text:
\section{Introduction}

The problem of multiple outliers identification received attention of many
authors. The majority of  outlier identification methods define  rules  for the rejection of the most extreme observations.
The bulk of publications  have been concentrated on the normal distribution (see \cite{bolshev_chauvenets_1975,davies_identification_1993,dixon_analysis_1950,grubbs_sample_1950,rosner_detection_1975,tietjen_grubbs-type_1972},  see surveys in \cite{barnett1974outliers,zerbet_statistical_2002}. For non-normal case,
the most of the literature pertains to the exponential and gamma distributions, see
\cite{chikkagoudar_distributions_1983,kabe_testing_1970,kimber_testing_1988,lalitha_multiple_2012,lewis_recursive_1979,likes_distribution_1967,lin_exact_2009,lin_tests_2014,zerbet_new_2003}.

Constructing outlier identification methods, the most of authors suppose that the number $s$ of  observations suspected to be outliers  is specified. These methods have a serious drawback:
only two possible conclusions are done:  exactly $s$ observations are admitted as  outliers or it is concluded that outliers are absent. More natural is to consider methods
which do not specify  the number of
 suspected observations or at least specify the upper limit $s$ for it. Such methods are not very numerous and they concern  normal or exponential samples. These are  \cite{rosner_detection_1975,bolshev_chauvenets_1975,hawkins_identification_1980} methods for normal samples,
 \cite{kimber_tests_1982,lin_exact_2009,lin_tests_2014}) methods for exponential samples. The only method which does not specify the upper limit $s$  is the  \cite{davies_identification_1993} method for normal samples.

We give a competitive and simple method for outlier identification in samples from location-scale and shape-scale families of probability distributions. The upper limit $s$ is not specified, as in the the case of Davies-Gather method.   The method is based on a theorem giving  asymptotic properties of extreme z-scores. Robust estimators of model parameters are used defining z-scores.

In Section \ref{sec:outliers}  we present a short  overview of the notion of the outlier region given by  \cite{davies_identification_1993}. In Section \ref{sec:new_method} we give asymptotic properties of extreme z-scores based on equivariant estimators of model parameters, and
 introduce  a new outlier identification method for  parametric models based on  the asymptotic result and robust  estimators.  In section \ref{sec:davies} we consider rather evident generalizations of Davies-Gather tests for normal data to location-scale families. In Section \ref{sec:compare} we give a short overview of known multiple outlier identification methods for normal samples which do not specify an exact number of suspected outliers.
 In Section \ref{sec:simul} we compare performance of the new and existing methods.

\section{Outliers and outlier regions}
\label{sec:outliers}
Suppose that  data are independent random variables
$X_1,\ldots,X_n$. Denote by $F_{i}(x)$ the c.d.f. of $X_i$.

Let
${\cal F}_0=\{F(x,\theta ), \theta \in \Theta \subset
\mathbf{R}^m\} \label{eq:param_family}
$
be a parametric family of absolutely continuous cumulative distribution functions with continuous unimodal densities $f$  on the support
$\supp(F)$  of the c.d.f. $F$.

\vskip 0.2cm \noindent Suppose that if the data are not contaminated with unusual observations, then  the following  null  hypothesis $H_0$ is true: there exist $\theta\in \Theta$ such that
\begin{equation}
F_{1}(x)=\ldots=F_{n}(x)=F(x,\theta).\label{eq:null hypothesis}
\end{equation}

There are two different definitions of an outlier. In the first case
outlier is an observation which falls  into some outlier region $out(X)$. The outlier region is a set such   that the probability
for at least one observation from a sample to  fall into it is small if the hypothesis $H_0$ is true. In such a case the probability that a specified observation $X_i$  falls into $out(X)$ is very small. If an observation $X_i$ has  distribution different from that  under $H_0$ then this probability may be considerably higher.

In the second case the value $x_i$ of $X_i$ is an outlier if the probability distribution of $X_i$ is different from that under $H_0$, formally $F_i \neq F(x,\theta)$. In this case outliers are often called {\it contaminants}.

So in the first case exists a very small probability to have
an outlier under $H_0$. In the second case outliers (contaminants) do not exist under $H_0$ and  outliers (contaminants)  do not necessary fall into the outlier region. Both definitions give approximately the same outliers if the alternative distribution is concentrated in the outlier region. Namely such contaminants can be called outliers in the sense that outliers are anomalous extreme observations. In such a case it is possible to compare outlier and contaminant search methods.

In this paper, we consider  location-scale and shape-scale families. Location-scale families have the form ${\cal F}_{ls}=\{F_0((x-\mu)/\sigma), \mu \in {\bf R},\sigma>0\}$   with the completely specified baseline
c.d.f $F_0$ and  p.d.f. $f_0$. Shape-scale families have the form ${\cal F}_{ls}=\{G_0(((x/\theta)^\nu), \theta, \nu >0\}$   with completely specified baseline
c.d.f $G_0$ and  p.d.f. $g_0$. By logarithmic transformation the shape-scale families are transformed to  location-scale family, so we concentrate on location-scale families. Methods for such families are easily modified to methods for shape-scale families.

The right-sided $\alpha$-outlier region for a location-scale family is
\begin{equation*}
out_r(\alpha_n,F)=\{x\in {\bf R}:x>\mu+\sigma F_0^{-1}(1-\alpha)\}
\end{equation*}
and the left-sided $\alpha$-outlier region is
\begin{equation*}
out_l(\alpha_n,F)=\{x\in {\bf R}:x<\mu+\sigma F_0^{-1}(\alpha)\}.
\end{equation*}
The two-sided $\alpha$-outlier region has the form
\begin{equation}
out(\alpha,F)=\{x\in {\bf R}/[\mu+\sigma F_0^{-1}(\alpha/2),\mu+\sigma F_0^{-1}(1-\alpha/2)]\}.
\end{equation}

If $f_0$ is symmetric, then the two-sided outlier region is simpler:
\begin{equation*}
out(\alpha,F)=\{x\in {\bf R}:|x-\mu|>\sigma F_0^{-1}(1-\alpha/2)\}.
\end{equation*}
The value of $\alpha$
is chosen depending on the size $n$ of a  sample:
$\alpha=\alpha_n$. The choice is based on assumption that
under $H_0$ for some $\bar\alpha$  close to zero
\begin{eqnarray}
\PP\{\cap_{i=1}^n\{X_i \notin out(\alpha_n,F)\}\}= (\PP\{X_i
\notin out(\alpha_n,F)\})^n =1-\bar \alpha. \label{eq:out_reg_p}
\end{eqnarray}
 The equality
\eqref{eq:out_reg_p} means that under $H_0$ the probability that
{\it none} of $X_i$ falls into $\alpha_n$ - outlier region is
$1-\bar\alpha$. It implies that
\begin{equation} \alpha_n=1-(1-\bar\alpha)^{1/n}.\label{eq:alpha n}\end{equation} The sequence
$\alpha_n$ decreases from $\bar \alpha$ to  $0$ as $n$ goes from
$1$ to $\infty$.

The first definition of an outlier is as follows: for a sample size $n$ a realization $x_i$ of $X_i$ is called
{\it outlier} if $x_i\in out(\alpha_n,F)$;   $x_i$  is called
{\it right  outlier} if $x_i\in out_r(\alpha_n,F)$.

The number of outliers $D_n$ under $H_0$ has the binomial
distribution $B(n,\alpha_n)$ and the expected number of outliers
in the sample under $H_0$ is $\EE D_n=n\alpha_n.$ Note that $\EE D_n\to
-\ln(1- \bar\alpha)\approx \bar\alpha$ as $n\to\infty$. For example, if
 $\bar\alpha=0.05$, then $\ln(1-\bar\alpha)\approx
0.05129$ and for  $n\geq 10$ the expected number of outliers  is
approximately $0.051$, i.e. it practically does not depend on $n$. So  under $H_0$  the expected
number of outliers  $0.051$ is  negligible with respect to the sample size $n$.

\section{New method}
\label{sec:new_method}
\subsection{Preliminary results}

Suppose that a c.d.f. $F\in {\cal F}_{ls}$
belongs also to the domain of attraction ${\cal G}_\gamma$, $\gamma\geq  0$ (see \cite{de2007extreme}).

If $F\in {\cal G}_0\cap {\cal F}_{ls}$ , then  there exist normalizing constants $a_n>0 $
and $b_n\in {\bf R}$ such that $\lim_{n\to\infty} F_0^n(a_n x+b_n) = e^{-e^{-x}}.$  Similarly, if
$F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma>0$,  then $\lim_{n\to\infty} F_0^n(a_n x+b_n) =  e^{-(-x)^{-1/\gamma}}, \quad x<0$, $\lim_{n\to\infty} F^n(a_n x+b_n) = 1, x\geq 0.$

One of possible choices of the sequences  $\{b_n\}$ and $\{a_n\}$ is
\begin{equation}
 b_n=F_0^{-1}(1-\frac1n),\quad a_n=1/(nf_0(b_n)). \label{eq:anbn}
\end{equation}
In the particular case of the normal distribution equivalent form $a_n=1/b_n$ can be used. Expressions of $b_n$ and $a_n$ for some most used distributions are given in Table \ref{tab:an_bn}.

\begin{table}
 \caption{Expressions of $b_n$ and $a_n$}\label{tab:an_bn}
 \centering
\begin{tabular}{|c|c|c|c|}
  \hline

  Distribution& $F_0(x)$ & $b_n$ & $a_n$  \\
  \hline
  Normal&$\Phi(x)$ &$\Phi^{-1}(1-1/n)$ &$1/b_n$ \\
  Type I extreme value&$1-e^{-e^{x}}$ &$\ln\ln n$&$e^{-b_n}$ \\
  Type II extreme value& $e^{-e^{-x}}$  &$\ln(-\ln(1-1/ n))$ & $e^{b_n}/(n-1)$ \\
  Logistic&$ \frac1{1+e^{-x}}$  &$  \ln(n-1)$ &$ {n}/({n-1})$ \\
  Laplace & $\frac12+\frac12\text{sign}(x)(1-e^{-|x|})$ & $ \ln({n}/2)$& $1$ \\
  Cauchy & $\frac12+\frac1{\pi}\arctan (x)$  & $\cot(\frac\pi{n})$ & $\frac{\pi}{n}/\sin^2(\frac\pi{n})$
\\
\hline
\end{tabular}

\end{table}

\begin{cond} Consider a model that satisfies the following conditions:\label{cond:cond}
\begin{enumerate}[label=\alph*)]
\item  $\hat\mu$ and $\hat \sigma$ are consistent estimators of $\mu$ and $\sigma$;\label{cond:a}

\item the limit distribution of ($\sqrt{n}(\hat\mu-\mu), \sqrt{n}(\hat\sigma-\sigma))$ is non-degenerate;\label{cond:b}

\item
\begin{equation*}
    \lim_{x\to \infty} \frac{xf_0(x)}{\sqrt{1-F_0(x)}}= 0.
\end{equation*}\label{cond:c}
\end{enumerate}
\end{cond}

\hyperref[cond:cond]{Condition A}  \ref{cond:c} is satisfied for many location-scale models including  the normal, type I extreme value, type II extreme value, logistic, Laplace ($F\in {\cal G}_0$), Cauchy
($F\in {\cal G}_1$).

Set $Y_i=(X_i-\mu)/{\sigma}$, $\hat Y_i=(X_i-\hat\mu)/{\hat\sigma}$. The random variables $\hat Y_i$ are called z-scores. Denote by $Y_{(1)}\leq Y_{(2)}\leq \ldots \leq Y_{(n)}$ and $\hat Y_{(1)}\leq \ldots \leq \hat Y_{(n)}$ the respective order statistics

 The following theorem is useful for  right outliers detection test construction.
\begin{theorem}\label{theo:teorem1}

If  $F\in {\cal G}_0\cap {\cal F}_{ls}$ and \hyperref[cond:cond]{Conditions A} hold, then for fixed
$s$
$$
((\hat Y_{(n)}-b_n)/{a_n},(\hat Y_{(n-1)}- b_n)/{ a_n}
,...,(\hat Y_{(n-s+1)}- b_n)/{ a_n})\stackrel{d}\rightarrow$$$$
L_0=(-\ln E_1,-\ln (E_1+E_2),...,-\ln (E_1+...+E_s))
$$
as $
n\to\infty$, where $E_1,...,E_s$ are i.i.d. standard
exponential random variables.

