 0.208)  & 0.042(-0.139, 0.033)  \\
\multicolumn{1}{l|}{N4$(+)$}  & 0.237( 0.115, 0.310)  & 0.568(-0.670,-0.474)  & 0.001(-0.069, 0.125)  & 0.066(-0.049, 0.148)  & 0.011(-0.113, 0.079)  \\
\multicolumn{1}{l|}{N5(-)}  & 0.004(-0.073, 0.138)  & -0.471( 0.357, 0.567) & -0.041(-0.163, 0.046) & -0.110(-0.208, 0.006) & -0.225( 0.135, 0.345) \\
\multicolumn{1}{l|}{N6(-)}  & -0.147(-0.202,-0.022) & -0.808( 0.717, 0.905) & 0.259( 0.145, 0.327)  & -0.025(-0.106, 0.074) & -0.132( 0.053, 0.227) \\
\multicolumn{1}{l|}{N7(-)}  & 0.000(-0.048, 0.123)  & -0.962( 0.883, 1.064) & -0.111(-0.226,-0.055) & -0.044(-0.102, 0.064) & 0.004(-0.071, 0.089)  \\
\multicolumn{1}{l|}{N8(-)}  & -0.026(-0.080, 0.106) & -1.130( 1.046, 1.240) & -0.133(-0.261,-0.080) & -0.076(-0.140, 0.036) & 0.000(-0.065, 0.103)  \\
\multicolumn{1}{l|}{N9(-)}  & -0.051(-0.131, 0.048) & -0.866( 0.768, 0.956) & -0.292(-0.399,-0.214) & 0.171( 0.092, 0.267)  & -0.018(-0.068, 0.104) \\
\multicolumn{1}{l|}{N10(-)} & -0.364(-0.435,-0.256) & -0.842( 0.752, 0.940) & 0.070(-0.029, 0.155)  & -0.109(-0.181,-0.004) & 0.096(-0.187,-0.016)  \\
\multicolumn{1}{l|}{A1(-)}  & 0.000(-0.112, 0.088)  & -0.127( 0.013, 0.205) & -0.779(-0.876,-0.670) & 0.030(-0.071, 0.127)  & 0.044(-0.134, 0.060)  \\
\multicolumn{1}{l|}{A2$(+)$}  & 0.437( 0.359, 0.523)  & 0.000(-0.053, 0.099)  & 0.557( 0.464, 0.626)  & -0.043(-0.124, 0.034) & 0.046(-0.128, 0.026)  \\
\multicolumn{1}{l|}{A3(-)}  & 0.178( 0.080, 0.286)  & -0.571( 0.467, 0.674) & -0.565(-0.692,-0.480) & -0.077(-0.177, 0.030) & 0.138(-0.231,-0.029)  \\
\multicolumn{1}{l|}{A4$(+)$}  & 0.013(-0.034, 0.142)  & 0.000(-0.039, 0.100)  & 0.980( 0.895, 1.050)  & -0.015(-0.098, 0.032) & 0.000(-0.071, 0.060)  \\
\multicolumn{1}{l|}{A5(-)}  & -0.156(-0.248,-0.088) & -0.041(-0.047, 0.105) & -0.815(-0.894,-0.725) & 0.000(-0.078, 0.077)  & 0.086(-0.163,-0.011)  \\
\multicolumn{1}{l|}{A6$(+)$}  & -0.063(-0.157, 0.020) & -0.184( 0.102, 0.270) & 0.717( 0.630, 0.812)  & 0.007(-0.100, 0.073)  & 0.014(-0.096, 0.073)  \\
\multicolumn{1}{l|}{A7(-)}  & -0.367(-0.452,-0.297) & -0.095( 0.015, 0.161) & -0.733(-0.800,-0.640) & 0.057(-0.016, 0.133)  & 0.026(-0.098, 0.047)  \\
\multicolumn{1}{l|}{A8$(+)$}  & 0.110( 0.039, 0.185)  & -0.038(-0.011, 0.128) & 0.692( 0.616, 0.770)  & 0.079( 0.007, 0.149)  & 0.028(-0.106, 0.034)  \\
\multicolumn{1}{l|}{A9$(+)$}  & 0.119( 0.040, 0.199)  & -0.111( 0.055, 0.206) & 0.752( 0.667, 0.835)  & 0.064(-0.015, 0.140)  & 0.114(-0.194,-0.042)  \\
\multicolumn{1}{l|}{A10$(+)$} & 0.440( 0.352, 0.515)  & 0.075(-0.145, 0.008)  & 0.321( 0.244, 0.401)  & 0.125( 0.033, 0.193)  & 0.056(-0.136, 0.019)  \\
\multicolumn{1}{l|}{C1$(+)$}  & 0.111( 0.002, 0.180)  & 0.101(-0.179, 0.003)  & -0.037(-0.108, 0.079) & 0.686( 0.583, 0.771)  & 0.131(-0.244,-0.064)  \\
\multicolumn{1}{l|}{C2(-)}  & 0.070(-0.015, 0.191)  & -0.190( 0.078, 0.285) & 0.108(-0.020, 0.185)  & -0.658(-0.793,-0.581) & 0.145(-0.215,-0.012)  \\
\multicolumn{1}{l|}{C3$(+)$}  & 0.024(-0.078, 0.083)  & 0.000(-0.066, 0.094)  & 0.114( 0.046, 0.206)  & 0.401( 0.300, 0.465)  & 0.280(-0.379,-0.217)  \\
\multicolumn{1}{l|}{C4(-)}  & -0.140(-0.201,-0.036) & -0.607( 0.504, 0.677) & 0.049(-0.052, 0.116)  & -0.496(-0.576,-0.407) & -0.041( 0.006, 0.170) \\
\multicolumn{1}{l|}{C5$(+)$}  & 0.081( 0.001, 0.192)  & 0.046(-0.161, 0.047)  & 0.000(-0.053, 0.151)  & 0.779( 0.691, 0.887)  & -0.051(-0.066, 0.126) \\
\multicolumn{1}{l|}{C6(-)}  & 0.011(-0.088, 0.124)  & -0.184( 0.062, 0.277) & 0.046(-0.078, 0.139)  & -0.705(-0.841,-0.621) & 0.089(-0.188, 0.024)  \\
\multicolumn{1}{l|}{C7$(+)$}  & -0.129(-0.221,-0.044) & -0.139( 0.057, 0.235) & 0.110( 0.035, 0.214)  & 0.528( 0.434, 0.618)  & 0.038(-0.157, 0.021)  \\
\multicolumn{1}{l|}{C8(-)}  & -0.010(-0.087, 0.094) & -0.275( 0.173, 0.355) & -0.241(-0.356,-0.174) & -0.524(-0.608,-0.421) & 0.000(-0.059, 0.122)  \\
\multicolumn{1}{l|}{C9$(+)$}  & 0.036(-0.058, 0.130)  & -0.014(-0.067, 0.134) & 0.123( 0.052, 0.241)  & 0.723( 0.626, 0.821)  & -0.076(-0.040, 0.149) \\
\multicolumn{1}{l|}{C10$(+)$} & 0.000(-0.107, 0.059)  & -0.016(-0.051, 0.117) & 0.128( 0.065, 0.231)  & 0.524( 0.416, 0.587)  & 0.234(-0.337,-0.172)  \\
\multicolumn{1}{l|}{O1$(+)$}  & 0.079(-0.021, 0.145)  & 0.000(-0.057, 0.109)  & -0.045(-0.146, 0.014) & 0.000(-0.108, 0.056)  & 0.623(-0.711,-0.544)  \\
\multicolumn{1}{l|}{O2(-)}  & 0.000(-0.051, 0.121)  & -0.220( 0.114, 0.283) & -0.087(-0.183,-0.015) & 0.009(-0.055, 0.117)  & -0.581( 0.502, 0.675) \\
\multicolumn{1}{l|}{O3$(+)$}  & 0.067(-0.026, 0.135)  & -0.153( 0.092, 0.251) & 0.025(-0.061, 0.096)  & 0.002(-0.098, 0.063)  & 0.589(-0.665,-0.502)  \\
\multicolumn{1}{l|}{O4(-)}  & 0.024(-0.027, 0.144)  & -0.202( 0.098, 0.267) & -0.154(-0.230,-0.064) & 0.036(-0.022, 0.148)  & -0.571( 0.487, 0.660) \\
\multicolumn{1}{l|}{O5$(+)$}  & 0.237( 0.133, 0.271)  & 0.067(-0.137, 0.002)  & -0.058(-0.132, 0.002) & 0.176( 0.069, 0.205)  & 0.581(-0.651,-0.509)  \\
\multicolumn{1}{l|}{O6(-)}  & -0.211(-0.272,-0.102) & -0.003(-0.108, 0.060) & -0.047(-0.121, 0.044) & 0.028(-0.038, 0.133)  & -0.510( 0.434, 0.605) \\
\multicolumn{1}{l|}{O7$(+)$}  & 0.076(-0.017, 0.123)  & 0.166(-0.221,-0.084)  & -0.035(-0.100, 0.036) & 0.097( 0.012, 0.153)  & 0.450(-0.524,-0.382)  \\
\multicolumn{1}{l|}{O8$(+)$}  & -0.024(-0.163, 0.021) & -0.165( 0.090, 0.270) & -0.107(-0.197,-0.020) & 0.000(-0.123, 0.060)  & 0.656(-0.764,-0.577)  \\
\multicolumn{1}{l|}{O9$(+)$}  & -0.063(-0.153, 0.008) & -0.217( 0.151, 0.309) & 0.240( 0.156, 0.315)  & 0.107( 0.017, 0.179)  & 0.258(-0.343,-0.183)  \\
\multicolumn{1}{l|}{O10$(+)$} & 0.240( 0.132, 0.275)  & -0.004(-0.036, 0.113) & 0.000(-0.072, 0.068)  & 0.097( 0.012, 0.155)  & 0.685(-0.762,-0.613)  \\ \hline
\end{longtable}
\end{comment}

