text:
\section{Introduction}

Let $\{X_i\}_{i=1}^n$ be a sample of i.i.d.\ nonnegative random variables with common distribution function $F$ that is known to be concave. The seminal work by \citet{Grenander1956II} establish that the nonparametric maximum likelihood estimator of $F$ is given by the least concave majorant $\hat{\mathbb F}_n$ of the empirical distribution function $\mathbb F_n$. In this context, the present paper provides new results on two aspects of the problem.

The first set of results concerns the closeness between $\hat{\mathbb F}_n$ and $\mathbb F_n$ relative to the uniform norm $\|\cdot\|_\infty$. The celebrated Marshall's lemma indicates that $\hat{\mathbb F}_n$ is $\sqrt{n}$-consistent for $F$ relative to $\|\cdot\|_\infty$---specifically, for each $n\in\mathbf N$,
\[
\|\hat{\mathbb F}_n-F\|_\infty\le\|\mathbb F_n-F\|_\infty~.
\]
\citet{Kiefer_Wolfowitz1976minimax} show that $\hat{\mathbb F}_n$ in fact is asymptotically equivalent to $\mathbb F_n$ by essentially establishing the following result:
\begin{thm}\label{Thm: KW}
Let $\alpha_{F}\equiv\inf\{x\in\mathbf R: F(x)=1\}$ and $F$ be twice continuously differentiable on $[0,\alpha_{F}]$ with $\sup\{x\in\mathbf R: F(x)=0\}=0$. If $\alpha_{F}<\infty$ and
\begin{align}
\beta_{F} & \equiv\inf_{0<x<\alpha_{F}}\frac{-f'(x)}{f^2(x)}>0~,\label{Eqn: KW beta}\\
\gamma_{F} & \equiv\frac{\sup_{0<x<\alpha_{F}}-f'(x)}{\inf_{0<x<\alpha_{F}}f^2(x)}<\infty~,\label{Eqn: KW gamma}
\end{align}
then it follows that
\[
\sup_{0\le x\le\alpha_{F}}|\hat{\mathbb F}_n(x)-\mathbb F_n(x)|=O(n^{-2/3}(\log n)^{2/3})\text{ almost surely}~.
\]
\end{thm}
\noindent Since $F(0)=0$ by assumption, it follows from $\alpha_{F}<\infty$ that the distribution associated with $F$ has bounded support. In turn, the condition \eqref{Eqn: KW beta} implies that $f'(x)<0$ for all $x\in(0,\alpha_{F})$ and hence that $F$ is strictly concave on the support $[0,\alpha_{F}]$. Finally, the condition \eqref{Eqn: KW gamma} demands that the derivative $f'$ of the density be small relative to the minimum of $f$ on its support. This paper provides refinements of Theorem \ref{Thm: KW} along several dimensions. First, we consider situations where $f$ has potentially unbounded support, i.e., $\alpha_{F}=\infty$, an extension which appears to be nontrivial---see Remark 5 in \citet[p.82]{Kiefer_Wolfowitz1976minimax} and also  \citet{Kiefer_Wolfowitz1977minimax}. In doing so, we necessarily have $\gamma_{F}=\infty$ (so that \eqref{Eqn: KW gamma} is being violated) because the density $f$ must approach zero along the right tail. Our second refinement, which may or may not be surprising, allows $F$ to be non-strictly concave or even non-concave but requires that it be ``asymptotically'' strictly concave as the sample size tends to infinity. In any case, the above asymptotic order result is uncovered under weaker regularity conditions.

 %

The second set of results concerns inference aspect of the problem. In particular, we develop a test for the hypothesis that $F$ is concave or equivalently $f$ is nonincreasing. The insight we exploit here is that the hypothesis can be equivalently formulated as:
\[
\mathrm H_0: \|\hat F-F\|_p=0 \qquad \text{v.s.} \qquad \mathrm H_1: \|\hat F-F\|_p>0 ~,
\]
where $\hat F$ is the least concave majorant of $F$, and $\|\cdot\|_p$ is an $L^p$ norm (with appropriate weighting) for $p\in[1,\infty]$. Thus, it is natural for us to employ the test statistic $\sqrt n\|\hat{\mathbb F}_n-\mathbb F_n\|_p$. There are several technical challenges, however, in establishing statistical properties of our test. First, as demonstrated in \citet{BeareandMoon2015} and \citet{Beare_Fang2016Grenander}, the least concave majorant operator $F\mapsto \hat F$ is not fully differentiable if $F$ is concave but not strictly concave, and hence the conventional Delta method is inapplicable in establishing the asymptotic distribution of $\sqrt n\|\hat{\mathbb F}_n-\mathbb F_n\|_p$. However, the same authors have also shown that it is Hadamard directionally differentiable, a weaker notion of differentiability under which the Delta method is in fact preserved \citep{Shapiro1991,Dumbgen1993}. Second, as shown by \citet{Dumbgen1993} and \citet{FangSantos2018HDD}, even though the Delta method generalizes to deriving asymptotic distributions under Hadamard directional differentiability, it does not generalize to bootstrap consistency. This can be remedied by appealing to the rescaled bootstrap of \citet{Dumbgen1993}. Lastly, the weak limit of $\sqrt n\|\hat{\mathbb F}_n-\mathbb F_n\|_p$ is degenerate at zero when $F$ is strictly concave, a consequence of the Kiefer-Wolfowitz theorem. To build up a level $\alpha$ test, we leverage the asymptotic order results on $\|\hat{\mathbb F}_n-\mathbb F_n\|_\infty$ and propose a selection procedure we call the KW-selection that determines whether $F$ is strictly concave or not under the null. The idea is that the convergence rates of $\hat{\mathbb F}_n$ are different in these two cases, and hence we may introduce a suitable tuning parameter to identify the truth. Our test is in fact shown to have local size control which may be of interest since the weak limit of the statistic exhibits ``discontinuity'' with respect to the true distribution. Unfortunately, the local power will be poor under local (contiguous) alternatives that approach a strictly concave distribution.