If $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma> 0$ and \hyperref[cond:cond]{Conditions A}  hold, then the limit random vector is
$$
L_\gamma=( E_1^{-1}-1,(E_1+E_2)^{-1}-1,...,(E_1+...+E_s)^{-1}-1).
$$

\end{theorem}

\vskip 0.2cm
\begin{remark}

Note that  $2(E_1+...+E_i)\sim \chi^2(2i)$.
  It implies that if $F\in {\cal G}_0\cap {\cal F}_{ls}$, then for fixed $i$, $i=1,...,s$,
\begin{equation}
\PP\{(\hat Y_{(n-i+1)}-b_n)/a_n\leq x\}\to
1-F_{\chi^2_{2i}}(2e^{-x})\quad \mbox{as}\;\;n\to\infty.
\end{equation}
Similarly, if  $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma>0$,  then for fixed $i$, $i=1,...,s$,
\begin{equation}
\PP\{(\hat Y_{(n-i+1)}-b_n)/a_n\leq x\}\to
1-F_{\chi^2_{2i}}(\frac2{1+x
})\quad \mbox{as}\;\;n\to\infty.
\end{equation}

\end{remark}

The following theorem is useful for construction of outlier detection tests
in two-sided case when  $f_0$ is symmetric.  For any sequence $\zeta_1,...,\zeta_n$ denote by $|\zeta|_{(1)}\leq ... \leq |\zeta|_{(n)}$ the ordered absolute values $|\zeta_1|,..., |\zeta_n|$.
\begin{theorem}\label{theo:teorem2}

Suppose that the function $f_0$ is symmetric. If  $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma\geq 0$ and \hyperref[cond:cond]{Conditions A} hold, then for fixed
$s$
$$
((|\hat Y|_{(n)}-b_{2n})/{a_{2n}},(|\hat Y|_{(n-1)}-b_{2n})/{a_{2n}}
,...,(|\hat Y|_{(n-s+1)}-b_{2n})/{a_{2n}})\stackrel{d}\rightarrow L_\gamma
$$
as $
n\to\infty$.
\end{theorem}
\vskip 0.2cm

\begin{remark}\label{theo:remark2}
\hyperref[theo:teorem2]{Theorem 3.2} implies that if $F\in {\cal G}_0\cap {\cal F}_{ls}$, $n\to\infty$, then for fixed $i$, $i=1,...,s$,
\begin{equation}
\PP\{(|\hat Y|_{(n-i+1)}-b_{2n})/a_{2n}\leq x\}\to
1-F_{\chi^2_{2i}}(2e^{-x}),
\end{equation}
and if $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma>0$,  then
\begin{equation}
\PP\{(|\hat Y|_{(n-i+1)}-b_{2n})/a_{2n}\leq x\}\to 1-F_{\chi^2_{2i}}(2/(1+x)).
\end{equation}
\end{remark}
Suppose now that the function $f_0$ is not symmetric. Set  $Y^*_i=-(X_i-\mu)/\sigma$. The c.d.f. and p.d.f. of $Y^*_i$ are $1-F_0(-x)$
 and $f_0(-x)$, respectively. Set \begin{equation}\label{eq:normal_seq_star}
  b^*_{n}=-F_0^{-1}(1/n), \quad a^*_{n}=1/({nf_0(-b^*_{n})}).
\end{equation}
 For example, if type I extreme value distribution is considered, then
 $$
  b_{n}=\ln\ln n, \quad a_{n}=\frac1{\ln n},\quad   b^*_{n}=-\ln(-\ln(1-\frac1n)), \quad a^*_{n}=-\frac1{(n-1)\ln(1-\frac1n)}.
 $$
 For the type II extreme value distribution $a_n,b_n,a_n^*,b_n^*$ have the same expressions as $a^*_n,b^*_n,a_n,b_n$
 for the Type I extreme value distribution, respectively.

 \begin{remark}\label{theo:remark3}
 Similarly as in \hyperref[theo:teorem1]{Theorem 3.1} we have that if $s$ is fixed and $F\in {\cal G}_0\cap {\cal F}_{ls}$, then
  for fixed $i$, $i=1,...,s$,
\begin{equation}
\PP\{( Y_{(i)}+b^*_{n})/(-a^*_{n})\leq x\}=\PP\{(\hat Y^*_{(n-i+1)}-b^*_{n})/a^*_{n}\leq x\}\to
1-F_{\chi^2_{2i}}(2e^{-x}),
\end{equation}
and if  $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma> 0$, then
  for fixed $i$, $i=1,...,s$,
\begin{equation}\label{normal_const_left}
\PP\{( Y_{(i)}+b^*_{n})/(-a^*_{n})\leq x\}=\PP\{(\hat Y^*_{(n-i+1)}-b^*_{n})/a^*_{n}\leq x\}\to
1-F_{\chi^2_{2i}}(2/(1+x)).
\end{equation}
\end{remark}
\subsection{Robust estimators for location-shape distributions}
The choice of the estimators $\hat\mu$ and $\hat \sigma$ is important when outlier detection problem is considered.
The ML estimators from the complete sample  are not stable when
 outliers exist.

In the case of location-scale families highly efficient robust estimators of the location and scale parameters $\mu$ and  $\sigma$ are (see \cite{rousseeuw_alternatives_1993})
\begin{equation}
\hat\mu=MED-\hat\sigma F_0^{-1}(0.5),\quad \hat\sigma=Q_n=d \,W_{([0.25 n(n-1)/2])},\label{ref:robust estimators}
\end{equation}
where $MED$ is the empirical median, $W_{ij}=|X_i-X_j|,\quad 1\leq i<j\leq n$ are $C_n^2=n(n-1)/2$ absolute values of the differences $X_i-X_j$ and $W_{(l)}$ is
the $l$th order statistic from $W_{ij}$.

The constant $d$ has the form
$d=1/K_0^{-1}(5/8)$,
where $K^{-1}_0(x)$ is the inverse of the c.d.f. of $Y_1-Y_2$, $Y_i=(X_i-\mu)/\sigma\sim F_0(x)$.

Expressions of $K^{-1}_0(x)$ and values $d$ for some well-known  location-scale families are given in Table \ref{tab:cn_dn}.
\vskip 0.2cm
\begin{table}
 \caption{Values of $d$  for various probability distributions}\label{tab:cn_dn}
\centering
\begin{tabular}{|c|c|c|c|}
  \hline

  Distribution & $K_0(x)$ &d \\
  \hline
  Normal &$\Phi(x/\sqrt{2})$&2.2219 \quad \\
  Type I extr.val. &$1/(1+e^{-x})$&1.9576   \\
  Type II extr.val.  \quad & $1/(1+e^{-x})$&1.9576  \\
  Logistic   &$ 1-\frac{(x-1)e^x+1}{(e^x-1)^2}$ &1.3079 \\
  Laplace  & $1-\frac12 (1+\frac{x}2)e^{-x}$&1.9306\\
  Cauchy   & $\frac12+\frac1{\pi}\arctan (x/2)$\quad &1.2071 \\
  \hline
\end{tabular}
\end{table}

The above considered estimators are equivariant under $H_0$, i.e.
 for any $e\in {\bf R}, f>0$, the following
equalities hold:
$$
\hat\mu((X_1-e)/f,\ldots,(X_n-e)/f)=(\hat\mu(X_1,\ldots,X_n)-e)/f,
$$$$
\hat\sigma((X_1-e)/f,\ldots,(X_n-e)/f)=\hat\sigma(X_1,\ldots,X_n)/f.
$$
Equivariant estimators have the following property: the distribution of $(\hat\mu-\mu)/\sigma$, $\hat\sigma/\sigma$ and $(\hat\mu-\mu)/\hat\sigma$ is parameter-free.

\subsection{Right outliers identification method for location-scale families}

Suppose that $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma\geq 0$.
 Let $a_n,b_n$ be defined by
 \eqref{eq:anbn}.
Set
$$
 U^+_{(n-i+1)}(n)=1-F_{\chi^2_{2i}}(2e^{-(\hat Y_{(n-i+1)}- b_n)/ a_n}),\;\gamma=0,$$
 $$ U^+_{(n-i+1)}(n)=1-F_{\chi^2_{2i}}(2/(1+(\hat Y_{(n-i+1)}- b_n)/ a_n),\;\gamma>0, $$
\begin{equation}
U^+(n,s)=\max_{1\leq i\leq s}U^+_{(n-i+1)}(n).
\end{equation}

\begin{theorem}
The distribution of the statistic $U^+(n,s)$ is parameter-free for any fixed $n$
\end{theorem}

\vskip 0.2cm

Denote by $u^+_\alpha(n,s)$ the $\alpha$ critical value of the
statistic $U^+(n,s)$.  Note that it is exact, not asymptotic $\alpha$
critical value: $\PP\{U^+(n,s)\geq u^+_\alpha(n,s)\}=\alpha$ under
$H_0$.

\hyperref[theo:teorem1]{Theorem 3.1} implies that
 the limit distribution (as $n\to\infty$) of  the random variable $U^+(n,s)$ coincides with  the distribution of the random variable
$V^+(s)=\max_{1\leq i\leq s}V^+_i,$ where $ V^+_i=1-F_{\chi^2_{2i}}(2(E_1+...+E_i))$, $E_1,...,E_s$ are i.i.d. standard exponential  random variables. The random variables $V^+_1,...,V^+_s$ are dependent identically distributed and the distribution of each $V^+_i$ is uniform: $V^+_i\sim U(0,1)$.

Denote by  $v^+_{\alpha}(s)$ the $\alpha$ critical values of the random variable $V^+(s)$. They are easily found by simulation  many times generating $s$ i.i.d. standard exponential random variables and computing values of the random variables $V^+(s)$.

Our simulations showed that the  below proposed outlier identification methods based on exact and approximate critical values of the statistic $U^+(n,s)$  give practically the same results, so
 for samples of size $n\geq 20$ we recommend to  approximate the $\alpha$-critical level of the statistic  $U^+(n,s)$ by
 the critical values $v^+_\alpha(s)$ which depend only on $s$. We shall see that for the purpose of outlier identification only the critical values $v^+_\alpha(5)$ are needed.  We found that the critical values $v^+_{\alpha}(5)$ are: $v^+_{0.1}(5)=0.9677$,  $v^+_{0.05}(5)=0.9853$,  $v^+_{0.01}(5)=0.9975$.

Our simulations showed that the performances of the below proposed outlier identification method based on exact and approximate critical values of the statistic $U^+(n,5)$  are similar for samples of size $n\geq 20$.

We write shortly BP-method for the below considered method.
\vskip 0.2cm
{\it BP  method for right outliers.} Begin outlier search using observations corresponding to the largest  values of $\hat Y_i$. We recommend begin with five  largest. So take $s=5$ and
compute  the values of the statistics $$U^+(n,5)=\max_{1\leq i\leq 5}U^{+}_{(n-i+1)}(n).$$

 If $U^+(n,5) \leq v^+_\alpha(5)$,  then it is concluded that outliers  are
absent and no further investigation is done. Under $H_0$ the probability of such event is approximately $1-\alpha$.

 If $U^+(n,5) > v^+_\alpha(5)$,  then it is concluded that outliers exist.

 Note that (see the classification scheme below) that if $U^+(n,5) > v^+_\alpha(5)$,  then minimum one observation is declared as an outlier. So the probability to declare  absence of outliers does not depend on the following classification scheme.

If it is concluded that outliers exist then search of outliers is done using the following steps.