\begin{table}[h]
\caption{Point estimate and confidence interval constructed by $L^0.5$}
\label{tab:empCI1}
\begin{tabular}{lccccc}
\hline
 &
  E &
  N &
  A &
  C &
  O \\ \hline
\multicolumn{1}{l|}{E1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.882\\      (  0.793,  0.980)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.055\\      (-0.161,  0.003)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.068\\      (-0.183,-0.017)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.015\\      (-0.078,  0.100)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.087\\      (  0.005,  0.170)\end{tabular} \\
\multicolumn{1}{l|}{E2(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.845\\      (-0.962,-0.760)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.117\\      (-0.228,-0.048)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.126,  0.057)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.031\\      (-0.061,  0.132)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.020\\      (-0.079,  0.104)\end{tabular} \\
\multicolumn{1}{l|}{E3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.786\\      (  0.702,  0.876)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.286\\      (-0.347,-0.192)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.203\\      (  0.120,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123\\      (  0.050,  0.213)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088\\      (-0.175,-0.020)\end{tabular} \\
\multicolumn{1}{l|}{E4(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.913\\      (-1.019,-0.842)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.072\\      (-0.011,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.022\\      (-0.075,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.099,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.084\\      (-0.184,-0.033)\end{tabular} \\
\multicolumn{1}{l|}{E5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.895\\      (  0.812,  0.987)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.012\\      (-0.032,  0.118)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.155\\      (  0.059,  0.209)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.104\\      (  0.023,  0.183)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085\\      (  0.021,  0.172)\end{tabular} \\
\multicolumn{1}{l|}{E6(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.751\\      (-0.849,-0.660)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.011\\      (-0.103,  0.069)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.088\\      (-0.163,  0.010)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062\\      (-0.161,  0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.141\\      (  0.052,  0.226)\end{tabular} \\
\multicolumn{1}{l|}{E7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}1.114\\      (  1.020,  1.221)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.062\\      (  0.018,  0.189)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.083\\      (  0.001,  0.172)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.105\\      (  0.020,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.052,  0.117)\end{tabular} \\
\multicolumn{1}{l|}{E8(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.722\\      (-0.823,-0.630)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.099\\      (  0.000,  0.176)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.118\\      (  0.024,  0.210)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.069\\      (-0.022,  0.157)\end{tabular} \\
\multicolumn{1}{l|}{E9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.858\\      (  0.748,  0.953)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.131,  0.054)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.111,  0.075)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.013\\      (-0.110,  0.086)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.246\\      (-0.332,-0.145)\end{tabular} \\
\multicolumn{1}{l|}{E10(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.829\\      (-0.930,-0.741)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.128\\      (  0.025,  0.193)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.048\\      (-0.123,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.121\\      (-0.215,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.092,  0.078)\end{tabular} \\
\multicolumn{1}{l|}{N1(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.152\\      (-0.210,-0.028)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.975\\      (  0.875,  1.073)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.008\\      (-0.121,  0.078)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.081\\      (  0.016,  0.200)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.104\\      (  0.017,  0.193)\end{tabular} \\
\multicolumn{1}{l|}{N2$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.161\\      (  0.039,  0.215)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.675\\      (-0.771,-0.589)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.064,  0.115)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.084\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.085\\      (-0.185,-0.012)\end{tabular} \\
\multicolumn{1}{l|}{N3(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.203\\      (-0.271,-0.093)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.786\\      (  0.696,  0.881)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.232\\      (  0.124,  0.304)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.122\\      (  0.031,  0.208)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.042\\      (-0.139,  0.033)\end{tabular} \\
\multicolumn{1}{l|}{N4$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.237\\      (  0.115,  0.310)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.568\\      (-0.670,-0.474)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.001\\      (-0.069,  0.125)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.066\\      (-0.049,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.113,  0.079)\end{tabular} \\
\multicolumn{1}{l|}{N5(-)} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.073,  0.138)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.471\\      (  0.357,  0.567)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.163,  0.046)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.110\\      (-0.208,  0.006)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.225\\      (  0.135,  0.345)\end{tabular} \\
\multicolumn{1}{l|}{N6(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.147\\      (-0.202,-0.022)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.808\\      (  0.717,  0.905)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.259\\      (  0.145,  0.327)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.025\\      (-0.106,  0.074)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.132\\      (  0.053,  0.227)\end{tabular} \\
\multicolumn{1}{l|}{N7(-)} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.048,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.962\\      (  0.883,  1.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111\\      (-0.226,-0.055)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.044\\      (-0.102,  0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.004\\      (-0.071,  0.089)\end{tabular} \\
\multicolumn{1}{l|}{N8(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.026\\      (-0.080,  0.106)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-1.130\\      (  1.046,  1.240)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.133\\      (-0.261,-0.080)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.140,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.065,  0.103)\end{tabular} \\
\hline
\end{tabular}%