The literature on testing concavity of the cumulative distribution function (or equivalently monotonicity of the density function) is surprisingly limited. In Section \ref{Sec: comparison}, we review related studies in this regard and compare them to our test. Overall, there are roughly two types of tests in the density context. The first type is concerned with special classes of alternatives. This includes the work of \citet{Woodroofe_Sun1999} who test uniformity against monotonically increasing (but not uniform) densities, or uniformity against convex cdfs. Therefore, these tests are powerful in detecting these special classes by design, but may perform poorly for other alternatives, which is confirmed in our simulation studies. The second type remains agnostic about the particular natures of alternative hypothesis, but relies on critical values from the asymptotically least favorable distributions. This includes the tests of \citet{Durot2003testing} and \citet{Kulikov_Lopuhaa2004testing}. While the use of least favorable distributions is common practice in some other settings \citep{Carolan_Tebbs2005,Delgado_Escanciano2012monotonicity,DelgadoEscanciano2013ConditionalSD,DelgadoEscanciano2016ConditionalMI}, the resulting tests may be too conservative and cause substantial power loss. Finally, in similar contexts, there are recent studies such as \citet{BeareShi2019DRO} and \citet{Seo2018SM} that aim to improve power by bootstrap. However, due to the lack of Kiefer-Wolfowitz type results, it is unclear what happens to their tests when the limiting distributions of the test statistics are degenerate. We shall remove the dependence on least favorable distributions by bootstrap, and control the size of our test (both pointwise and locally) even when degeneracy occurs by utilizing the Kiefer-Wolfowitz's theorem.

We now introduce some notation. We denote by $\mathbf R_+$ the set of nonnegative real numbers. For a sequence $\{Z_n\}_{n=1}^\infty$ of random variables, we write $Z_n=O_{a.s.}(a_n)$ for a deterministic sequence $\{a_n\}_{n=1}^\infty$ if $Z_n/a_n=O(1)$ as $n\to\infty$ almost surely. Here and elsewhere in this paper, for a sequence $\{b_n\}$ of real numbers, $b_n=O(1)$ as $n\to\infty$ means that $\{b_n\}$ is bounded. For two sequences of real numbers $\{a_n\}$ and $\{b_n\}$, we write $a_n\lesssim b_n$ if $a_n/b_n =O(1)$ as $n\to\infty$, and write $a_n\gtrsim b_n$ if $b_n\lesssim a_n$. Moreover, the notation $a_n\asymp b_n$ means that $a_n/b_n=O(1)$ and $b_n/a_n=O(1)$ as $n\to\infty$. Analogously, for two functions $f,g:\mathbf R\to\mathbf R_+$, we write $f(\epsilon)\asymp g(\epsilon)$ as $\epsilon\downarrow 0$ if $f(\epsilon)/g(\epsilon)=O(1)$ and $g(\epsilon)/f(\epsilon)=O(1)$ as $\epsilon\downarrow 0$. For an arbitrary nonempty set $T$, $\ell^\infty(T)$ is the space of bounded real-valued functions on $\mathbf R_+$ equipped with the uniform norm $\|\cdot\|_\infty$, i.e., $\|f\|_\infty\equiv \sup_{x\in T}|f(x)|$. The space $C_0(\mathbf R_+)$ is the family of all real-valued continuous functions on $\mathbf R_+$ that vanish at infinity. We denote by $\xrightarrow{L}$ the weak convergence in the sense of Hoffmann-J{\o}rgensen \citep{Vaart1996}.

The remainder of the paper is structured as follows. Section \ref{Sec: KW} presents refinements of Theorem \ref{Thm: KW}. Section \ref{Sec: tests} formally develops our test and then compare to some existing monotonicity tests from the literature. Section \ref{Sec: simulation} conducts simulation studies, with some of the results relegated to Appendix \ref{App: Simulation}. Section \ref{Sec: conclusion} concludes. All proofs are collected in Appendix \ref{Sec: proofs}, while extensions of the test results to a general setup are discussed in Appendix \ref{App: extension}.