{\bf Step 1.} Set $d_1=\max\{i\in\{1,...,5\}:U^+_{(n-i+1)}(n)> v^+_\alpha(5)\}$. Note that the maximum $d_1>0$ exists because $U^+(n,5) > v^+_\alpha(5)$.

  If $d_1< 5$, then classification is finished at this step:  $d_1$ observations  are declared as right outliers because if
the value of  $X_{(n-d_1)}$ is declared as an outlier, then it is natural  to declare values of $X_{(n)},...,X_{(n-d_1+1)}$  as  outliers, too.

 If $d_1=5$,  then it is possible that the number of outliers is higher than $5$.

 Then the observation corresponding to $i=1$ (i.e corresponding to $X_{(n)}$) is declared as an outlier and we proceed to the step 2.

 {\bf Step 2.} The above written procedure is repeated
 taking $U^+(n-1,5)=\max_{1\leq i\leq 5}U^+_{(n-i)}(n-1)$ instead of $U^+(n,5)$; here
$$
U^+_{(n-i)}(n-1)=1-F_{\chi^2_{2i}}(2e^{-(\hat Y_{(n-i)}- b_{n-1})/ a_{n-1}}),\quad i=1,...,5,
$$

 Set $d_2=\max\{i\in\{1,...,5\}:U^+_{(n-i)}(n-1)> v^+_\alpha(5)\}$.  If  $d_2<5$, the classification is finished  and $d_2 + 1$ observations are declared as  outliers.

 If $d_2=5$,  then it is possible that the number of  outliers is higher than $6$.
 Then the observation corresponding to the largest  $\hat Y_{(n-1)}$ is declared as an outlier, in total $2$ observations (i.e.  the observations corresponding to $i=1,2$ (i.e corresponding to $X_{(n)}$ and $X_{(n-1)}$) are declared as  outliers and we proceed to the step 3. And so on.
 Classification  finishes at the $l$th step when $d_l<5$. So we declare $(l-1)$  outliers in the previous steps and $d_l$  outliers in the last one. The total number of observations declared as  outliers is $l-1+d_l$.
 These observations are values of $X_{(n)},...,X_{(n-d_l-l+2)}$.

\subsection{Left outliers identification method for location-scale families}
Let $a_n^*,b_n^*$ be the normalizing constants defined by \eqref{eq:normal_seq_star}. If $F\in {\cal G}_0\cap {\cal F}_{ls}$,  $i=1,...,s$, then  set
$$
  U^-_{(i)}(n)=1-F_{\chi^2_{2i}}(2e^{(\hat Y_{(i)}+ b^*_n)/ a^*_n}),\quad
 U^-(n,s)=\max_{1\leq i\leq s}U^-_{(i)}(n).
$$
If $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma> 0$, then replace $e^{(\hat Y_{(i)}+ b^*_n)/ a^*_n}$ by $1/{(1+(\hat Y_{(i)}+ b^*_n)/ a^*_n})$.
Denote by $u^{-}_\alpha(n,s)$   the $\alpha$ critical value of the
statistic $U^-(n,s)$.

\hyperref[theo:teorem1]{Theorem 3.1}  and \hyperref[theo:remark3]{Remark 3}  imply that
 the limit distribution (as $n\to\infty$) of  the random variable $U^-(n,s)$ coincides with  the distribution of the random variable
$V^+(s)$. So the critical values $u^{-}_\alpha(n,s)$ are approximated by the critical values  $v^-_\alpha(s)=v^+_{\alpha}(s)$.

The left outliers search method coincides with the right outliers search method replacing $+$ to $-$ in all formulas.
\subsection{Outlier detection tests for location-scale families: two-sided alternative, symmetric distributions}

Let $a_n,b_n$ be defined by \eqref{eq:anbn}.
If $F\in {\cal G}_0\cap {\cal F}_{ls}$,  $i=1,...,s$, then  set
 $$U_{(n-i+1)}(n)=1-F_{\chi^2_{2i}}(2e^{-(|\hat Y|_{(n-i+1)}- b_{2n})/ a_{2n}}),
 \quad U(n,s)=\max_{1\leq i\leq s}U_{(n-i+1)}(n)
.$$
If $F\in {\cal G}_\gamma\cap {\cal F}_{ls}$, $\gamma> 0$, then replace $e^{(\hat Y_{(i)}+ b^*_n)/ a^*_n}$ by $1/{(1+(\hat Y_{(i)}+ b^*_n)/ a^*_n})$. Denote by $u_\alpha(n,s)$ the $\alpha$ critical value of the
statistic $U(n,s)$.

\hyperref[theo:teorem1]{Theorem 3.1} and \hyperref[theo:remark2]{Remark 2} imply that
 the limit distribution (as $n\to\infty$) of  the random variable $U(n,s)$ coincides with  the distribution of the random variable
$V^+(s)$. So the critical values $u_\alpha(n,s)$ are approximated by the critical values  $v_\alpha(s)=v^+_{\alpha}(s)$.

The outliers search method coincides with the right outliers search method skipping upper index $+$  in all formulas.

 \subsection{Outlier detection tests for location-scale families: two-sided alternative, non-symmetric distributions}

Suppose now that the function $f_0$ is not symmetric.
  Let $a_n,b_n,a^*_n,b^*_n$ be defined by \eqref{eq:normal_seq_star}.

Begin outlier search using observations corresponding to the largest and the smallest values of $\hat Y_i$.
We recommend begin with five smallest and
five largest. So
compute  the values of the statistics $U^-(n,5)$ and
$U^+(n,5)$.
 If $U^-(n,5) \leq v_{\alpha/2}(5)$ and $U^+(n,5) \leq v_{\alpha/2}(5)$,  then it is concluded that outliers  are
absent and no further investigation is done.

 If $U^-(n,5) > v_{\alpha/2}(5)$ or $U^-(n,5) > v_{\alpha/2}(5)$,  then it is concluded that outliers exist.
 If $U^-(n,5) > v_{\alpha/2}(5)$, then left outliers are searched as in Section 3.3. If $U^+(n,5) > v_{\alpha/2}(5)$, then right outliers are searched as in Section  3.2. The only difference is that $\alpha$ is replaced by $\alpha/2$ in all formulas.

\subsection{Outlier identification method for shape-scale families}
If shape-scale families of the form $\{F(t;\theta,\nu)=G_0((t/\theta)^\nu),\;\theta,\nu> 0\}$ with
specified $G_0$ are considered then the above given tests for location-scale families could be used because
if $X_1,...,X_n$ is a sample from shape scale family then $Z_1,...,Z_n$, $Z_i=\ln X_i$, is a sample from location-scale
family $\{F_0((x-\mu)/\sigma,\mu\in {\bf R}, \sigma>0\})$ with $\mu=\ln \theta$, $\sigma=1/\nu$, $F_0(x)=G_0(e^x)$.

\subsection{Illustrative example}
To illustrate simplicity of the  BP-method, let us consider an illustrative example of its application (sample size  $n=20$,
$r=7$ outliers).  The sample of size $n=20$ from standard normal distribution  was generated. The 1st-3rd and 17th-20th observations were  replaced by outliers.  The observations $x_i$, the absolute values $|\hat{Y}_i|$ of the z-scores $\hat{Y}_i$, and the ranks $(i)$ of  $|\hat{Y}_i|$ are presented in Table \ref{table:example1}.

\begin{table}[t]
\caption{Illustrative sample ($n=20, r=7$)}\label{table:example1}
\centering
\begin{tabular}{|rrrrr|rrrr|}
  \hline
 & i & $x_i$ & $|\hat{Y}_i|$ & (i) & i & $x_i$ & $|\hat{Y}_i|$ & (i) \\
  \hline
 & 1 & 6.10 & 3.18 & 16 & 11 & -0.69 & 0.28 & 9 \\
 & 2 & 10 & 5.17 & 18 & 12 & -0 & 0.07 & 5 \\
 & 3 & 6.20 & 3.23 & 17 & 13 & 0.05 & 0.10 & 6 \\
 & 4 & -0.08 & 0.03 & 2 & 14 & -0.20 & 0.03 & 1 \\
 & 5 & 0.63 & 0.39 & 11 & 15 & -0.25 & 0.06 & 4 \\
 & 6 & -0.54 & 0.21 & 7 & 16 & -0.64 & 0.25 & 8 \\
 & 7 & 1.37 & 0.77 & 13 & 17 & -6.30 & 3.14 & 15 \\
 & 8 & 0.46 & 0.30 & 10 & 18 & -5.50 & 2.73 & 14 \\
 & 9 & -0.22 & 0.04 & 3 & 19 & -12.10 & 6.10 & 19 \\
 & 10 & 0.94 & 0.55 & 12 & 20 & -20 & 10.13 & 20 \\
   \hline
\end{tabular}
\end{table}

\begin{table}[t]
\caption{Illustrative example of BP test observations classification.}\label{table:example11}
\centering
\begin{tabular}{|rrrrrr|r|}
  \hline
 & $U_{(20)}(20)$ & $U_{(19)}(20)$ & $U_{(18)}(20)$ & $U_{(17)}(20)$ & $U_{(16)}(20)$ & $U(20, 5)$\\
 & \bf  1.000000 & 1.000000 & 1.000000 & 0.999998 & 1.000000  & 1.000000\\ \hline
  & $U_{(19)}(19)$ & $U_{(18)}(19)$ & $U_{(17)}(19)$ & $U_{(16)}(19)$ & $U_{(15)}(19)$ & $U(19, 5)$\\
  & \bf 0.999685 & 0.999998 & 0.999916 & 0.999998 & 1.000000 & 1.000000\\  \hline
    & $U_{(18)}(18)$ & $U_{(17)}(18)$ & $U_{(16)}(18)$ & $U_{(15)}(18)$ & $U_{(14)}(18)$ & $U(18, 5)$\\
  & \bf 0.998046 & 0.996970 & 0.999893 & 0.999997 & 0.999997 & 0.999997\\ \hline
    & $U_{(17)}(17)$ & $U_{(16)}(17)$ & $U_{(15)}(17)$ & $U_{(14)}(17)$ & $U_{(13)}(17)$ & $U(17, 5)$\\
  &\bf 0.924219 &\bf  0.996446 &\bf  0.999871 &\bf  0.999940 & 0.084290 & 0.999940\\ \hline
   \hline
\end{tabular}
\end{table}

In Table \ref{table:example11} we present steps of the classification procedure by the BP method. First, we compute (see line 1 of Table \ref{table:example11})  value of the statistic\footnote{In fact the value of the statistic is 0.999999999.  It is rounded to 1.} $U(20, 5)= \max_{1\leq i\leq 5}U_{(20-i+1)}(20)= 1$. Since $U(20, 5)=1 > 0.9853 = v_{0.05}(5)$,  we reject the null hypothesis, conclude that outliers exist and begin the search of outliers.

{\it Step 1}.  The inequality  $U_{(16)}(20) = 1.0000 > 0.9853 = v_{0.05}(5)$ (note that $U_{(16)}(20)$ corresponds to the fifth largest observation in absolute value) implies that  $d_1=5$. So it is possible that the number of outliers might be greater than 5. We reject the largest in absolute value 20th observation as an outlier and continue the search of outliers.

{\it Step 2.} The inequality $ U_{(15)}(19) = 1.0000 > 0.9853 = v_{0.05}(5)$ (note that $U_{(15)}(19)$ corresponds to the fifth largest observation in absolute value from the remaining 19 observations) implies that $d_2=5$. So it is possible that the number of outliers might be greater than 6. We declare the second largest in absolute value  observation as an outlier. So two observations (19th and 20th) are declared as outliers. We continue the search of outliers.

 {\it Step 3.}  The inequality  $U_{(14)}(18) = 0.999997 > 0.9853 = v_{0.05}(5)$ implies that $d_3=5$.  We declare the third largest in absolute value  observation as an outlier. So three observations (2nd, 19th and 20th) are declared as outliers. We continue the search of outliers.