\end{table}

\begin{table}[h]
\caption{Point estimate and confidence interval constructed by $L^0.5$}
\label{tab:empCI1}

\begin{tabular}{lccccc}
\hline
 &
  E &
  N &
  A &
  C &
  O \\ \hline

\multicolumn{1}{l|}{N9(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.131,  0.048)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.866\\      (  0.768,  0.956)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.292\\      (-0.399,-0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.171\\      (  0.092,  0.267)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.018\\      (-0.068,  0.104)\end{tabular} \\
\multicolumn{1}{l|}{N10(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.364\\      (-0.435,-0.256)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.842\\      (  0.752,  0.940)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.029,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.109\\      (-0.181,-0.004)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.096\\      (-0.187,-0.016)\end{tabular} \\
\multicolumn{1}{l|}{A1(-)} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.112,  0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.127\\      (  0.013,  0.205)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.779\\      (-0.876,-0.670)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.030\\      (-0.071,  0.127)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.044\\      (-0.134,  0.060)\end{tabular} \\
\multicolumn{1}{l|}{A2$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.437\\      (  0.359,  0.523)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.053,  0.099)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.557\\      (  0.464,  0.626)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.043\\      (-0.124,  0.034)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.128,  0.026)\end{tabular} \\
\multicolumn{1}{l|}{A3(-)} &
  \begin{tabular}[c]{@{}c@{}}0.178\\      (  0.080,  0.286)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571\\      (  0.467,  0.674)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.565\\      (-0.692,-0.480)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.077\\      (-0.177,  0.030)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.138\\      (-0.231,-0.029)\end{tabular} \\
\multicolumn{1}{l|}{A4$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.013\\      (-0.034,  0.142)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.039,  0.100)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.980\\      (  0.895,  1.050)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.015\\      (-0.098,  0.032)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.071,  0.060)\end{tabular} \\
\multicolumn{1}{l|}{A5(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.156\\      (-0.248,-0.088)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (-0.047,  0.105)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.815\\      (-0.894,-0.725)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.078,  0.077)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.086\\      (-0.163,-0.011)\end{tabular} \\
\multicolumn{1}{l|}{A6$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.157,  0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184\\      (  0.102,  0.270)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.717\\      (  0.630,  0.812)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.007\\      (-0.100,  0.073)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.014\\      (-0.096,  0.073)\end{tabular} \\
\multicolumn{1}{l|}{A7(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.367\\      (-0.452,-0.297)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.095\\      (  0.015,  0.161)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.733\\      (-0.800,-0.640)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.057\\      (-0.016,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.026\\      (-0.098,  0.047)\end{tabular} \\
\multicolumn{1}{l|}{A8$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.110\\      (  0.039,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.038\\      (-0.011,  0.128)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.692\\      (  0.616,  0.770)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.079\\      (  0.007,  0.149)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.106,  0.034)\end{tabular} \\
\multicolumn{1}{l|}{A9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.119\\      (  0.040,  0.199)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.111\\      (  0.055,  0.206)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.752\\      (  0.667,  0.835)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.064\\      (-0.015,  0.140)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114\\      (-0.194,-0.042)\end{tabular} \\
\multicolumn{1}{l|}{A10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.440\\      (  0.352,  0.515)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.075\\      (-0.145,  0.008)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.321\\      (  0.244,  0.401)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.125\\      (  0.033,  0.193)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.056\\      (-0.136,  0.019)\end{tabular} \\
\multicolumn{1}{l|}{C1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.111\\      (  0.002,  0.180)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.101\\      (-0.179,  0.003)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.037\\      (-0.108,  0.079)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.686\\      (  0.583,  0.771)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.131\\      (-0.244,-0.064)\end{tabular} \\
\multicolumn{1}{l|}{C2(-)} &
  \begin{tabular}[c]{@{}c@{}}0.070\\      (-0.015,  0.191)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.190\\      (  0.078,  0.285)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.108\\      (-0.020,  0.185)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.658\\      (-0.793,-0.581)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.145\\      (-0.215,-0.012)\end{tabular} \\
\multicolumn{1}{l|}{C3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.078,  0.083)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.066,  0.094)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.114\\      (  0.046,  0.206)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.401\\      (  0.300,  0.465)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.280\\      (-0.379,-0.217)\end{tabular} \\
\multicolumn{1}{l|}{C4(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.140\\      (-0.201,-0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.607\\      (  0.504,  0.677)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.049\\      (-0.052,  0.116)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.496\\      (-0.576,-0.407)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.041\\      (  0.006,  0.170)\end{tabular} \\
\multicolumn{1}{l|}{C5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.081\\      (  0.001,  0.192)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.161,  0.047)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.053,  0.151)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.779\\      (  0.691,  0.887)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.051\\      (-0.066,  0.126)\end{tabular} \\
\multicolumn{1}{l|}{C6(-)} &
  \begin{tabular}[c]{@{}c@{}}0.011\\      (-0.088,  0.124)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.184\\      (  0.062,  0.277)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.046\\      (-0.078,  0.139)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.705\\      (-0.841,-0.621)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.089\\      (-0.188,  0.024)\end{tabular} \\
\hline
\end{tabular}%
\end{table}