\section{The Kiefer-Wolfowitz Theorems}\label{Sec: KW}

This section presents refinements of the Kiefer-Wolfowitz theorem that allow the support of the density function $f$ to be unbounded. Throughout, we think of the cdf $F$ as a function on $\mathbf R_+$ (rather than on $\mathbf R$). We proceed with the following assumption.

\begin{ass}\label{Ass: setup}
(i) $\{X_i\}_{i=1}^n$ is an i.i.d.\ sample with common distribution function $F$, and (ii) $F$ is strictly concave on $\mathbf R_+$.
\end{ass}

Assumption \ref{Ass: setup} simply formalizes the i.i.d.\ setup and strict concavity of the distribution function $F$. To establish the refinements, we need to impose further regularity conditions on $F$. In particular, we need to introduce an analog of $\beta_F$ in \eqref{Eqn: KW beta}. Specifically, for any $\epsilon\in[0,1]$, we define
\begin{align}\label{Eqn: beta}
\bar\beta(\epsilon)\equiv \inf_{0\le x\le F^{-1}(1-\epsilon)}\frac{-f'(x)}{f^2(x)}~,
\end{align}
where we suppress the dependence of $\bar\beta(\epsilon)$ on the underling distribution for simplicity. Thus, $\bar\beta(\epsilon)$ may be viewed as a truncated version of $\beta_F$ in \eqref{Eqn: KW beta}. As it turns out, while $\bar\beta(\epsilon)$ is allowed to approach zero as $\epsilon\to 0$, the speed at which it approaches zero determines the asymptotic order of $\|\hat{\mathbb F}_n-\mathbb F_n\|_\infty$. To formalize our discussions, we now impose:

\begin{ass}\label{Ass: equivalence}
(i) $F$ is twice continuously differentiable on $\mathbf R_+$, (ii) $\bar\beta(\epsilon) \asymp \epsilon^\tau$ as $\epsilon\to 0$ for some constant $\tau>-1$, and (iii)
\[
\|F''\|_{1/2}\equiv\left[\int_0^\infty \sqrt{|F''(x)|}\,\mathrm dx\right]^2<\infty~.
\]
\end{ass}

Assumption \ref{Ass: equivalence}(i) is the same smoothness condition required by Theorem \ref{Thm: KW}. Assumption \ref{Ass: equivalence}(ii) characterizes the exact rates at which $\bar\beta(\epsilon)$ is allowed to approach zero as $\epsilon\to 0$. The special case $\tau=0$ implies $\inf_{0\le x<\infty}\{-f'(x)/f^2(x)\}>0$, which is exactly the condition \eqref{Eqn: KW beta} in the case of unbounded support. If $\tau>0$, then $\bar\beta(\epsilon)\to 0$ as $\epsilon\to 0$; if $\tau\in(-1,0)$, then $\bar\beta(\epsilon)\to \infty$ as $\epsilon\to 0$ which still fulfills \eqref{Eqn: KW beta}. Assumption \ref{Ass: equivalence}(iii) is a technical condition that serves to control the interpolation error for the cdf $F$. When the support of the density function $f$ is bounded, Assumption \ref{Ass: equivalence}(iii) is automatically fulfilled because $F''$ as a continuous function is bounded. Thus, our assumptions are strictly weaker than those imposed in \citet{Kiefer_Wolfowitz1976minimax}.

Given Assumptions \ref{Ass: setup} and \ref{Ass: equivalence}, we now present the first refinement.

\begin{thm}\label{Thm: Equivalence}
If Assumptions \ref{Ass: setup} and \ref{Ass: equivalence} hold, then, as $n\to\infty$,
\[
\|\hat{\mathbb F}_n-\mathbb F_n\|_\infty=\begin{cases}
O_{a.s.}\big(((\log n)/n)^{\frac{2}{2\tau+3}}\big) & \text{ if }\tau\ge 0\\
O_{a.s.}\big(((\log n)/n)^{\frac{\tau+2}{2\tau+3}}\big) & \text{ if }\tau\in (-1,0)\\
\end{cases}~.
\]
\end{thm}

Theorem \ref{Thm: Equivalence} delivers the asymptotic order of $\|\hat{\mathbb F}_n-\mathbb F_n\|_\infty$, which crucially depends on the parameter $\tau$. The slower $\bar\beta(\epsilon)$ approaches zero (as $\epsilon\to 0$) or the smaller $\tau$ is, the closer $\hat{\mathbb F}_n$ and $\mathbb F_n$ is (asymptotically). If $\bar\beta(\epsilon)$ approaches zero too fast or $\tau$ is too large, then the asymptotic equivalence of $\hat{\mathbb F}_n$ and $\mathbb F_n$ are no longer implied. This happens precisely when $\tau\ge 1/2$, in which case
\begin{align}
\sqrt n ((\log n)/n)^{\frac{2}{2\tau+3}}= n^{\frac{2\tau-1}{2\tau+3}} (\log n)^{\frac{2}{2\tau+3}}\to\infty\text{ as } n\to\infty~.
\end{align}
The special case $\tau=0$ leads to $\|\hat{\mathbb F}_n-\mathbb F_n\|_\infty=O_{a.s.}\big(((\log n)/n)^{2/3}\big)$, which is exactly the result in Theorem \ref{Thm: KW}. There are two major new ingredients in establishing the theorem, compared to the proof of Theorem \ref{Thm: KW}. One is controlling the variation of $\mathbb F_n$ over a small upper quantile region---see the treatment of the $T_{n,k_n+1}$ term and the $B_n$ term in the proof. The other is bounding the interpolation error for $F$ by the $L_{1/2}$-integral of $F''$ based on a result from approximation theory \citep{Burchard1974splines}. The latter allows us to dispense with the regularity condition $\gamma_F<\infty$ \citep[p.82]{Kiefer_Wolfowitz1976minimax}.