 {\it Step 4.} The inequalities $U_{(13)}(17) = 0.084290 < 0.9853 = v_{0.05}(5)$ and
$U_{(14)}(17) = 0.999940 > 0.9853 = v_{0.05}(5)$ imply that $d_4 = 4$. So four
additional observations (the fourth, fifth, sixth and seventh largest in absolute value observations), namely the 3d, 1st, 17th, and 7th are declared as outliers, The outlier search is finished. In all,   7 observations were declared as outliers: 1-3,17-20, as was expected. Note that  since the outlier search procedure was done after rejection of the null hypothesis, the significance level did not change.

We created R package \href{https://github.com/linas-p/outliersTests}{outliersTests}\footnote{
The R package outliersTest package can be accessed:
\href{https://github.com/linas-p/outliersTests}{https://github.com/linas-p/outliersTests}
} to be able to use the proposed BP test in practice within R package.

\section{Generalization of Davies-Gather outlier identification method }
\label{sec:davies}

Let us consider location-scale families.
 Following the idea of Davies-Gather \cite{davies_identification_1993}  define an empirical analogue
of the right outlier region  as a random region
\begin{equation}
OR_r(\alpha_n)=\{x:\,x>\hat\mu+\hat\sigma g_{n.\alpha}\},
\end{equation} where $g_{n.\alpha}$ is found using the condition
\begin{equation}\label{eq:g radimas}
\PP\{X_i\bar{\in}OR_r(\alpha_n),i=1,\ldots,n|H_0\}=1-\alpha,
\end{equation}
 and $\hat\mu,\hat\sigma$ are robust equivariant
estimators of the parameters $\mu,\sigma$.

Set $$
\hat Y_{(n)}=({X}_{(n)}-\hat\mu)/\hat\sigma.$$
The distribution of $\hat Y_{(n)}$ is parameter-free under $H_0$.

The equation \eqref{eq:g radimas} is equivalent to the equation
equation
$$
\PP\{\hat Y_{(n)}\leq g_{n,\alpha}\}|H_0\}=1-\alpha.
$$
So $g_{n,\alpha}$ is the upper
$\alpha$ critical value of the random variable $\hat Y_{(n)}$. It is easily
computed  by simulation.
\vskip 0.2cm
\noindent {\it Generalized Davies-Gather method  for right outliers identification:} if $\hat Y_{(n)}\leq g_{n,\alpha}$, then it is concluded that right outliers are absent. The probability of such event is $\alpha$.
 If $\hat Y_{(n)}>g_{n,\alpha}$, then it is concluded that right outliers exist.
 The value $x_i$ of the
random variable $X_i$ is admitted as an outlier if $x_i\in
OR_r(\alpha_n)$, i.e. if
$x_i>\hat\mu+\hat\sigma g_{n,\alpha}$. Otherwise it is admitted as a
non-outlier. \vskip 0.2cm

An empirical analogue
of the left outlier region  as a random region
\begin{equation}
OR_l(\alpha_n)=\{x:\,x<\hat\mu+\hat\sigma h_{n.1-\alpha}\},
\end{equation} where $h_{n.1-\alpha}$ is found using the condition
\begin{equation}\label{eq:g radimas 2}
\PP\{X_i\bar{\in}OR_l(\alpha_n),i=1,\ldots,n|H_0\}=1-\alpha,
\end{equation}
Set $$
\hat Y_{(1)}=({X}_{(1)}-\hat\mu)/\hat\sigma.$$
The distribution of $\hat Y_{(1)}$ is parameter-free under $H_0$.

The equation \eqref{eq:g radimas 2} is equivalent to the equation
equation
$$
\PP\{\hat Y_{(1)}\geq h_{n,1-\alpha}|H_0\}=1-\alpha.
$$
So $h_{n,\alpha}$ is the upper
$1-\alpha$ critical value of the random variable $\hat Y_{(1)}$. It is easily
computed  by simulation.
\vskip 0.2cm
\noindent {\it Generalized Davies-Gather method  for left outliers identification:} if $\hat Y_{(1)}\geq h_{n,1-\alpha}$, then it is concluded that left outliers are absent. The probability of such event is $\alpha$.
 If $\hat Y_{(1)}<h_{n,\alpha}$, then it is concluded that left outliers exist.
 The value $x_i$ of the
random variable $X_i$ is admitted as an outlier if $x_i\in
OR_l(\alpha_n)$, i.e. if
$x_i<\hat\mu+\hat\sigma h_{n,1-\alpha}$. Otherwise it is admitted as a
non-outlier. \vskip 0.2cm

Let us consider two-sided case.

If the distribution of $X_i$ is symmetric,
 then the empirical analogue
of the  outlier region  is the random region
\begin{equation}
OR(\alpha_n)=\{x:|x-\hat\mu|>\hat\sigma g_{n.\alpha/2}\}.
\end{equation}
In this case
$$1-\alpha=\PP\{X_i\in OR(\alpha_n),i=1,...,n|H_0\}=\PP\{|\hat Y|_{(n)}\leq g_{n.\alpha/2}\}.
$$
{\it Generalized Davies-Gather method for left and right outliers identification (symmetric distributions):} if
$|\hat Y|_{(n)}\leq g_{n.\alpha/2}$, then it is concluded that  outliers are absent. The probability of such event is $\alpha$.
 If $|\hat Y|_{(n)}> g_{n.\alpha/2}$, then it is concluded that  outliers exist.
 The value $x_i$ of the
random variable $X_i$ is admitted as a left outlier if $x_i<\hat\mu-\hat\sigma g_{n,\alpha/2}$, it is admitted as a right outlier if
$x_i>\hat\mu+\hat\sigma g_{n,\alpha/2}$. Otherwise it is admitted as a
non-outlier.

If distribution of $X_i$ is non-symmetric, then   the empirical analogue
of the  outlier region  is defined as follows:
\begin{equation*}
OR(\alpha_n)=\{x\in {\bf R}/[\hat\mu+\hat\sigma g_{n,1-\alpha/2},\hat\mu+\hat\sigma g_{n,\alpha/2})]\},
\end{equation*}
In this case
$$
    1-\alpha=\PP\{X_i{\in}[\hat\mu+\hat\sigma h_{n,1-\alpha/2},\,\hat\mu+\hat\sigma g_{n,\alpha/2}],i=1,\ldots,n|H_0\}=
$$$$\PP\{h_{n,1-\alpha/2}\leq \hat Y_{(1)}\leq \hat Y_{(n)}\leq g_{n,\alpha/2})|H_0\}.
$$
\vskip 0.2cm
{\it Generalized Davies-Gather method for left and right outliers identification (non-symmetric distributions):} if $\hat Y_{(1)}\geq h_{n,1-\alpha/2}$
and $\hat Y_{(n)}\leq g_{n,\alpha/2}$, then it is concluded that  outliers are absent. The probability of such event is $\alpha$.
 If $\hat Y_{(1)}< h_{n,1-\alpha/2}$ or $\hat Y_{(n)}> g_{n,\alpha/2}$, then it is concluded that  outliers exist.
 The value $x_i$ of the
random variable $X_i$ is admitted as a left outlier if $x_i<\hat\mu+\hat\sigma h_{n,1-\alpha/2}$, it is admitted as a right outlier if
$x_i>\hat\mu+\hat\sigma g_{n,\alpha/2}$. Otherwise it is admitted as a
non-outlier.

\section{Short survey of multiple outlier identification methods for normal data   }
\label{sec:compare}
\subsection{Rosner's method}  Let us formulate Rosner's method
in the form mostly used in practice. Suppose that the number of outliers does not exceed $s$ and
the two-sided alternative  is considered.  Set (see \cite{rosner_detection_1975,rosner1977percentage})
\begin{eqnarray*}
R_{1} =\max_{1\leqslant j \leqslant n} |\tilde Y_j| =
\max_{1\leqslant j \leqslant n} {|X_j - \bar{X}| }/{S_X}, \ \
\ S_X^2 = \sum_{j=1}^{n}(X_{(j)} - \bar{X})^2/{(n-1)}.
\end{eqnarray*}
$|\tilde Y_j|=|(X_j-\bar X)/S_X|$ may be interpreted as a distance
between $X_j$ and $\bar X$. Remove the observation $X_{j_1}$ which
is most distant from $\bar X$. This maximal  distance is $R_{1}$.
The value of $X_{j_1}$ is a possible candidate for contaminant.

Recompute the statistic using $n-1$ remaining observations and
denote by $R_{2}$ the obtained statistic. Remove the observation
$X_{j_2}$ which is most distant from the new empirical mean. The
value of $X_{j_2}$ is also possible candidate for contaminant.
Repeat the procedure until the statistics  $R_{1}, \cdots, R_{s}$
are computed. So we obtain all possible candidates for
contaminants. They are values of $X_{j_1},\ldots,X_{j_s}$

Fix $\alpha$ and find $\lambda_{in}$ such that
\begin{equation*}
 \PP \{ R_{1} > \lambda_{in}  | H_0\}=...= \PP \{ R_{s} > \lambda_{in}  | H_0\},\quad \PP \{ \cup_{i=1}^{s} \{R_{i} > \lambda_{in} \} | H_0\} = \alpha.
\end{equation*}
If $n>25$, then the  approximations
\begin{eqnarray*}
\lambda_{in}\approx t_{\frac{\alpha}{2(
n-i-1)}}(n-i+1)\sqrt{\frac{n-i }{n-i-1+t^2_{\frac{\alpha}{2(
n-i-1)}}(n-i+1)}}\sqrt{1-\frac{1}{n-i+1}},
\end{eqnarray*}
are recommended  (see \cite{rosner_detection_1975}); here $t_{p}(\nu)$ is the $p$ critical value of the Student
distribution with $\nu$ degrees of freedom.
\vskip 0.2cm
 {\it Rosner's method for left and right outliers identification}: if $R_{i}\leqslant \lambda_{in}$ for all
$i=1,\cdots, s$, then it is concluded that outliers are
absent.
If there exists $i_0\in
\{1,\ldots,s\}$ such that $R_{i_0}> \lambda_{i_0n}$, i.e. the
event $ \cup_{i=1}^{s} \{R_{i} > \lambda_{in} \}$ occurs, then it is concluded that outliers exist. In this case
classification of observations to outliers and non-outliers is done in the following way: if $R_{s}
> \lambda_{sn}$, then it is concluded that there are $s$ outliers and they are values of  $X_{j_1},\ldots,X_{j_s}$. If
$R_j \leqslant \lambda_{jn}$ for $j=s,s-1,\ldots,i+1$, and $R_{i} >
\lambda_{in}$,  then it is concluded that there are $i$
outliers and
 they are values of   $X_{j_1},\ldots,X_{j_i}$.

If   right outliers are searched, then define
 $R^+_{1} =\max_{1\leqslant i \leqslant n}\tilde Y_i,
$ and repeat the above procedure taking approximations
\begin{eqnarray*}
\lambda^+_{in} \approx t_{\frac{\alpha}{
n-i-1}}(n-i+1)\sqrt{\frac{n-i }{n-i-1+t^2_{\frac{\alpha}{
n-i-1}}(n-i+1)}}\sqrt{1-\frac{1}{n-i+1}}.
\end{eqnarray*}

\begin{figure}[t]
  \centering
  \includegraphics[width=0.7\linewidth]{figure1
    \caption{The true values of the significance level of Rosner's and BP tests in function of $n$ for different values of $s$ ($\alpha=0.05$ is used in approximations).
	}\label{fig:hrejectionv2}

\end{figure}

Denote by $R_s$ the Rosner's test with a fixed upper limit  $s$.
Our simulation results confirm  that  the true significance level is different from the level $\alpha$ suggested by the approximation  when $n$ is not large. Nevertheless, it is approaching
$\alpha$ as $n$ increases, see    Figure \ref{fig:hrejectionv2}. The true significance value of the BP test, which uses asymptotic values of the test statistic are
also presented in Figure \ref{fig:hrejectionv2}.