\begin{table}[h]
\caption{Point estimate and confidence interval constructed by $L^0.5$}
\label{tab:empCI1}

\begin{tabular}{lccccc}
\hline
 &
  E &
  N &
  A &
  C &
  O \\ \hline

\multicolumn{1}{l|}{C7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.129\\      (-0.221,-0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.139\\      (  0.057,  0.235)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.110\\      (  0.035,  0.214)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.528\\      (  0.434,  0.618)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.038\\      (-0.157,  0.021)\end{tabular} \\
\multicolumn{1}{l|}{C8(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.010\\      (-0.087,  0.094)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.275\\      (  0.173,  0.355)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.241\\      (-0.356,-0.174)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.524\\      (-0.608,-0.421)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.059,  0.122)\end{tabular} \\
\multicolumn{1}{l|}{C9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.058,  0.130)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.014\\      (-0.067,  0.134)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.123\\      (  0.052,  0.241)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.723\\      (  0.626,  0.821)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.076\\      (-0.040,  0.149)\end{tabular} \\
\multicolumn{1}{l|}{C10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.107,  0.059)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.016\\      (-0.051,  0.117)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.128\\      (  0.065,  0.231)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.524\\      (  0.416,  0.587)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.234\\      (-0.337,-0.172)\end{tabular} \\
\multicolumn{1}{l|}{O1$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.079\\      (-0.021,  0.145)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.057,  0.109)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.045\\      (-0.146,  0.014)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.108,  0.056)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.623\\      (-0.711,-0.544)\end{tabular} \\
\multicolumn{1}{l|}{O2(-)} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.051,  0.121)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.220\\      (  0.114,  0.283)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.087\\      (-0.183,-0.015)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.009\\      (-0.055,  0.117)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.581\\      (  0.502,  0.675)\end{tabular} \\
\multicolumn{1}{l|}{O3$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.026,  0.135)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.153\\      (  0.092,  0.251)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.025\\      (-0.061,  0.096)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.002\\      (-0.098,  0.063)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.589\\      (-0.665,-0.502)\end{tabular} \\
\multicolumn{1}{l|}{O4(-)} &
  \begin{tabular}[c]{@{}c@{}}0.024\\      (-0.027,  0.144)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.202\\      (  0.098,  0.267)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.154\\      (-0.230,-0.064)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.036\\      (-0.022,  0.148)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.571\\      (  0.487,  0.660)\end{tabular} \\
\multicolumn{1}{l|}{O5$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.237\\      (  0.133,  0.271)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.067\\      (-0.137,  0.002)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.058\\      (-0.132,  0.002)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.176\\      (  0.069,  0.205)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.581\\      (-0.651,-0.509)\end{tabular} \\
\multicolumn{1}{l|}{O6(-)} &
  \begin{tabular}[c]{@{}c@{}}-0.211\\      (-0.272,-0.102)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.003\\      (-0.108,  0.060)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.047\\      (-0.121,  0.044)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.028\\      (-0.038,  0.133)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.510\\      (  0.434,  0.605)\end{tabular} \\
\multicolumn{1}{l|}{O7$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.076\\      (-0.017,  0.123)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.166\\      (-0.221,-0.084)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.035\\      (-0.100,  0.036)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097\\      (  0.012,  0.153)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.450\\      (-0.524,-0.382)\end{tabular} \\
\multicolumn{1}{l|}{O8$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.024\\      (-0.163,  0.021)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.165\\      (  0.090,  0.270)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.107\\      (-0.197,-0.020)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.123,  0.060)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.656\\      (-0.764,-0.577)\end{tabular} \\
\multicolumn{1}{l|}{O9$(+)$} &
  \begin{tabular}[c]{@{}c@{}}-0.063\\      (-0.153,  0.008)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.217\\      (  0.151,  0.309)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.240\\      (  0.156,  0.315)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.107\\      (  0.017,  0.179)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.258\\      (-0.343,-0.183)\end{tabular} \\
\multicolumn{1}{l|}{O10$(+)$} &
  \begin{tabular}[c]{@{}c@{}}0.240\\      (  0.132,  0.275)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}-0.004\\      (-0.036,  0.113)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.000\\      (-0.072,  0.068)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.097\\      (  0.012,  0.155)\end{tabular} &
  \begin{tabular}[c]{@{}c@{}}0.685\\      (-0.762,-0.613)\end{tabular} \\ \hline
\end{tabular}%
\end{table}