Our second refinement allows the common distribution function $F$ that the finite sample $X_1,\ldots,X_n$ share to be not strictly concave or even non-concave. However, we do require that it vary with the sample size $n$ in such a way that it approaches (in a suitable sense to be specified) a strictly concave distribution function as $n\to\infty$, a notion we call ``local to concavity.'' In order to formalize our local analysis, we denote by $\mathbf P$ the set of probability measures with the support contained in $\mathbf R_+$ that possibly govern the data:
\[
\mathbf P\equiv\{P\in\mathbf M: P(\mathbf R_+)=1\}~,
\]
where $\mathbf M$ is the set of all Borel probability measures on $\mathbf R$ that are dominated by the Lebesgue measure on $\mathbf R$. Further, we think of the distribution function $F\equiv F(P)$ as a map $F:\mathbf P\to\ell^\infty(\mathbf R_+)$ defined by
\[
F(P)(x)\equiv P((-\infty,x])~,\,\forall\,x\in\mathbf R_+~.
\]
We may now formalize the precise meaning of ``local'' as follows.

\iffalse
As standard in the framework of limits of experiments \citep{LeCam1972experiments,Vaart1991representation}, asymptotic local analysis entails local perturbations to the true distribution $P$ in the sense of Fr\'{e}chet differentiability of one dimensional submodels contained in $\mathbf P$. This notion is formalized as follows.
\fi

\begin{defn}\label{Defn: differentiable path}
A function $t\mapsto P_t$ mapping a neighborhood $(-\epsilon,\epsilon)$ of zero into $\mathbf P$ is called a differentiable path passing through $P$ if, for $P_0=P$ and some function $h: \mathbf R_+\to\mathbf R$,
\begin{align}\label{Eqn: differentiable path}
\lim_{t\to 0}\int\left[\frac{\mathrm dP_t^{1/2}-\mathrm dP^{1/2}}{t}-\frac{1}{2}h\,\mathrm dP^{1/2}\right]^2= 0~.
\end{align}
\end{defn}

The notation $\mathrm dP_t$ and $\mathrm dP$ may be understood as the densities of $P_t$ and $P$ with respect to some dominating measure $\mu_t$ (for each $t$), though the integral in \eqref{Eqn: differentiable path} does not depend on the choice of $\mu_t$ \citep[p.362]{Vaart1998}. Loosely speaking, \eqref{Eqn: differentiable path} implies that $P_t$ gets closer and closer to $P_0$ ``on average'', as $t\to 0$. The function $h$ is referred to as the score function of $P$ and satisfies $\int h\,\mathrm dP=0$ and $h\in L^2(P)$---see, for example, Lemma 25.14 in \citet{Vaart1998}. The term ``score function'' makes sense in view of the relation $h(x)=\frac{\mathrm d}{\mathrm dt}\log \mathrm dP_t(x)\big|_{t=0}$ (which is the usual definition of score function) under regularity conditions---see, for example, Lemma 7.6 in \citet{Vaart1998}.

If $\{P_t: |t|<\epsilon\}$ is a differentiable path, then we obtain by Example 5.3.1 in \citet{BKRW993Efficient} that
\begin{align}\label{Eqn: F diff}
\lim_{t\to 0}\|\frac{F(P_t)-F(P)}{t}-\dot F(h)\|_\infty=0~.
\end{align}
where
\[
\dot F(h)(x)\equiv\int_{-\infty}^x h(u)\,P(\mathrm du)~.
\]
Intuitively, \eqref{Eqn: F diff} means that the distribution function $F_t\equiv F(P_t)$ smoothly passes through $F\equiv F(P_0)$ at the same speed as the underlying probability measure $P_t$ passes through $P_0$. For a differentiable path $\{P_t\}$ satisfying \eqref{Eqn: differentiable path} and $F_t\equiv F(P_t)$, we say $\{F_t\}$ is {\it local to concavity} if $F$ is concave and is {\it local to strict concavity} if $F$ is strictly concave.