\subsection{Bolshev's method}

Suppose that the number of contaminants does not
exceed $s$. For $i = 1,\cdots,n$ set
$$
\hat Y_i=(X_i-\bar X)/s,\quad \tau_i^+= n\cdot (1-T_{n-2}(\hat Y_i)) ,\quad \tau_i= n\cdot (1-T_{n-2}(|\hat Y_i|))
$$
where $\bar X$ and $s$ are the empirical mean and standard deviation, $T_{n-2}(x)$ is the c.d.f. of Thompson's distribution with $n-2$ degrees of freedom.

 Let us consider search for right outliers. Note that
 the
largest  $s$ observations $X_{(n-s+1)},...,X_{(n)}$ define the
smallest $s$ order statistics $\tau^+_{(1)}\leq...\leq
\tau^+_{(n)}$. Possible candidates for outliers are namely the
values of  $X_{(n-s+1)},...,X_{(n)}$.

Set $\tau^+ = \min_{1\leqslant i \leqslant s}{\tau_{(i)}^+}/{i}.$
\vskip 0.2cm \noindent {\it Bolshev's method for right outliers search}. If $\tau^+ \geq
\tau_{1-\alpha}^+(n,s)$, then it is concluded that outliers are
absent;
 here
$\tau_{1-\alpha}^+(n,s)$ is the $1-\alpha$ critical value of the
test statistic under $H_0$. If $\tau^+ < \tau_{1-\alpha}^+(n,s)$, then it is concluded that outliers exist.
In such a case outliers  are
selected in the following way: if  ${\tau^+ _i}/{i} <
\tau_{1-\alpha}^+(n,s)$ then the value of the order statistic
$X_{(n-i+1)}$ is  admitted as an outlier, $i=1,...,s$. \vskip
0.2cm

In the case of left and right outliers search  Bolshev's method uses
$\tau_{(i)}$ instead of $\tau^+_{(i)}$, defining the statistic
 $ \tau = \min_{1\leqslant i \leqslant
s}{\tau_{(i)}}/{i}.$
\vskip 0.2cm \noindent {\it Bolshev's method for left and right outliers search}. If $\tau \geq
\tau_{1-\alpha}(n,s)$, then it is concluded that outliers are
absent;
 here
$\tau_{1-\alpha}(n,s)$ is the $1-\alpha$ critical value of the
 statistic $\tau$ under $H_0$.  If $\tau < \tau_{1-\alpha}(n,s)$, then it is concluded that outliers
exist. In such a case they are selected in
the following way: if  ${\tau _i}/{i} < \tau_{1-\alpha}(n,s)$
then the observation corresponding to $\tau_i$ is  admitted
as an outlier, $i=1,...,s$. \vskip 0.2cm

\subsection{Hawking's method }

Suppose that the number of contaminants does not
exceed $s$. Let us consider the search for right outliers.  For $k=1,...,s$ set
 \begin{equation*}
b^+_{k} = \frac{1}{\sqrt{k(n-k)}}\sum_{i=1}^k \tilde Y_{(n-i+1)}=
\frac{1}{\sqrt{k(n-k)}}\sum_{i=1}^k (X_{(n-i+1)}-
\bar{X})/S_X.
\end{equation*}
$b^+_{k}$ proportional to the sum of $k$ largest $\tilde Y_{(n-i+1)}$.  Set $B^+= \max_{1\leqslant k \leqslant s } b^+_{k}.$
\vskip 0.2cm \noindent {\it Hawking's method}. If $B^+ \leq
B^+_{\alpha}(n,s)$ then it is concluded that outliers are absent;
 here
$B^+_{\alpha}(n,s)$ is the $\alpha$ critical value of the
statistic under $H_0$. If
$B^+ > B^+_{\alpha}(n,s)$, then it is concluded that outliers exist. In such a case  outliers are selected in the
following way: if  $b^+_{i} > B^+_{\alpha}(n,s)$, then the value of
the order statistic $X_{(n-i+1)}$ is  admitted as an outlier,
$i=1,...,s$.

\section{Comparative analysis of outlier identification methods by simulation}
\label{sec:simul}

In the case of location-scale classes  probability distribution of all considered test statistics does not depend on $\mu$ and $\sigma$, so we generated samples of various sizes $n$ with $n-r$
observations having the c.d.f. $F_0$  and  $r$ observations
having  various alternative distributions concentrated in the outlier region. We shall call such observations "contaminant outliers", shortly
$c-outliers$. As was mentioned, outliers which are not c-outliers, i.e. outliers from regular observations with the c.d.f. $F_0$,  are very rare.

 We repeated simulations $M=100000$ times and  using various methods we classified
observations to outliers and non-outliers and computed the mean number $ D_{O_cO}$ of correctly identified
c-outliers, the  mean number $ D_{ON}$ of  c-outliers which were not
 identified,   the mean number $ D_{NO}$ of  non c-outliers
admitted as outliers, and  the  mean number $ D_{NN}$ of non c-outliers admitted as
non-outliers.

An outlier identification method is ideal if each outlier is detected and each non-outlier is declared as a  non-outlier.
In practice it is impossible to do with the probability one. Two errors are possible: a) an outlier is not declared as such (masking effect); b)  a non-outlier is
declared as an outlier (swamping effect). We shall write shortly "masking value" for the mean number of non-detected c-outliers  and
"swamping value" for the mean number of "normal" observations declared as outliers in the simulated samples.

If  swamping is small for two tests  then  a test with smaller masking effect should be
preferred because in this case the distribution of the data remaining after excluding of suspected outliers should be closer
to the distribution of non-outlier data.

From the other side, if swamping for  Method 1 is considerably bigger
 than swamping of  Method 2 and  masking is smaller for Method 1, then  it does not
mean that Method 1 is better because this method rejects many extreme
non-outliers from the tails of the regular distribution $F_0$ and the sample remaining after
classification may be not treated as a sample from this regular
distribution even if all c-outliers are eliminated.

For various  families of distributions, sample sizes $n$, and alternatives we
 compared Davies-Gather (DG) and new (BP) methods performance. In the case of normal distribution
we also compared them with Rosner's, Bolshev's and Hawking's methods.

We used two different classes of alternatives:
in the first case c-outliers are spread widely in the outlier region around the mean, in the second case c-outliers are concentrated in a very
short interval laying in the outlier region. More precisely, if right outliers were searched, then
we simulated $r$ observations concentrated in in the right
outlier region $out_r(\alpha_n,F_0)=\{x:x>x_{\alpha_n}\}$ using the following alternative families of distribution:

1) Two parameter exponential distribution
$\mathcal{E}(\theta,x_{{\alpha_n}})$ with the scale parameter $\theta$.
 If $\theta$ is small, then outliers are concentrated near the border of the outlier region. If $\theta$  is large then  outliers are widely spread in the outlier region. If $\theta$ increases, then the mean of outlier distribution increases. Note that even if $\theta$ is very near 0 and the true number of outliers $r$ is large, these
 outliers may corrupt strongly the data making tails of histogram two heavy.

 2) Truncated normal distribution
$\mathcal{T}\mathcal{N}(x_{\alpha_n}, \mu, \rho)$ with the location and scale parameters $\mu,\rho$ ($\mu>x_{\alpha_n})$.  If $\rho$ is small then  this distribution is concentrated in a small interval around $\mu$. If $\mu$ increases, then the mean of outlier distribution increases.

For lack of place we present a small part of our investigations. Note that the results are very
similar for all sample sizes $n\geq 20$. Multiple outlier problem is not very relevant for smaller sample sizes.

\subsection{Investigation of outlier identification methods  for normal data}

We  use notation  $B,H, R, DG$, and $BP$ for the Bolshev's, Hawking's, Rosner's, Davies-Gather's, and the new methods, respectively. If $DG$ method is based on maximum likelihood estimators, then we write $DG_{ml}$ method, if it is based on robust estimators, we write $DH_{rob}$ method.

For comparison of above considered methods we fixed the significance level $\alpha=0.05$.  We remind that the significance level $\alpha$ is the probability to reject minimum one observation as an outlier under the hypothesis $H_0$ which means that all observations are realizations of i.i.d. having the same normal distribution. The only test, namely R method uses approximate critical values of the test statistic, so the significance values for this test is only approximately $0.05$ and depends on $s$ and $n$. In Figure 1 the true significance level value for $s=5,15$ and $[0.4 n]$ in function of $n$ are given.

The $B,H$,  and $R$ tests  methods have a drawback that the upper bound for the possible number of outliers $s$ must be fixed. The BP and DG tests have an advantage that they do not
require it.

Our investigations showed that H,B and $DG_{ml}$ methods have other serious drawbacks.
 So firstly let us look closer at these methods.

\begin{figure}[t]
    \centering
  \includegraphics[width=0.6\linewidth]{figure2}
  \caption{Hawkin's method: the values of $D_{NO} + D_{OO}$ in function of $\mu$ and $r$ ($n=100$, $s=5$).}\label{fig:hawkins1}

\end{figure}

\vskip 0.2cm

\begin{table}[ht]
\caption{Hawkin's method: the values of $D_{NO} + D_{OO}$ in function of $\mu$ and $r$ ($n=100$, $s=5$).}\label{tab:hawkins1}
\centering
\begin{tabular}{lllll}
  \hline
r $\backslash$ $\mu$ & 0.1 & 1 & 6.3 & 10 \\
  \hline
 $1$ & 0.31 + 0.00 & 0.66 + 0.00 & 3.93 + 1.00 & 3.99 + 1.00 \\
$2$ & 0.87 + 0.00 & 2.15 + 0.06 & 3.00 + 1.21 & 3.00 + 2.00 \\
 $3$& 1.33 + 0.08 & 1.99 + 0.84 & 2.00 + 2.00 & 2.00 + 2.00 \\
$4$ & 0.89 + 0.58 & 1.00 + 1.42 & 1.00 + 3.00 & 1.00 + 3.00 \\
$5$ & 0.01 + 1.15 & 0.00 + 2.03 & 0.00 + 3.02 & 0.00 + 3.96 \\
   \hline
\end{tabular}

\end{table}

If the true number of c-outliers $r$ exceeds $s$, then the B and H methods can not find them even if they are very far from the limits of the outlier region.
Nevertheless, suppose that $r$ does not exceed $s$ and look at the performance of the H method. Set $n=100$, $s=5$, and suppose that c-outliers are generated by
right-truncated normal distribution $\mathcal{T}\mathcal{N}(x_{\alpha_n}, \mu, \rho)$ with fixed $\rho$ and increasing $\mu$. Note that the true number of c-outliers is supposed to be unknown but do not exceed $s=5$. In Figure \ref{fig:hawkins1} the mean numbers of rejected  non-c-outliers $D_{NO}$ are given in function of the parameter $\mu$  (the value of the parameter $\rho=0.1^2$ is fixed) for fixed values of  $r$ see Figure \ref{fig:hawkins1}. In Table \ref{tab:hawkins1}  the values of $D_{NO}$ plus the values of the mean numbers of truly rejected c-outliers are given.   The Table \ref{tab:hawkins1} shows that if  $r=1$, then if $\mu$ is sufficiently large, the c-outlier is found but the number of rejected non-c-outliers $D_{NO}$ increases to 4, so swamping is very large. Similarly, if $r=2$, then  $D_{NO}$ increases to 3, so swamping is large. Beginning from $r=3$ not all c-outliers are found even for large $\mu$.  Swamping is smallest if the true value $r$  coincides with $s$ but even in this case one c-outlier is not found even for large $\mu$.
 Taking into account that the true number $r$ of c-outliers is not known in real data, the performance of the H methos is very poor. Results are similar for other  values of $n$, $s$, and  distributions of c-outliers. As a rule, H mehod finds rather well the c-outliers but  swamping is very large because this method has a tendency to reject a number near $s$ of observations for remote alternatives.  which is good if $r=s$ but is bad if $r$ is different from $s$.