}

\begin{comment}
\begin{longtable}[c]{lllllllllll}
\caption{p=1, 95\% Confidence Interval}
\label{tab:ci1}\\
\hline
                            & \multicolumn{10}{c}{p=0.5, 95\% Confidence  Interval}                                                        \\
\endfirsthead
\endhead
\hline
\endfoot
\endlastfoot
                            & \multicolumn{5}{c}{lower bound}                                 & \multicolumn{5}{c}{upper bound}            \\
                            & V1     & V2     & V3     & V4     & V5                          & V1     & V2     & V3     & V4     & V5     \\ \hline
\multicolumn{1}{l|}{row 1}  & 0.792  & -0.161 & -0.180 & -0.119 & \multicolumn{1}{l|}{0.006}  & 0.982  & 0.007  & -0.014 & 0.054  & 0.171  \\
\multicolumn{1}{l|}{row 2}  & -0.974 & -0.235 & -0.126 & -0.042 & \multicolumn{1}{l|}{-0.081} & -0.770 & -0.050 & 0.056  & 0.150  & 0.102  \\
\multicolumn{1}{l|}{row 3}  & 0.692  & -0.354 & 0.118  & -0.004 & \multicolumn{1}{l|}{-0.173} & 0.867  & -0.194 & 0.274  & 0.158  & -0.018 \\
\multicolumn{1}{l|}{row 4}  & -1.025 & -0.015 & -0.073 & -0.050 & \multicolumn{1}{l|}{-0.187} & -0.847 & 0.140  & 0.079  & 0.109  & -0.035 \\
\multicolumn{1}{l|}{row 5}  & 0.810  & -0.032 & 0.061  & -0.002 & \multicolumn{1}{l|}{0.022}  & 0.987  & 0.123  & 0.211  & 0.157  & 0.173  \\
\multicolumn{1}{l|}{row 6}  & -0.853 & -0.115 & -0.160 & -0.143 & \multicolumn{1}{l|}{0.050}  & -0.662 & 0.061  & 0.012  & 0.038  & 0.225  \\
\multicolumn{1}{l|}{row 7}  & 1.024  & 0.019  & 0.003  & -0.008 & \multicolumn{1}{l|}{-0.052} & 1.228  & 0.195  & 0.174  & 0.170  & 0.117  \\
\multicolumn{1}{l|}{row 8}  & -0.823 & 0.000  & -0.035 & 0.047  & \multicolumn{1}{l|}{-0.024} & -0.629 & 0.181  & 0.143  & 0.234  & 0.155  \\
\multicolumn{1}{l|}{row 9}  & 0.737  & -0.142 & -0.107 & -0.158 & \multicolumn{1}{l|}{-0.334} & 0.944  & 0.048  & 0.078  & 0.037  & -0.146 \\
\multicolumn{1}{l|}{row 10} & -0.930 & 0.018  & -0.120 & -0.178 & \multicolumn{1}{l|}{-0.092} & -0.739 & 0.190  & 0.048  & -0.001 & 0.078  \\
\multicolumn{1}{l|}{row 11} & -0.185 & 0.895  & -0.117 & 0.063  & \multicolumn{1}{l|}{0.019}  & -0.004 & 1.097  & 0.079  & 0.249  & 0.194  \\
\multicolumn{1}{l|}{row 12} & 0.002  & -0.801 & -0.065 & -0.250 & \multicolumn{1}{l|}{-0.192} & 0.176  & -0.614 & 0.112  & -0.069 & -0.020 \\
\multicolumn{1}{l|}{row 13} & -0.231 & 0.725  & 0.131  & 0.102  & \multicolumn{1}{l|}{-0.135} & -0.057 & 0.915  & 0.308  & 0.283  & 0.037  \\
\multicolumn{1}{l|}{row 14} & 0.089  & -0.686 & -0.075 & -0.105 & \multicolumn{1}{l|}{-0.116} & 0.285  & -0.485 & 0.118  & 0.096  & 0.077  \\
\multicolumn{1}{l|}{row 15} & -0.046 & 0.359  & -0.158 & -0.192 & \multicolumn{1}{l|}{0.138}  & 0.167  & 0.575  & 0.049  & 0.026  & 0.348  \\
\multicolumn{1}{l|}{row 16} & -0.172 & 0.735  & 0.154  & -0.034 & \multicolumn{1}{l|}{0.056}  & 0.007  & 0.928  & 0.333  & 0.151  & 0.230  \\
\multicolumn{1}{l|}{row 17} & -0.008 & 0.888  & -0.212 & -0.065 & \multicolumn{1}{l|}{-0.072} & 0.158  & 1.076  & -0.043 & 0.109  & 0.090  \\
\multicolumn{1}{l|}{row 18} & -0.021 & 1.055  & -0.245 & -0.085 & \multicolumn{1}{l|}{-0.066} & 0.155  & 1.257  & -0.068 & 0.100  & 0.104  \\
\multicolumn{1}{l|}{row 19} & -0.085 & 0.795  & -0.388 & 0.142  & \multicolumn{1}{l|}{-0.066} & 0.092  & 0.989  & -0.208 & 0.322  & 0.107  \\
\multicolumn{1}{l|}{row 20} & -0.398 & 0.763  & -0.019 & -0.127 & \multicolumn{1}{l|}{-0.184} & -0.221 & 0.955  & 0.162  & 0.055  & -0.012 \\
\multicolumn{1}{l|}{row 21} & -0.118 & 0.011  & -0.872 & -0.081 & \multicolumn{1}{l|}{-0.136} & 0.084  & 0.208  & -0.666 & 0.121  & 0.059  \\
\multicolumn{1}{l|}{row 22} & 0.362  & -0.063 & 0.465  & -0.149 & \multicolumn{1}{l|}{-0.129} & 0.527  & 0.093  & 0.626  & 0.011  & 0.025  \\
\multicolumn{1}{l|}{row 23} & 0.090  & 0.465  & -0.681 & -0.168 & \multicolumn{1}{l|}{-0.232} & 0.299  & 0.678  & -0.471 & 0.044  & -0.029 \\