We next show that the conclusion in Theorem \ref{Thm: KW} is preserved under local to strict concavity. To this end, we impose the following assumption.

\begin{ass}\label{Ass: local}
(i) $\{X_i\}_{i=1}^n$ is an i.i.d.\ sample with common probability measure $P_{1/\sqrt{n}}$, and (ii) $\{P_{1/\sqrt{n}}\}\subset\mathbf P$ corresponds to the differentiable path $\{P_t\}$ passing through $P_0\equiv P$ in the sense of Definition \ref{Defn: differentiable path}.
\end{ass}

\begin{thm}\label{Thm: Equivalence, local}
If Assumptions \ref{Ass: setup}(ii), \ref{Ass: equivalence} and \ref{Ass: local} hold, then, as $n\to\infty$,
\[
\|\hat{\mathbb F}_n-\mathbb F_n\|_\infty=\begin{cases}
O_{a.s.}\big(((\log n)/n)^{\frac{2}{2\tau+3}}\big) & \text{ if }\tau\ge 0\\
O_{a.s.}\big(((\log n)/n)^{\frac{\tau+2}{2\tau+3}}\big) & \text{ if }\tau\in (-1,0)\\
\end{cases}~.
\]
\end{thm}

Theorem \ref{Thm: Equivalence, local} is not much deeper than Theorem \ref{Thm: KW}, but it has implications for the problem of testing concavity as we shall elaborate in Section \ref{Sec: tests}.

\section{Testing Concavity}\label{Sec: tests}

In this section, we develop a test of concavity that controls size regardless of whether concavity is strict or not. This is accomplished by building on the asymptotic order results established previously. We shall also compare our concavity test with existing monotonicity tests. For simplicity, in what follows, we focus on the canonical case $\tau=0$ in Assumption \ref{Ass: equivalence}(ii). The general case makes no essential differences---see Appendix \ref{App: extension} for extensions.

\subsection{The Test Statistic}

First, we introduce our test statistic and derive its asymptotic distribution. To this end, we introduce the least concave majorant (LCM) operator following \citet{Beare_Fang2016Grenander}---see also \citet{BeareandMoon2015}.

\begin{defn}
Given a convex set $T\subseteq \mathbf R_+$, the LCM over $T$ is the operator $\mathcal M_T: \ell^\infty(\mathbf R_+)\to\ell^\infty(T)$ that maps each $\theta\in\ell^\infty(\mathbf R_+)$ to the function
\[
\mathcal M_T\theta(x)\equiv\inf\{g(x): g\in \ell^\infty(T)\text{, }g\text{ is concave, and }\theta\le g\text{ on }T\},\quad \forall \, x\in T ~.
\]
If $T=\mathbf R_+$, then we write $\mathcal M\equiv\mathcal M_{\mathbf R_+}$.
\end{defn}

The hypothesis of our interest can now be formulated as:
\[
\mathrm H_0: \phi(F)=0\quad \text{v.s.}\quad \mathrm H_1: \phi(F)>0~,
\]
where $\phi: \ell^\infty(\mathbf R_+)\to\mathbf R$ is defined by
\[
\phi(F)=\|\mathcal MF-F\|_p=\begin{cases}
[\int_{\mathbf R_+}(\mathcal MF(x)-F(x))^p g(x)\,\mathrm dx]^{1/p} & \text{ if }p\in[1,\infty)\\
\sup_{x\in\mathbf R_+}|\mathcal MF(x)-F(x)| & \text{ if }p=\infty
\end{cases}~,
\]
with $g\in L^1(\mathbf R_+)$ some known positive weighting function. Here and throughout, we work with an arbitrarily fixed $p\in[1,\infty]$, and thus suppress the dependence of $\phi$ on $p$ for notational simplicity. Given the above formulation, we employ the statistic $\sqrt n \phi(\mathbb F_n)$. To derive the weak limit of $\sqrt n \phi(\mathbb F_n)$, note that under the null hypothesis, $\mathcal MF=F$ and hence we may rewrite:
\begin{align}\label{Eqn: test statistic using D}
\sqrt n \phi(\mathbb F_n)=\sqrt n\|\mathcal M\mathbb F_n-\mathbb F_n\|_p=\|\sqrt n\{\mathcal D\mathbb F_n-\mathcal D F\}\|_p~,
\end{align}
where $\mathcal D \equiv\mathcal M-\mathcal I$ with $\mathcal I$ the identity operator on $\ell^\infty(\mathbf R_+)$. Thus, the asymptotic distribution of $\sqrt n \phi(\mathbb F_n)$ would be an implication of the continuous mapping theorem and the Delta method, if we could show that $\mathcal M$ (and hence $\mathcal D$) is Hadamard differentiable \citep[p.372-374]{Vaart1996}. Unfortunately, as demonstrated by \citet{BeareandMoon2015} and \citet{Beare_Fang2016Grenander}, the LCM operator fails to be fully differentiable but only Hadamard directionally differentiable in general \citep{Shapiro1990}. Nonetheless, this type of directional differentiability suffices for applying a generalized version of the Delta method \citep{Shapiro1991,Dumbgen1993}.