The B and $DG_{ml}$ tests have a drawback that they use maximum likelihood estimators which are not robust and estimate parameters badly in presence of outliers.
Once more, set $n=100$, $s=5$, and suppose that c-outliers are generated by
two-parameters exponential distribution $\mathcal{T}\mathcal{E}(x_{\alpha_n}, \theta)$ with increasing $\theta$. Swamping values are negligible in, so only masking values( mean numbers of non-rejected  c-outliers $D_{ON}$) are important. In  Figure  3 the masking values  in function of the parameter $\theta$  are given  for fixed values of  $r$.

Both methods perform very similarly. The  masking values are   large for every value of $r>1$. If $r$ increases, then masking values increase, too. For example, if $r=5$, then almost 3 c-outliers from 5  are not rejected in average even for large values of $\theta$.

Similar results hold taking other values of $n$, $s$ and various distributions of c-outliers.

The above analysis shows that the  B, H, $DG_{ml}$ methods have serious drawbacks,  so we exclude these methods from further consideration.

\begin{figure}[t]

\centering
  \includegraphics[width=0.5\linewidth]{figure3}

	\caption{The number of outliers rejected as non-outliers ($D_{ON}$). The alternative: two-sided, the outliers generated by two-parameters exponential distribution on both sides. }\label{fig:dgb}
\end{figure}

\vskip 0.2cm

Let us consider  the remaining three methods: R, DG, and BP. For small $n$ the true significance level of Rosner's test differ considerably from the
 suggested, so we present comparisons of tests performance for $n=50,100,1000$ (see Tables \ref{tab:norm50}-\ref{tab:norm1000}). Truncated exponential distribution was used for outliers simulation. Remoteness
 of the mean of outliers from the border of the outlier region is characterized by the parameter $\theta$.

  \begin{table}[h]
	\caption{The masking values $D_{ON}$ ($n=50$ and $n=100$).}\label{tab:norm50}
	\centering
\begin{tabular}{|l|l|lllll|}
  \hline
  & $n = 50$ & &&&& \\
  \hline
r&Method$\backslash\theta$ & 0.1 & 0.4 & 1 & 4 & 10 \\
  \hline
2&Rosner\_5 & 1.36 & 0.95 & 0.51 & 0.15& 0.06 \\
&  Rosner\_15 & 1.36 & 0.95 & 0.51 & 0.15& 0.06 \\
&  Rosner\_[0.4n] & 1.36 & 0.95 & 0.51 & 0.15& 0.06 \\
&  DGr\_rob & 1.56 & 1.17 & 0.71 & 0.24 & 0.10 \\
&  BP & 0.92 & 0.66 & 0.37& 0.10 & 0.04 \\
   \hline
5&Rosner\_5 & 3.79 & 3.31 & 2.11 & 0.48 & 0.16 \\
&  Rosner\_15 & 3.66 & 3.21 &2.04 & 0.46 & 0.16 \\
&  Rosner\_[0.4n] & 3.66 & 3.21 &2.04 & 0.46 & 0.16 \\
&  DG\_rob &4.70 & 4.10& 2.90 & 1.09 & 0.48 \\
&  BP & 2.00 & 1.68 & 1.18 & 0.40 & 0.15 \\
   \hline
8&Rosner\_5 & 8.00& 7.97 & 7.54 & 3.70 & 3.06 \\
&  Rosner\_15 &  5.70& 5.48 & 4.52 & 1.00 & 0.29 \\
&  Rosner\_[0.4n]& 5.70& 5.48 & 4.52 & 1.00 & 0.29 \\
&  DG\_rob & 7.90 & 7.49 & 6.10 & 2.67 & 1.24 \\
&  BP & 4.27 & 3.84 & 3.25 & 1.47 & 0.57 \\
   \hline
\end{tabular}\begin{tabular}{|l|lllll|}
  \hline
    & $n = 100$ & &&& \\
  \hline
r & 0.1 & 0.4 & 1 & 4 & 10 \\
  \hline
2 & 1.19 & 0.71 & 0.33 & 0.09 & 0.04 \\
 &  1.19 & 0.71 & 0.33 & 0.09 & 0.04 \\
 &  1.19 & 0.71 & 0.33 & 0.09 & 0.04 \\
& 1.31 & 0.84 & 0.44 & 0.13 & 0.06 \\
 & 0.50 & 0.32 & 0.15 & 0.04 & 0.02 \\
   \hline
5 & 3.52 & 2.57 & 1.27 & 0.27 & 0.10 \\
 & 3.43 & 2.52 & 1.24 & 0.26 & 0.10 \\
 & 3.43 & 2.52 & 1.24 & 0.26 & 0.10 \\
 & 4.23 & 3.01 & 1.81 & 0.57 & 0.25 \\
 & 0.78 & 0.60 & 0.43 & 0.15 & 0.07 \\
   \hline
10 & 10.0  & 9.90 & 8.21 & 5.10 & 5.00 \\
 & 6.88 & 6.54 & 4.36 & 0.69 & 0.22 \\
 & 6.88 & 6.54 & 4.36 & 0.69 & 0.22 \\
 & 9.74 & 8.38 & 5.78 & 2.12 & 0.92 \\
 & 2.21 & 1.90 & 1.73 & 0.74 & 0.30 \\
   \hline
\end{tabular}

\end{table}

\begin{table}[h]
	\caption{The masking values $D_{ON}$ ($n=1000$).}\label{tab:norm1000}
\centering
\begin{tabular}{|l|l|lllll|}
  \hline
r&Method$\backslash\theta$ & 0.1 & 0.4 & 1 & 4 & 1000 \\
  \hline
5&Rosner\_5 & 2.15 & 0.69 & 0.29 & 0.07 & 0.00 \\
 & Rosner\_15 & 2.12 & 0.66 & 0.27 & 0.07 & 0.00 \\
 & Rosner\_[0.4n] & 2.12 & 0.66 & 0.27 & 0.07 & 0.00 \\
 & DG\_rob & 1.99 & 0.78 & 0.35
 & 0.09 & 0.00 \\
 & BP &0.25 & 0.23 & 0.22 & 0.11 & 0.00 \\
   \hline
20&Rosner\_5 & 19.0 & 15.8 & 15.0 & 15.0 & 15.0 \\
 & Rosner\_15 & 19.2 & 10.9 & 5.52 & 5.00 & 5.00 \\
 & Rosner\_[0.4n] & 12.7 & 6.94 & 1.76 & 0.30 & 0.00 \\
&  DG\_rob & 14.8 & 6.97 & 3.32 & 1.93 & 0.00 \\
&  BP & 0.29 & 0.26 & 0.23 & 0.18  & 0.00 \\
   \hline
100&Rosner\_5 & 100 & 99.9 & 96.7 & 95.0 & 95.0 \\
 & Rosner\_15 & 100  & 99.92 &  96.4  & 85.0 & 85.0\\
 & Rosner\_[0.4n]  & 55.8 & 56.8 & 50.4 & 4.43 & 0.01 \\
 & DG
\_rob & 100 & 89.9 & 61.6 & 22.2 & 0.1 \\
 & BP & 4.72 & 4.00 & 3.95 &3.58 & 0.04 \\
   \hline
\end{tabular}

\end{table}

 Swamping values $D_{NO}$  (the mean numbers of non-c-outliers declared as outliers) are very small for all tests. For example, even if $n=1000$, the  R and DG methods reject in average  as outliers only  0.05 from $n-r=995, 980, 900$  non-c-outliers. For the BP method this number is $0.25, 0.19, 0.05$ from $995, 980$, and 900 non-c-outliers, respectively. So only masking values $D_{ON}$ (the mean numbers of c-outliers declared as non-outliers) are important for outlier identification methods comparison.

 Necessity to guess the upper limit $s$ for a possible number of outliers is considered as  a drawback of the Rosner's method. Indeed, if the
 true number of outliers $r$ is greater than the chosen upper limit $s$, then $r-s$ outliers are not identified with the probability one.  And
  even if $r\leq s$, it is  not clear how important is closeness of $r$ to $s$. So first we investigated the problem of the upper limit choice.

  Here we present masking values $D_{ON}$ of the  Rosner's tests for $s=5,15$ and $[0.4n]$. Similar results are obtained for other values of $s$.

  Our investigations show that it is sufficient to fix $s=[0.4n]$, which  is clearly larger than it can be expected in real data. Indeed, Tables \ref{tab:norm50}-\ref{tab:norm1000} show that for $r>s$ $Rosner_5$ and $Rosner_{15}$   do not find $r-s$ outliers
  even if they are very remote, as it should be. Nevertheless, we see that even if the true number of outliers $r$  is much smaller than $[0.4n]$,  for any  considered $n$, $r\leq s=5,15$ the masking values of the $Rosner_{[0.4n]}$ test are  approximately the same (even a little smaller) as  the masking values of  the tests $Rosner_5$ and $Rosner_{15}$,  for $r>s$  they  are clearly smaller.

  Hence, $s=[0.4n]$ should be recommended for Rosner's test application, and performance of $Rosner_{[0.4]}$, Davies-Gather robust ($DG_{rob})$ and the proposed $BP$ methods should be compared.

  All three methods find all c-outliers if they are sufficiently remote.
  For $n=50$ the BP method gives uniformly smallest masking values and the DG method gives uniformly largest masking values for any considered $r$ in all diapason of alternatives. For $n=100$ and $r=2,5$ the result is the same. For
  $n=100$ and $r=10$ (it means that even for very small $\theta$ the data is seriously corrupted) the BP method is also the best except that for the  most remote alternatives the $Rosner_{[0.4n]}$ method slightly outperforms the BP method. For $n=1000$ and the most of alternatives the BP method strongly outperforms other methods, except the most remote alternatives.

  The DG and Rosner's methods have very large masking if many outliers are concentrated near the outlier region border. In this case data is seriously corrupted, however,
  these methods do not see outliers.

Conclusion: in most considered situations  the BP method is the best outlier identification method. The second is Rosner's method with $s=[0.4]$, and the third is the Davies-Gather method based on robust estimation. Other methods have poor performance.

\subsection{Investigation of outlier identification methods  for other location-scale models}

We investigated performance of the new method for location-scale families different from normal. We compare
the BP method with the generalized Davies-Gather method for logistic, Laplace (symmetric, $F\in {\cal G}_0\cap {\cal F}_{ls}$), extreme values
(non-symmetric $F\in {\cal G}_0\cap {\cal F}_{ls}$), and Cauchy (symmetric, $F\in {\cal G}_1\cap {\cal F}_{ls}$) families. C-outliers were generating using truncated
exponential distribution concentrated in two-sided outlier region.  Swamping values being small, masking value, see Table \ref{tab:other} and differences between the true number of c-outliers and the number of rejected observations, see Figures
\ref{fig:logist}-\ref{fig:cauchy}, were compared. The BP and $DG_{rob}$ methods find very well the most remote outliers, meanwhile, the BP method identifies much better  closer outliers.  The $DG_{rob}$ method identifies badly multiple outliers concentrated near the border of the outlier region, whereas the BP method does well. The $DG_{ML}$ is not appropriate for multiple outlier search.

\begin{figure}[t]
\centering
\subfloat[Logistic distribution.]{%
\resizebox*{5cm}{!}{\includegraphics{figure4a}}}\hspace{5pt}
\subfloat[Laplace distribution.]{%
\resizebox*{5cm}{!}{\includegraphics{figure4b}}}

	\caption{The difference between number outliers and rejected observations   given that sample size $n=100$ and $r=10$ outliers. }\label{fig:logist}
\end{figure}

\begin{figure}[t]
\centering
\subfloat[Extreme value II distribution.]{%
\resizebox*{5cm}{!}{\includegraphics{figure5a}}}\hspace{5pt}
\subfloat[Cauchy distribution.]{%
\resizebox*{5cm}{!}{\includegraphics{figure5b}}}

	\caption{The difference between number outliers and rejected observations   given that sample size $n=100$ and $r=10$ outliers. }\label{fig:cauchy}
\end{figure}

\begin{table}[ht]
\caption{Masking values for logistic, Laplace, extreme value II and Cauchy distribution, when $n=100$, $r=5$.}\label{tab:other}
\centering
\begin{tabular}{|l|clll|clll|}
  \hline
  & Logistic & & & & Laplace & & &\\ \hline
Method$\backslash \theta$ & 0.1 & 1 & 6.3 & 10 & 0.1 & 1 & 6.3 & 10 \\
  \hline
DaviesGather\_ML & 5 & 4.89 & 3.64 & 3.42 & 5 & 4.96 & 3.98 & 3.78 \\
  DaviesGather\_Robust & 4.21 & 2.69 & 0.76 & 0.51 & 4.27 & 2.98 & 0.87 & 0.59 \\
  BP & 1.3 & 1.13 & 0.78 & 0.64 & 1.31 & 1.21 & 0.8 & 0.66 \\
   \hline
  \hline
   & Extreme value II &&&& Cauchy &&&\\ \hline
Method$\backslash \theta$ & 0.1 & 1 & 6.3 & 10 & 1 & 100 & 1000 & $10^5$ \\
  \hline
DaviesGather\_ML & 4.96 & 4.19 & 3 & 2.9 & 5 & 5 & 5 & 5 \\
  DaviesGather\_Robust & 4.29 & 2.25 & 0.59 & 0.4  & 3.81 & 2.89 & 0.8 & 0.01 \\
  BP & 1.25 & 0.56 & 0.14 & 0.11 & 0.38 & 0.4 & 0.39 & 0.13 \\
   \hline
\end{tabular}

\end{table}

\newpage

\section{Conclusion}
In many situations, the proposed  outlier identification method  has superior performance as compared to  existing methods. The method is based on an asymptotic result, so it should be not applied for samples of very small size $n\leq 15$.