\multicolumn{1}{l|}{row 24} & -0.006 & -0.047 & 0.892  & -0.094 & \multicolumn{1}{l|}{-0.069} & 0.166  & 0.096  & 1.045  & 0.038  & 0.062  \\
\multicolumn{1}{l|}{row 25} & -0.257 & -0.049 & -0.892 & -0.077 & \multicolumn{1}{l|}{-0.164} & -0.097 & 0.107 & -0.723 & 0.080 & -0.012 \\
\multicolumn{1}{l|}{row 26} & -0.136 & 0.104  & 0.629  & -0.085 & \multicolumn{1}{l|}{-0.094} & 0.041  & 0.276  & 0.810  & 0.091  & 0.076  \\
\multicolumn{1}{l|}{row 27} & -0.457 & 0.019  & -0.797 & -0.001 & \multicolumn{1}{l|}{-0.099} & -0.300 & 0.168  & -0.638 & 0.150  & 0.047  \\
\multicolumn{1}{l|}{row 28} & 0.043  & -0.012 & 0.618  & 0.002  & \multicolumn{1}{l|}{-0.107} & 0.190  & 0.130  & 0.771  & 0.147  & 0.033  \\
\multicolumn{1}{l|}{row 29} & 0.046  & 0.056  & 0.669  & -0.011 & \multicolumn{1}{l|}{-0.195} & 0.207  & 0.211  & 0.836  & 0.148  & -0.042 \\
\multicolumn{1}{l|}{row 30} & 0.348  & -0.148 & 0.245  & 0.005  & \multicolumn{1}{l|}{-0.138} & 0.512  & 0.009  & 0.402  & 0.167  & 0.018  \\
\multicolumn{1}{l|}{row 31} & -0.003 & -0.180 & -0.099 & 0.566  & \multicolumn{1}{l|}{-0.247} & 0.177  & 0.006  & 0.087  & 0.756  & -0.067 \\
\multicolumn{1}{l|}{row 32} & 0.000  & 0.050  & -0.021 & -0.799 & \multicolumn{1}{l|}{-0.211} & 0.206  & 0.258  & 0.183  & -0.584 & -0.008 \\
\multicolumn{1}{l|}{row 33} & -0.091 & -0.051 & 0.049  & 0.303  & \multicolumn{1}{l|}{-0.383} & 0.071  & 0.112  & 0.209  & 0.472  & -0.220 \\
\multicolumn{1}{l|}{row 34} & -0.168 & 0.494  & -0.046 & -0.549 & \multicolumn{1}{l|}{0.010}  & -0.005 & 0.667  & 0.123  & -0.377 & 0.175  \\
\multicolumn{1}{l|}{row 35} & -0.020 & -0.110 & -0.049 & 0.680  & \multicolumn{1}{l|}{-0.075} & 0.169  & 0.092  & 0.152  & 0.881  & 0.118  \\
\multicolumn{1}{l|}{row 36} & -0.074 & 0.035  & -0.079 & -0.838 & \multicolumn{1}{l|}{-0.184} & 0.139  & 0.251  & 0.137  & -0.615 & 0.029  \\
\multicolumn{1}{l|}{row 37} & -0.218 & 0.086  & 0.038  & 0.460  & \multicolumn{1}{l|}{-0.159} & -0.041 & 0.266  & 0.215  & 0.648  & 0.019  \\
\multicolumn{1}{l|}{row 38} & -0.073 & 0.154  & -0.354 & -0.606 & \multicolumn{1}{l|}{-0.057} & 0.109  & 0.339  & -0.172 & -0.416 & 0.125  \\
\multicolumn{1}{l|}{row 39} & -0.062 & -0.023 & 0.054  & 0.637  & \multicolumn{1}{l|}{-0.043} & 0.125  & 0.176  & 0.242  & 0.837  & 0.146  \\
\multicolumn{1}{l|}{row 40} & -0.116 & -0.025 & 0.068  & 0.428  & \multicolumn{1}{l|}{-0.341} & 0.050  & 0.145  & 0.233  & 0.604  & -0.175 \\
\multicolumn{1}{l|}{row 41} & -0.037 & -0.059 & -0.146 & -0.107 & \multicolumn{1}{l|}{-0.711} & 0.132  & 0.111  & 0.014  & 0.061  & -0.543 \\
\multicolumn{1}{l|}{row 42} & -0.020 & 0.122  & -0.184 & -0.047 & \multicolumn{1}{l|}{0.506}  & 0.156  & 0.296  & -0.017 & 0.128  & 0.681  \\
\multicolumn{1}{l|}{row 43} & -0.036 & 0.092  & -0.058 & -0.086 & \multicolumn{1}{l|}{-0.666} & 0.129  & 0.254  & 0.099  & 0.078  & -0.503 \\
\multicolumn{1}{l|}{row 44} & -0.020 & 0.099  & -0.225 & -0.026 & \multicolumn{1}{l|}{0.484}  & 0.153  & 0.273  & -0.060 & 0.148  & 0.658  \\
\multicolumn{1}{l|}{row 45} & 0.110  & -0.138 & -0.127 & 0.054  & \multicolumn{1}{l|}{-0.655} & 0.250  & 0.003  & 0.006  & 0.192  & -0.513 \\
\multicolumn{1}{l|}{row 46} & -0.263 & -0.108 & -0.121 & -0.037 & \multicolumn{1}{l|}{0.434}  & -0.091 & 0.064  & 0.043  & 0.136  & 0.606  \\
\multicolumn{1}{l|}{row 47} & -0.043 & -0.229 & -0.099 & -0.007 & \multicolumn{1}{l|}{-0.527} & 0.099  & -0.087 & 0.037  & 0.136  & -0.385 \\
\multicolumn{1}{l|}{row 48} & -0.178 & 0.089  & -0.194 & -0.104 & \multicolumn{1}{l|}{-0.764} & 0.012  & 0.274  & -0.017 & 0.083  & -0.577 \\
\multicolumn{1}{l|}{row 49} & -0.148 & 0.158  & 0.159  & 0.037  & \multicolumn{1}{l|}{-0.343} & 0.014  & 0.320  & 0.318  & 0.203  & -0.183 \\
\multicolumn{1}{l|}{row 50} & 0.111  & -0.039 & -0.068 & 0.006  & \multicolumn{1}{l|}{-0.766} & 0.258  & 0.114  & 0.071  & 0.152  & -0.616 \\ \hline
\end{longtable}