\begin{defn}\label{Defn: HDD}
Let $\mathbb D$ and $\mathbb E$ be normed spaces equipped with norms $\|\cdot\|_{\mathbb D}$ and $\|\cdot\|_{\mathbb E}$ respectively, and $\phi:\mathbb D_\phi\subseteq \mathbb D\to\mathbb E$. The map $\phi$ is said to be {\it Hadamard directionally differentiable} at $\theta \in\mathbb D_\phi$ {\it tangentially} to $\mathbb D_0\subseteq\mathbb D$, if there is a map $\phi_\theta':\mathbb D_0\to\mathbb E$ such that:
\begin{equation}\label{Eqn: HDD}
\lim_{n\rightarrow \infty}\|\frac{\phi(\theta +t_n h_n)-\phi(\theta)}{t_n} -\phi_\theta'(h)\|_{\mathbb E} = 0 ~,
\end{equation}
for all sequences $\{h_n\}\subset\mathbb D$ and $\{t_n\}\subset\mathbf R_+$ such that $t_n\downarrow 0$, $h_n\to h\in\mathbb D_0$ as $n\to\infty$ and $\theta+t_nh_n\in\mathbb D_\phi$ for all $n$.
\end{defn}

The defining feature of Hadamard directional differentiability is that, unlike Hadamard (full) differentiability, the directional derivative is in general nonlinear though necessarily continuous and positively homogeneous of degree one \citep{Shapiro1990}.\footnote{Let $\mathbb D$ be a vector space. A function $f: \mathbb D\to\mathbf R$ is said to be positively homogeneous of degree one if and only if $f(ax) = af(x)$ for all $x\in\mathbb D$ and all $a\in\mathbf R_+$.} We refer the reader to \citet{Shapiro1990} and a more recent review by \citet{FangSantos2018HDD} for additional discussions. Proposition 2.1 in \citet{Beare_Fang2016Grenander} implies that $\mathcal M$ is Hadamard directionally differentiable at any concave $F\in\ell^\infty(\mathbf R_+)$ tangentially to the set $C_0(\mathbf R_+)$ with the derivative $\mathcal M_F': C_0(\mathbf R_+)\to\ell^\infty(\mathbf R_+)$ given by: for any $h\in C_0(\mathbf R_+)$ and $x\in \mathbf R_+$,
\[
(\mathcal M_F'h)(x)=(\mathcal M_{T_{F,x}}h)(x)~,
\]
where $T_{F,x}=\{x\}\cup U_{F,x}$ with $U_{F,x}$ the union of all open intervals $A\subseteq \mathbf R_+$ such that (i) $x\in A$, and (ii) $F$ is affine over $A$. We emphasize that $\mathcal M_F'$ is equal to the identity operator on $C_0(\mathbf R_+)$ if and only if $F$ is strictly concave, in which case $\mathcal M$ is Hadamard differentiable at $F$ tangentially to $C_0(\mathbf R_+)$---see Proposition 2.2 in \citet{Beare_Fang2016Grenander} for more details.

We are now in a position to state the asymptotic distribution of our statistic $\sqrt n \phi(\mathbb F_n)$ by invoking the generalized Delta method.

\begin{lem}\label{Lem: weak limit}
If Assumption \ref{Ass: setup}(i) holds, then $\sqrt n \phi(\mathbb F_n)\xrightarrow{L} \|\mathcal M_F'\mathbb G-\mathbb G\|_p$ under $\mathrm H_0$, where $\mathbb G\equiv\mathbb B\circ F$ and $\mathbb B$ is the standard Brownian bridge on $[0,1]$.
\end{lem}

Lemma \ref{Lem: weak limit} establishes the weak limit of the test statistic $\sqrt n \phi(\mathbb F_n)$ under the null. The Brownian bridge $t\mapsto\mathbb G(t)\equiv\mathbb B(F(t))$ is a zero mean Gaussian process with covariance function: for all $s,t\in[0,1]$,
\begin{align}
\mathrm{Cov}(\mathbb G(t),\mathbb G(s))=F(\min(s,t))-F(t)F(s)~.
\end{align}
The limiting distribution in Lemma \ref{Lem: weak limit} is not pivotal because it depends on $F$ through the process $\mathbb G$ and, critically, the derivative $\mathcal M_F'$.

\subsection{The Critical Values}

Towards constructing critical values for our test, we next aim to estimate the law of $\|\mathcal M_F'\mathbb G-\mathbb G\|_p$ through bootstrap. There are, however, two complications involved, as we now elaborate.

First, when $F$ is non-strictly concave in which case $\mathcal M$ is only Hadamard directionally differentiable, the standard bootstrap (compare to \eqref{Eqn: test statistic using D}),
\begin{align}\label{Eqn: standard bootstrap}
\|\sqrt n\{\mathcal D\mathbb F_n^*-\mathcal D\mathbb F_n\}\|_p~,
\end{align}
is necessarily inconsistent. This is a consequence of Proposition 1 in \citet{Dumbgen1993}, which has been formalized by Theorem A.1 in \citet{FangSantos2018HDD}. Second, when $F$ is strictly concave, the weak limit of $\sqrt n \phi(\mathbb F_n)$ is degenerate at zero since $\mathcal M_F'=\mathcal I$, which is not surprising in view of Theorem \ref{Thm: Equivalence}. The first issue can be resolved by appealing to the rescaled bootstrap in \citet{Dumbgen1993}. As shall be seen shortly, the rescaled bootstrap is connected in a subtle way to the modified bootstrap in \citet{FangSantos2018HDD}; namely, it amounts to composing a suitable derivative estimator (see \eqref{Eqn: rescaled bootsrap2}) with some bootstrap process. The second issue is more challenging, which we shall fix by leveraging Theorems \ref{Thm: Equivalence} and \ref{Thm: Equivalence, local} as we describe now.