The formulated approach could be extended and used for  probability
distribution families different from location-scale and shape-scale.

The R package outliersTests was created for the practical usage of proposed test.

\section*{Disclosure statement}
None

\section*{Funding}
This research did not receive any specific grant from funding agencies in the public, commercial, or non-profit sectors.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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Greg Comestock with Kerry Paddock, Patterson Ranch, Gillette, Wyoming.
Katie Ocshner, Lou Leland, Frank Wedel and Mel Leland at the 36th Annual Schuler Red Angus Production Sale.
Ranchers enjoyed a beautiful day to appraise the offering at the Schuler Red Angus 36th Annual Bull Sale, March 22nd at the ranch near Bridgeport, Nebraska. The large crowd was joined by active online bidding to make a successful event for this Annual Bull Sale. The Schuler program is built upon the reproductive and maternal traits required to "Get the Cow Right First," but their genetics have earned numerous Red Angus "GridMaster" awards for the product traits of both the Schulers' and their customers' finished cattle. Schuler Red Angus' leadership in carcass data collection is well into its third decade; with approximately 20 percent of Red Angus' high accuracy carcass sires proven through Schulers' progeny testing program. Annually, the Schuler program tests young sires for both carcass and feed efficiency data to ensure the bulls they offer produce offspring demanded by downstream beef industry customers.
A large crowd of ranchers from Nebraska and surrounding states competed for both Red Angus and Schuler Red Composites, giving the Schuler family a great sale.
The high selling bull was lot 5, Schuler Bonafide E538. This son of Schuler Confidence C810 was out of a 2-year-old daughter of Schuler Envoy 2115Z and brought $16,500 for one half interest and full possession, going to Leland Red Angus of Sidney, Montana. Bonafide packaged powerful phenotype with a 118 ADG ratio and 124 IMF ratio.
Schuler Bourne 111C-7177E, Lot 18, was the 2nd high selling bulling. He sold to Select Sires of Plains City, OH at $13,500 for one half interest and full possession.
Lot 4, Schuler Headliner E548, a 2/17 son of Schuler Confidence, registration #3779021 sold to George Lee, Nebraska.
The Schuler Red Composite program is now in its third decade of providing ranchers with a user-friendly solution to profit utilizing heterosis and breed complementarity without increasing mature size. Buyers were able to select between uniform sire groups that varied from one-half to seven-eighths Red Angus genetics.
The high selling Schuler Red Composite bull was lot 124, Schuler Red 62 18 Karat-E937, going to long time Schuler customer Bryan Wilson of Alliance, Nebraska.
Repeat customers of the Schuler program, Kenansville Cattle Company, Florida and Cullan Farms, Nebraska, returned to make volume selections from the 36th annual offering.


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United States: Charlotte ( Nc) Nagaoka Guinea- Bissau; Czestochowa, fred, Gianni ( Italy) Electronic Designs ( 1977) Futuribile - The Life To Come ( 1978) Gianni Safred Alle TastiereS) This electronic / jazz / disco record ( Futuribile) was released on Italian " Music Scene" library label , Japan; Cachoeiro De Itapemirim, Brazil; Bissau later re- released by Folkways. 网易云音乐是一款专注于发现与分享的音乐产品, 依托专业音乐人、 dj、 好友推荐及社交功能, 为用户打造全新的音乐生活。.
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In the fifth year of elementary school inspired by the surf , Hirasawa took up the electric guitar, on TV, instrumental rock bands he heard on the radio later joining his junior high school' s band. Easy- listening but with cool synth sounds a warm " space age" atmosphere. 5mb) Brad Mackay did the opening and Dustin Harbin hosted the ceremony.
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Aussi en Streaming! Susumu Hirasawa ( 平沢 進) ( born April 2 1954) is a Japanese musician composer. Talks performances, more at the Smithsonian' s museums , tours Zoo. We' re all guilty of skipping through the opening songs of animes every now and again. The documentaries cover everything from music religion, to literature, cinema, politics physics. But even those who have no patience to sit through roughly a minute and a half of jpop/ jrock tunes have to admit that some anime legacies are defined by their choice of opening song.
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These selections are solely based on. Sparing some love for Prologue. Released as a single in 1972, Prologue is my personal favourite piece of music by Renaissance - a brilliant composition that was the first track on the album of the same name and often used as the opening number at the band' s live p 08, · Index Page for Mp3 albus Releases Last Update: 09/ 08/ Demon' s Crest ( SNES) soundtracks to your PC in MP3 format.
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text:
Josie and the Pussycats är en amerikansk komedifilm från 2001, regisserad av Harry Elfont och Deborah Kaplan, vilka även skrivit filmens manus.

Rollista (urval)
Rachael Leigh Cook - Josie McCoy
Tara Reid - Melody Valentine
Rosario Dawson - Valerie Brown
Alan Cumming - Wyatt Frame
Parker Posey - Fiona
Gabriel Mann - Alan M.
Paulo Costanzo - Alexander Cabot
Missi Pyle - Alexandra Cabot
Tom Butler - Kelly, agent
Alexander Martin - Les (medlem i bandet Du Jour)

Externa länkar

Amerikanska komedifilmer
Filmer 2001

redpajama_set_name: RedPajamaWikipedia


Today, anything that smells even slightly of comic books gets a bad rap in some circles, which is why I've put together a list of 11 literary graphic novels all book nerds should read. Even if you've never picked up an Archie comic, these graphic novels will show you just how versatile the medium can be.
Thankfully, kids these days have much better access to graphic novels than I did. Graphically-inclined series like Babymouse and Captain Underpants help ease kids into chapter-book reading with visual storytelling. A number of books for children and teens — including Walter Dean Myers' Monster and Gertrude Chandler Warner's The Boxcar Children — are available in graphic novel formats that motivate reluctant and learning readers to embrace literature.
If you've never read a graphic novel, there's no need to jump into Watchmen or From Hell, although those are both fantastic works of graphic literature. Try out one of the 11 literary graphic novels on the list below, and I'm sure you'll be coming back to the comics shop for more.
This graphic memoir chronicles author Alison Bechdel's relationship with her father, a closeted gay man who may have died by suicide. Fun Home won numerous awards, including the Lambda Literary Award in the Lesbian Memoir/Biography category and the Eisner Award for Best Reality-Based Work.
Iranian-French author Marjane Satrapi grew up during the Iranian Revolution of the 1970s. Her early-2000s graphic memoirs, Persepolis and Persepolis 2, recount how the world changed around her progressive, middle-class family in Iran, how she came to live in France, and how she returned to her homeland. The two volumes are combined in The Complete Persepolis. Persepolis won the Angoulême International Comics Festival Prize for First Comic Book in 2001.
Growing up on the Lower East Side in the early 20th century, Russian-Jewish twins Esther and Fanya Feinberg navigate life as first-generation citizens in this gorgeous and gritty graphic novel from Leela Corman.
Octavia E. Butler's 1979 speculative fiction novel Kindred got the graphic novel treatment in 2017. Both books center on Dana, a black woman who slips through time to antebellum Maryland, where she meets her ancestors — white and black — and examines what it truly means to be black in the U.S.
This collection of graphic short stories from This One Summer co-author Jillian Tamaki highlights the anxieties of contemporary women as they seek to improve themselves — for better or worse.
Perfect for any history buff or literature fan, Kate Beaton's Hark! A Vagrant is chock full of inside jokes that will have you cackling for days.
Based on his experiences working as a liaison between French and North Korean production companies, Guy Delisle's Pyongyang examines the curiously sinister nature of the North Korean capital, where everyone is happy and the old and sick don't exist.
While her friends gathered interviews from people living in Turkey, Syria, and Iraq, graphic novelist Sarah Glidden shadowed them, creating a work of meta-journalistic art that documents how journalism is done.
Daniel Clowes' Patience is subtitled: "A COSMIC TIMEWARP DEATHTRIP TO THE PRIMORDIAL INFINITE OF EVERLASTING LOVE." So, yeah, it's weird. The story centers on a man whose wife and unborn child have been murdered, and who, decades later, finds a way to travel back in time to save them.
Speaking of weird, have you read Agony? This little graphic novel has a nihilist-grunge aesthetic that's somehow both '90s and now. When you need to chuckle at someone else's misfortune, this is the book you want to read.
In this retelling of the 1,000 Nights, Scheherazade becomes Cherry: a young woman married to a cruel king who allows his friend the chance to seduce her for his kingdom and her hand. But Cherry is in love with her handmaid, Hero, and together they hatch a plan to distract Cherry's would-be suitor with a barrage of stories.


text:
Q: how to extract image from word file and save image in another location in php I want the images form word file and save those picture in another folder.
$document = 'hindiquestion.docx';
function readZippedImages($filename) {

    /*Create a new ZIP archive object*/
    $zip = new ZipArchive;
    /*Open the received archive file*/
    if (true === $zip->open($filename)) {
         for ($i=0; $i<$zip->numFiles;$i++) {
            /*Loop via all the files to check for image files*/
            $zip_element = $zip->statIndex($i);

           /*Check for images*/
           if(preg_match("([^\s]+(\.(?i)(jpg|jpeg|png|gif|bmp))$)",$zip_element['name'])) {
              saveImage('display.php?filename=".$filename."&index=".$i."');/* here this function save file in folder but not with image only corrupt file save in folder*/
              echo "<image src='display.php?filename=".$filename."&index=".$i."' /><hr />";/*Display images if present by using display.php*/
           }//image serach
        }
    }
}

function saveImage($path)
{
    copy($path, rand().'-img.jpg');
}

readZippedImages($document);
?>

A: You are adding display.php?.. to the path passed to saveImage. saveImage expects a plain path.
Wrong:
saveImage('display.php?filename=".$filename."&index=".$i."');

Correct:
saveImage($filename);

Also, if the image is a png, you save it as .jpg which is not good (unless it's just for testing).