\begin{longtable}[c]{lllllllllll}
\caption{p=0.5, 95\% Confidence Interval}
\label{tab:ci0.5}\\
\hline
                            & \multicolumn{10}{c}{p=0.5, 95\% Confidence  Interval}                                                        \\
\endfirsthead
\endhead
\hline
\endfoot
\endlastfoot
                            & \multicolumn{5}{c}{lower bound}                                 & \multicolumn{5}{c}{upper bound}            \\
                            & V1     & V2     & V3     & V4     & V5                          & V1     & V2     & V3     & V4     & V5     \\ \hline
\multicolumn{1}{l|}{row 1}  & 0.793  & -0.161 & -0.183 & -0.078 & \multicolumn{1}{l|}{0.005}  & 0.980  & 0.003  & -0.017 & 0.100  & 0.170  \\
\multicolumn{1}{l|}{row 2}  & -0.962 & -0.228 & -0.126 & -0.061 & \multicolumn{1}{l|}{-0.079} & -0.760 & -0.048 & 0.057  & 0.132  & 0.104  \\
\multicolumn{1}{l|}{row 3}  & 0.702  & -0.347 & 0.120  & 0.050  & \multicolumn{1}{l|}{-0.175} & 0.876  & -0.192 & 0.275  & 0.213  & -0.020 \\
\multicolumn{1}{l|}{row 4}  & -1.019 & -0.011 & -0.075 & -0.099 & \multicolumn{1}{l|}{-0.184} & -0.842 & 0.140  & 0.077  & 0.068  & -0.033 \\
\multicolumn{1}{l|}{row 5}  & 0.812  & -0.032 & 0.059  & 0.023  & \multicolumn{1}{l|}{0.021}  & 0.987  & 0.118  & 0.209  & 0.183  & 0.172  \\
\multicolumn{1}{l|}{row 6}  & -0.849 & -0.103 & -0.163 & -0.161 & \multicolumn{1}{l|}{0.052}  & -0.660 & 0.069  & 0.010  & 0.022  & 0.226  \\
\multicolumn{1}{l|}{row 7}  & 1.020  & 0.018  & 0.001  & 0.020  & \multicolumn{1}{l|}{-0.052} & 1.221  & 0.189  & 0.172  & 0.199  & 0.117  \\
\multicolumn{1}{l|}{row 8}  & -0.823 & 0.000  & -0.038 & 0.024  & \multicolumn{1}{l|}{-0.022} & -0.630 & 0.176  & 0.140  & 0.210  & 0.157  \\
\multicolumn{1}{l|}{row 9}  & 0.748  & -0.131 & -0.111 & -0.110 & \multicolumn{1}{l|}{-0.332} & 0.953  & 0.054  & 0.075  & 0.086  & -0.145 \\
\multicolumn{1}{l|}{row 10} & -0.930 & 0.025  & -0.123 & -0.215 & \multicolumn{1}{l|}{-0.092} & -0.741 & 0.193  & 0.047  & -0.036 & 0.078  \\
\multicolumn{1}{l|}{row 11} & -0.210 & 0.875  & -0.121 & 0.016  & \multicolumn{1}{l|}{0.017}  & -0.028 & 1.073  & 0.078  & 0.200  & 0.193  \\
\multicolumn{1}{l|}{row 12} & 0.039  & -0.771 & -0.064 & -0.197 & \multicolumn{1}{l|}{-0.185} & 0.215  & -0.589 & 0.115  & -0.020 & -0.012 \\
\multicolumn{1}{l|}{row 13} & -0.271 & 0.696  & 0.124  & 0.031  & \multicolumn{1}{l|}{-0.139} & -0.093 & 0.881  & 0.304  & 0.208  & 0.033  \\
\multicolumn{1}{l|}{row 14} & 0.115  & -0.670 & -0.069 & -0.049 & \multicolumn{1}{l|}{-0.113} & 0.310  & -0.474 & 0.125  & 0.148  & 0.079  \\
\multicolumn{1}{l|}{row 15} & -0.073 & 0.357  & -0.163 & -0.208 & \multicolumn{1}{l|}{0.135}  & 0.138  & 0.567  & 0.046  & 0.006  & 0.345  \\
\multicolumn{1}{l|}{row 16} & -0.202 & 0.717  & 0.145  & -0.106 & \multicolumn{1}{l|}{0.053}  & -0.022 & 0.905  & 0.327  & 0.074  & 0.227  \\
\multicolumn{1}{l|}{row 17} & -0.048 & 0.883  & -0.226 & -0.102 & \multicolumn{1}{l|}{-0.071} & 0.123  & 1.064  & -0.055 & 0.064  & 0.089  \\
\multicolumn{1}{l|}{row 18} & -0.080 & 1.046  & -0.261 & -0.140 & \multicolumn{1}{l|}{-0.065} & 0.106  & 1.240  & -0.080 & 0.036  & 0.103  \\
\multicolumn{1}{l|}{row 19} & -0.131 & 0.768  & -0.399 & 0.092  & \multicolumn{1}{l|}{-0.068} & 0.048  & 0.956  & -0.214 & 0.267  & 0.104  \\
\multicolumn{1}{l|}{row 20} & -0.435 & 0.752  & -0.029 & -0.181 & \multicolumn{1}{l|}{-0.187} & -0.256 & 0.940  & 0.155  & -0.004 & -0.016 \\
\multicolumn{1}{l|}{row 21} & -0.112 & 0.013  & -0.876 & -0.071 & \multicolumn{1}{l|}{-0.134} & 0.088  & 0.205  & -0.670 & 0.127  & 0.060  \\
\multicolumn{1}{l|}{row 22} & 0.359  & -0.053 & 0.464  & -0.124 & \multicolumn{1}{l|}{-0.128} & 0.523  & 0.099  & 0.626  & 0.034  & 0.026  \\
\multicolumn{1}{l|}{row 23} & 0.080  & 0.467  & -0.692 & -0.177 & \multicolumn{1}{l|}{-0.231} & 0.286  & 0.674  & -0.480 & 0.030  & -0.029 \\