The key to the level control of our test is a selection procedure that determines whether the concavity of $F$ is strict or not by exploiting the fact that the convergence rates in these two cases are different (see Theorem \ref{Thm: Equivalence} and Lemma \ref{Lem: weak limit}). Specifically, let $\{\kappa_n\}$ be a sequence of positive scalars such that $\kappa_n=o(1)$ and $(\log n)^{2/3}n^{-1/6}/\kappa_n=o(1)$ as $n\to\infty$. For example, we may take $\kappa_n=n^{-1/7}$ or $(\log n)^{-1}$. Define for each $n\in\mathbf N$,
\begin{align}\label{Eqn: KW selection}
\xi_n\equiv \frac{\sqrt{n}\|\hat{\mathbb F}_n-\mathbb F_n\|_p}{\kappa_n}~.
\end{align}
Then $\xi_n\xrightarrow{p} 0$ if $F$ is strictly concave by Theorem \ref{Thm: Equivalence} and $\xi_n\xrightarrow{p}\infty$ if $F$ is non-strictly concave by Lemma \ref{Lem: weak limit}. Thus, if $\sqrt n\phi(\mathbb F_n)\le \kappa_n$, we may take $F$ to be strictly concave and non-strictly concave otherwise. We refer to such a selection procedure as the KW-selection. As a result, the asymptotic level of our test can be controlled in this case by choosing the critical value to be $\kappa_n$.

On the other hand, when $F$ is non-strictly concave, the nondegenerate limit $\|\mathcal M_F'(\mathbb G)-\mathbb G\|_p$ in Lemma \ref{Lem: weak limit} can be estimated by composing a suitable estimator $\hat{\mathcal M}_n'$ of $\mathcal M_F'$ with the nonparametric bootstrap estimator $\sqrt{n}\{\mathbb F_n^*-\mathbb F_n\}$ of the Gaussian process $\mathbb G$, i.e., we employ the bootstrap estimator
\begin{align}\label{Eqn: rescaled bootsrap}
\|\hat{\mathcal M}_n'(\sqrt{n}\{\mathbb F_n^*-\mathbb F_n\})-\sqrt{n}\{\mathbb F_n^*-\mathbb F_n\}\|_p~,
\end{align}
where $\mathbb F_n^*(x)=\frac{1}{n}\sum_{i=1}^{n}W_{ni}1\{X_i\le x\}$ for $x\in\mathbf R_+$, $W_n\equiv(W_{n1},\ldots,W_{nn})$ a multinomial random vector with $n$ categories and probabilities $(1/n,\ldots,1/n)$, and $\hat{\mathcal M}_n': \ell^\infty(\mathbf R_+)\to\ell^\infty(\mathbf R_+)$ is some appropriate derivative estimator. Such a bootstrap procedure is justified by Theorem 3.2 in \citet{FangSantos2018HDD}. It is in fact precisely the rescaled bootstrap proposed by \citet{Dumbgen1993}, which amounts to estimating $\mathcal M_F'$ by: for any $h\in\ell^\infty(\mathbf R_+)$,
\begin{align}\label{Eqn: rescaled bootsrap2}
\hat{\mathcal M}_n'(h)=\frac{\mathcal M(\mathbb F_n+t_nh)-\mathcal M(\mathbb F_n)}{t_n}~,
\end{align}
where $t_n\downarrow 0$ such that $t_n\sqrt n\to\infty$---see \citet[p.390-1]{FangSantos2018HDD}. In turn, we may choose the critical value in this case as: for $\alpha\in(0,1)$,
\[
\hat c_{1-\alpha}^*\equiv \inf\{c\in\mathbf R: P_W(\|\hat{\mathcal M}_n'(\sqrt{n}\{\mathbb F_n^*-\mathbb F_n\})-\sqrt{n}\{\mathbb F_n^*-\mathbb F_n\}\|_p\le c)\ge 1-\alpha\}~,
\]
where $P_W$ denotes the probability with respect to the bootstrap weights $\{W_{ni}\}_{i=1}^n$ holding the data $\{X_i\}_{i=1}^n$ fixed. In practice, $\hat c_{1-\alpha}^*$ can be estimated by Monte Carlo simulations by drawing a large number of bootstrap samples $\{X_{b,i}^*\}_{i=1}^n$ from $\mathbb F_n$ for $b=1,\ldots,B$ (with the data $\{X_i\}_{i=1}^n$ fixed).