A: For the scripts of save images from word file into a certain folder (PowerShell), see https://gallery.technet.microsoft.com/How-to-save-from-word-file-46b72800
Add-Type -Assembly "system.io.compression.filesystem"
#copy and renaming the doc to .zip file
$ZipFile = $env:temp + "\" + [guid]::NewGuid().ToString() + ".zip"
Copy-Item -Path $WordFilePath -Destination $ZipFile
#extract all file to tmp folder
$TmpFolder = $env:temp + "\" + [guid]::NewGuid().ToString()
[io.compression.zipfile]::ExtractToDirectory($ZipFile, $TmpFolder)
#copy image files from media folder to destination
Get-ChildItem "$TmpFolder\word\media\" -recurse | Copy-Item -destination $DestinationFolder

A: Note : first you need to create folder "docimages"
 <?php

    class DImages {
        private $file;
        private $indexes = [ ];
        /** Local directory name where images will be saved */
        private $savepath = 'docimages';
        public function __construct( $filePath ) {
            $this->file = $filePath;
            $this->extractImages();
        }
        function extractImages() {
            $ZipArchive = new ZipArchive;
            if ( true === $ZipArchive->open( $this->file ) ) {
                for ( $i = 0; $i < $ZipArchive->numFiles; $i ++ ) {
                    $zip_element = $ZipArchive->statIndex( $i );
                    if ( preg_match( "([^\s]+(\.(?i)(jpg|jpeg|png|gif|bmp))$)", $zip_element['name'] ) ) {
                        $imagename                   = explode( '/', $zip_element['name'] );
                        $imagename                   = end( $imagename );
                        $this->indexes[ $imagename ] = $i;
                    }
                }
            }
        }
        function saveAllImages() {
            if ( count( $this->indexes ) == 0 ) {
                echo 'No images found';
            }
            foreach ( $this->indexes as $key => $index ) {
                $zip = new ZipArchive;
                if ( true === $zip->open( $this->file ) ) {
                    file_put_contents( dirname( __FILE__ ) . '/' . $this->savepath . '/' . $key, $zip->getFromIndex( $index ) );
                }
                $zip->close();
            }
        }

    }
    $DImages = new DImages("testdoc.docx");

    ?>

redpajama_set_name: RedPajamaStackExchange


text:
Q: Restarting my Cocoa app in Mountain Lion results in strange NSScrollView and NSTableView behavior My Mountain Lion app contains an NSScrollView with an NSTableView using an NSTableViewCell. I wanted the ScrollView to have rounded corners, like Reminders. I found an answer here that said to use:
[listScrollView setWantsLayer:YES];
[listScrollView.layer setCornerRadius:10.0f];

When I first start the app, everything works perfectly. I scroll down the list to view items at the bottom and quit the app. When I restart the app the top of the list is shifted down to match the top of the scroll bar, which remembers where I was scrolled to when I quit. However, scrolling or resizing the window causes the list to draw properly. If I click on one of the list items, it selects the item that should be at that position.
Here is the bottom of the list just before quit
Here is what is displayed after restarting the app

A: I figured out the problem, or at least how to avoid the strange behavior.
The strange behavior was in viewing a NSTableView's rows inside a NSScrollView after quitting a Cocoa app after partially scrolling down the TableView's items and then relaunching right away. Mountain Lion remembers the app's state of partially scrolled down and started drawing the TableView's rows at the top of the scroller knob, which was slid down from the top of the list.
The problem was where I was setting the ScrollView's layer to display, which I was using to create rounded corners for the scroll view.
I was originally setting the ScrollView's Layer in the Document's windowControllerDidLoadNib method using an IBOUTLET called listScrollView.
I moved that code into the ScrollView's drawRect method:
if (!self.wantsLayer) {
    [self setWantsLayer:YES];
    [self.layer setCornerRadius:10.0f];
}

Viola, the ScrollView now has rounded corners, which clip ScrollView's corners and scroller knob slot and the TableView's row corners at the top and bottom of the ScrollView.


text:
Home News SURVEY: Revealing the Saddest Fans in the NFL
SURVEY: Revealing the Saddest Fans in the NFL
Comments Off on SURVEY: Revealing the Saddest Fans in the NFL
Every week can be a roller coaster of emotions for dedicated NFL fans.
Of course, those who take home a 'W' are riding high on happiness, but what about the other fans? Those with their hopes crushed?
We surveyed more than 2,000 NFL fans across the country to see which teams have the saddest fans and who is more likely to shed a tear.
Photo: Lineups
When it comes to NFL Fans most likely to cry, the Dallas Cowboys take the number one spot. As the second-most winningest team of all time and five-time Super Bowl champs, their fans are invested. Coming in second is the New England Patriots (could they still be shedding tears over Tom Brady and Gronk?)
Rounding out the top three is the Green Bay Packers – and with the seemingly constant drama surrounding QB 1 Aaron Rodgers, that isn't hard to imagine.
When it comes to those least likely to cry, the Indianapolis Colts, Los Angeles Chargers, and Los Angeles Rams take the top spots.
But what are the top reasons for the tears? 59% of respondents say losing a game starts the waterworks while 35% say losing THE game, the Super Bowl, is why they cry. On the flip side, some fans admit to happy tears with 36% crying over a Super Bowl win and 27% shedding tears over winning an important game.
Wins and losses aren't the only reasons for weeping, 17% of fans say they've cried when their favorite player left the team and 15% cried after losing to a rival.
As we all know, alcohol can bring out true emotions – 15% of fans say they've cried while watching the game drunk.
When it comes to overall unhappiness, 42% of fans admit to turning off a game because they were too sad. Additionally, 1 in 3 fans also says they posted about their sadness on social media.
If you're thinking these feelings are too weak for football, you may be wrong as 79% say it's OK to cry over a team.
It may come as no surprise at all, the Detroit Lions ranked first for most miserable fans in the NFL, but hey, at least they secured a win in 2021!
Losing is clearly a source of misery with Cleveland Browns and New York Jets fans taking the #2 and #3 spots.
The least miserable fans are those cheering for the Arizona Cardinals, Indianapolis Colts, and New Orleans Saints.
To determine our ranking, we surveyed 2,008 NFL fans across the country in November 2021 and asked them whether they've ever become emotionally upset while watching an NFL game.
Respondents were also asked to rank the most miserable fan bases in the league. Among respondents, 50% were male and 50% were female with an average age of 37.
Featured Image: Lineups
TagscowboysDallas CowboyseaglesGreen Bay PackersNew England PatriotsNFLPackerspatriotsPhiladelphia EaglesPittsburgh SteelersSteelersWegENT
Top Tags New England Patriots WegENT betting MMA Football Premier League UFC NCAA sports betting NHL Featured MLB Dallas Cowboys NFL NBA

redpajama_set_name: RedPajamaCommonCrawl


About game: This is a game about great gambit – or actually, the last stage of it – Caesar plans to conquer the Gaul and earns ever-lasting glory (plus political security 🙂 ) It depicts last chapters of that process – the most volatile period, as this is the time of the Great Revolt.
In a base game we have possibility to see years 54-51 BC depicted in three nice scenarios, with an expansion bringing years 58-55 BC to the table.
The game follows well proven COIN (COunter INsurgency) mechanic as well as its concept – we have ever-powerful Rome (Caesar), strong but weaker direct opponents (Arverni, Belgae, Germans) and humble, but playing their own agenda Aedui.
Who will outsmart others? Who will betray Allies in the best moment and turn against them? And who will be smart enough to keep their friends close but enemies even closer? These are the question which The Falling Sky – Gallic Revolt answers very well!
Number of players: 1-4 – four is perfect – if you can get enough friends to play. But version for 1-3 is also very well scripted and definitely COIN games are one of the best suited for solitaire play.
Playing time: We have here a bunch of scenarios, with base game bringing three and Ariovistus expansion another two (if you count The Gallic War one). Some of them also have shorter versions, so it can be anything from 2 to 8 hours, with an experience significantly decreasing the amount of time needed.
So yes, this is definitely game you can play in one evening, especially shorter set-ups.
Complexity: The basic rules are pretty straightforward and easy to learn. It is nuances – in abbreviated text or special exceptions – which might made you reach for rulebook pretty often in the beginning. The Player's Aid's are very helpful, but I will still rate this as medium-complex wargame.
Climate – as you can see based on games I play, my avatar/log and session reports – I am a fan of Ancient History. And that game really well depicts the time of Gallic Conquest!
We have multiple scenarios which portraits various stages of Gallic Revolt plus great expansion – Ariovistus.
With many scenarios comes also replayability – this is also thanks to ever changing deck of events.
I like the asymmetric factions – there are some similarities, but each fraction is very different and the usual COIN dynamic – with hegemon and "insurgents" – is very neatly implemented.
Different victory conditions – definitely, this is great enhancement to the game. Also, the levels of points needed to reach automatic victory are set in a way that there is no possibility of "winning by chance".
While in longer scenarios (like 54 BC and The Gallic War) we are sure to use all or almost all cards, in shorter ones this can be only a fraction of available events. And it can happen, that this makes the game unbalanced towards one or the other side. I would rather see more balanced deck for shorter games.
Belgae in 52 scenario – well, just look at the Belgae situation. It is historical but gives few options for victory. On the other hand, this is a faction I am playing when teaching new players – short scenario, and I can just stay aside from main event.
The Gallic War switch from Germans to Arverni – yes, I know it has to happen, that is how decks are built. Still, this is not my favorite feature of the game.
All the nuances to the rules, abbreviations, commas, and's, or's, etc. I feel sometimes the form is more important then the merit.
For whom? Of course, first and foremost for COIN series fans – truly, this is in my opinion definitely the best usage of the system. Yes, I am biased to ancient history, but the way it was adopted for that time is great.
All wargamers who like medium-complex games, with some aspect of economy and fight will find that game very appealing.
While not the easiest of COIN games, I still think and recommend it also for gamers who would like to learn COIN system – this is the best adaptation for me, which can be used as hallmark.
And now let us have a look at the components – all pictures from my session reports.
The Falling Sky + Ariovistus expansion is a complete game. It has all the aspects I require from a wargame – climate, fight, interesting capabilities, replayability, suspense, well incorporated random factor, etc. And despite some drawbacks (in my humble opinion) this is a superb and well made game.
See you in another game review!
PS. Following Lucius Cornelius Sulla, a real example from whom Caesar draw, can you really be The Best Friend and The Worst Enemy to your opponents?


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This home is priced at $183/sqft, which is 4.2% less than similar homes in the 27519 zip code.
From Highway 55, turn on McCrimmon Pwky; Right on Green Level Church Rd; Left on Weycroft Avenue (Weycroft Subdivision); Right on Foxdale Ridge Drive. Home is on the right.
512 Foxdale Ridge Dr, Cary NC is a single family home of 3814 sqft and is currently priced at $699,000 . This single family home has 4 bedrooms and was built in 2010. A comparable home for sale at 136 Gravel Brook Ct in Cary is listed at $649,900. This home on 512 Foxdale Ridge Dr has not had a price change. In addition to single family homes, Highgarden also makes it easy to find Condos, New Homes and Foreclosures in Cary, NC. Magnolia Estates, Sedgefield and Toscana are nearby neighborhoods. MLS 2246677 has been posted on this site since 4/6/2019 (12 days ago).


text:
Sergio Escudero – piłkarz argentyńsko-japoński, ur. 1964
 Sergio Escudero – piłkarz japoński, ur. 1988
 Sergio Escudero – piłkarz hiszpański, ur. 1989


Russia is a great country. Russia is a land for alone and divorced men. It's a ground where all of men's inner desires come true. In many different movies, stories is shown how beautiful their country is and how amazing their girls are.
Those young men wish to find that young woman that will prepare their favorite food, smart and beautiful, that will be excellent mother for their kids. But most important what boys want is love that will give them that hot woman. Few men think that this unique girl you won't find anywhere and you'll loose a lot of time and money to locate one. Only that those who say that are mistaken and it is evident that all those young men don't know anything about beautiful Russian girls. All those characteristics are suited great Russian women. They were raised in that manner, to take care about their husband, to raise gentle children and do housecleaning. Hundreds of young men ask why they are so beautiful? That's because of their blood mix. In Russia many years were living different people from different places, therefore change in population is obvious. You must go to that country and to see for yourself what young girls live there. Foreigners turn their necks when see such ladies. It's sad that we do not have such young women, but the truth that they are is also nice.
So if you have problems in your life and you wish for changes? Travel to Russia and you will obtain your answers to all your questions.