\multicolumn{1}{l|}{row 24} & -0.034 & -0.039 & 0.895  & -0.098 & \multicolumn{1}{l|}{-0.071} & 0.142  & 0.100  & 1.050  & 0.032  & 0.060  \\
\multicolumn{1}{l|}{row 25} & -0.248 & -0.047 & -0.894 & -0.078 & \multicolumn{1}{l|}{-0.163} & -0.088 & 0.105 & -0.725 & 0.077 & -0.011 \\
\multicolumn{1}{l|}{row 26} & -0.157 & 0.102  & 0.630  & -0.100 & \multicolumn{1}{l|}{-0.096} & 0.020  & 0.270  & 0.812  & 0.073  & 0.073  \\
\multicolumn{1}{l|}{row 27} & -0.452 & 0.015  & -0.800 & -0.016 & \multicolumn{1}{l|}{-0.098} & -0.297 & 0.161  & -0.640 & 0.133  & 0.047  \\
\multicolumn{1}{l|}{row 28} & 0.039  & -0.011 & 0.616  & 0.007  & \multicolumn{1}{l|}{-0.106} & 0.185  & 0.128  & 0.770  & 0.149  & 0.034  \\
\multicolumn{1}{l|}{row 29} & 0.040  & 0.055  & 0.667  & -0.015 & \multicolumn{1}{l|}{-0.194} & 0.199  & 0.206  & 0.835  & 0.140  & -0.042 \\
\multicolumn{1}{l|}{row 30} & 0.352  & -0.145 & 0.244  & 0.033  & \multicolumn{1}{l|}{-0.136} & 0.515  & 0.008  & 0.401  & 0.193  & 0.019  \\
\multicolumn{1}{l|}{row 31} & 0.002  & -0.179 & -0.108 & 0.583  & \multicolumn{1}{l|}{-0.244} & 0.180  & 0.003  & 0.079  & 0.771  & -0.064 \\
\multicolumn{1}{l|}{row 32} & -0.015 & 0.078  & -0.020 & -0.793 & \multicolumn{1}{l|}{-0.215} & 0.191  & 0.285  & 0.185  & -0.581 & -0.012 \\
\multicolumn{1}{l|}{row 33} & -0.078 & -0.066 & 0.046  & 0.300  & \multicolumn{1}{l|}{-0.379} & 0.083  & 0.094  & 0.206  & 0.465  & -0.217 \\
\multicolumn{1}{l|}{row 34} & -0.201 & 0.504  & -0.052 & -0.576 & \multicolumn{1}{l|}{0.006}  & -0.036 & 0.677  & 0.116  & -0.407 & 0.170  \\
\multicolumn{1}{l|}{row 35} & 0.001  & -0.161 & -0.053 & 0.691  & \multicolumn{1}{l|}{-0.066} & 0.192  & 0.047  & 0.151  & 0.887  & 0.126  \\
\multicolumn{1}{l|}{row 36} & -0.088 & 0.062  & -0.078 & -0.841 & \multicolumn{1}{l|}{-0.188} & 0.124  & 0.277  & 0.139  & -0.621 & 0.024  \\
\multicolumn{1}{l|}{row 37} & -0.221 & 0.057  & 0.035  & 0.434  & \multicolumn{1}{l|}{-0.157} & -0.044 & 0.235  & 0.214  & 0.618  & 0.021  \\
\multicolumn{1}{l|}{row 38} & -0.087 & 0.173  & -0.356 & -0.608 & \multicolumn{1}{l|}{-0.059} & 0.094  & 0.355  & -0.174 & -0.421 & 0.122  \\
\multicolumn{1}{l|}{row 39} & -0.058 & -0.067 & 0.052  & 0.626  & \multicolumn{1}{l|}{-0.040} & 0.130  & 0.134  & 0.241  & 0.821  & 0.149  \\
\multicolumn{1}{l|}{row 40} & -0.107 & -0.051 & 0.065  & 0.416  & \multicolumn{1}{l|}{-0.337} & 0.059  & 0.117  & 0.231  & 0.587  & -0.172 \\
\multicolumn{1}{l|}{row 41} & -0.021 & -0.057 & -0.146 & -0.108 & \multicolumn{1}{l|}{-0.711} & 0.145  & 0.109  & 0.014  & 0.056  & -0.544 \\
\multicolumn{1}{l|}{row 42} & -0.051 & 0.114  & -0.183 & -0.055 & \multicolumn{1}{l|}{0.502}  & 0.121  & 0.283  & -0.015 & 0.117  & 0.675  \\
\multicolumn{1}{l|}{row 43} & -0.026 & 0.092  & -0.061 & -0.098 & \multicolumn{1}{l|}{-0.665} & 0.135  & 0.251  & 0.096  & 0.063  & -0.502 \\
\multicolumn{1}{l|}{row 44} & -0.027 & 0.098  & -0.230 & -0.022 & \multicolumn{1}{l|}{0.487}  & 0.144  & 0.267  & -0.064 & 0.148  & 0.660  \\
\multicolumn{1}{l|}{row 45} & 0.133  & -0.137 & -0.132 & 0.069  & \multicolumn{1}{l|}{-0.651} & 0.271  & 0.002  & 0.002  & 0.205  & -0.509 \\
\multicolumn{1}{l|}{row 46} & -0.272 & -0.108 & -0.121 & -0.038 & \multicolumn{1}{l|}{0.434}  & -0.102 & 0.060  & 0.044  & 0.133  & 0.605  \\
\multicolumn{1}{l|}{row 47} & -0.017 & -0.221 & -0.100 & 0.012  & \multicolumn{1}{l|}{-0.524} & 0.123  & -0.084 & 0.036  & 0.153  & -0.382 \\
\multicolumn{1}{l|}{row 48} & -0.163 & 0.090  & -0.197 & -0.123 & \multicolumn{1}{l|}{-0.764} & 0.021  & 0.270  & -0.020 & 0.060  & -0.577 \\
\multicolumn{1}{l|}{row 49} & -0.153 & 0.151  & 0.156  & 0.017  & \multicolumn{1}{l|}{-0.343} & 0.008  & 0.309  & 0.315  & 0.179  & -0.183 \\
\multicolumn{1}{l|}{row 50} & 0.132  & -0.036 & -0.072 & 0.012  & \multicolumn{1}{l|}{-0.762} & 0.275  & 0.113  & 0.068  & 0.155  & -0.613 \\ \hline
\end{longtable}

\end{comment}

\section{Discussions}

\clearpage

redpajama_set_name: RedPajamaArXiv