Now, for each fixed $\alpha\in(0,1)$ we set the critical value of our test as
\[
\hat c_{1-\alpha}\equiv\max\{\kappa_n,\hat c_{1-\alpha}^*\}~,
\]
which provides pointwise asymptotic level control as confirmed by the following theorem. Before stating the theorem, we formalize requirements on $t_n$, $\kappa_n$, and the $(1-\alpha)$th quantile $c_{1-\alpha}^*$ of the cdf of the weak limit $\|\mathcal M_F'(\mathbb G)-\mathbb G\|_p$.

\begin{ass}\label{Ass: kn}
(i) $\{t_n\}$ is a sequence of positive scalars such that $t_n\downarrow 0$ such that $t_n\sqrt n\to\infty$ as $n\to\infty$; (ii) $\{\kappa_n\}$ is a sequence of positive scalars such that $\kappa_n\downarrow 0$ and $(\log n)^{2/3}n^{-1/6}/\kappa_n\to 0$ as $n\to\infty$.
\end{ass}

\begin{ass}\label{Ass: quantile regularity}
The cdf of $\|\mathcal M_F'(\mathbb G)-\mathbb G\|_p$ is continuous and strictly increasing at its $(1-\alpha)$th quantile $c_{1-\alpha}^*$ when $F$ is non-strictly concave.
\end{ass}

Assumption \ref{Ass: kn} imposes the convergence rates on the tuning parameters $\kappa_n$ and $t_n$. We do not touch the challenging issue of optimal choices in this paper. Assumption \ref{Ass: quantile regularity} is a standard technical condition (often implicitly imposed) to ensure that consistent bootstrap can produce consistent critical values---see Lemma 11.2.1 in \citet{TSH2005}. In verifying Assumption \ref{Ass: quantile regularity}, we note that, by the proof of Theorem \ref{Thm: test concavity, local}, the map $g\mapsto \phi_F'(g)\equiv\|\mathcal M_F' g-g\|_p$ is subadditive\footnote{Let $\mathbb D$ be a vector space. A function $f: \mathbb D\to\mathbf R$ is said to be subadditive if and only if $f(x+y)\le f(x)+f(y)$ for all $x,y\in\mathbb D$.} and hence convex since it is obviously positively homogeneous of degree one. Hence, by Theorem 11.1 in \citet{Davydov1998local}, the distribution function $H$ of $\|\mathcal M_F'(\mathbb G)-\mathbb G\|_p$ is absolutely continuous and strictly increasing on $(r_0,\infty)$, with $r_0\equiv\inf\{r\in\mathbf R: H(r)>0\}$. Therefore, Assumption \ref{Ass: quantile regularity} holds whenever $c_{1-\alpha}^*>r_0$.

\iffalse

Consequently, Assumption \ref{Ass: quantile regularity} can be checked by looking at the smallest value of the corresponding bootstrap sequence. Since extreme statistics converge faster than $\sqrt n$, the asymptotic effect of such a pretest is presumably negligible; a similar insight has been exploited in \citet{Carson_Sun2007Tobit}.

\fi

We are now in a position to state the first main result of this paper.

\begin{thm}\label{Thm: test concavity, pointwise}
If Assumptions \ref{Ass: setup}(i), \ref{Ass: equivalence} with $\tau=0$, \ref{Ass: kn} and \ref{Ass: quantile regularity} hold, then it follows that, under $\mathrm H_0$,
\begin{align}
\limsup_{n\to\infty}P(\sqrt{n}\phi(\mathbb F_n)> \hat c_{1-\alpha})\le\alpha~,
\end{align}
and under $\mathrm H_1$,
\begin{align}
\lim_{n\to\infty}P(\sqrt{n}\phi(\mathbb F_n)> \hat c_{1-\alpha})=1~.
\end{align}
\end{thm}

Theorem \ref{Thm: test concavity, pointwise} shows that our test is pointwise (in $P$) asymptotically level $\alpha$ and consistent under any fixed alternative. However, in view of the irregularity of the problem and as argued in the literature \citep{TSH2005,AndrewsandGuggen2010ET,Romano_Shaikh2012uniform}, pointwise asymptotics can be unreliable in ``nonstandard'' settings. It is therefore of interest to investigate the uniform or at least local properties of our test.

To study both local size control and local power of our proposed test, we follow \citet[p.384-6]{Vaart1998} and consider differentiable paths in $\mathbf P$ passing through $P_0\equiv P$ that also belong to the set
\[
\mathbf H\equiv \{\{P_t\}: \text{(i)} \phi(F(P_t))=0 \text{ if }t\le 0\text{, and (ii)} \phi(F(P_t))>0 \text{ if }t> 0\}~.
\]
Thus, a differentiable path $\{P_t\}$ in $\mathbf H$ is such that if it satisfies the null hypothesis whenever $t\le 0$ but otherwise the alternative for all $t>0$. For a differentiable path $\{P_{\eta/\sqrt{n}}\}$ where $\eta\in\mathbf R$, we set $P^n\equiv \prod_{i=1}^n P$, $P_n^n\equiv\prod_{i=1}^n P_{\eta/\sqrt{n}}$ and define the power function of our test for sample size $n$ as
\[
\pi_n(P_{\eta/\sqrt{n}})\equiv P_n^n(\sqrt{n}\phi(\mathbb F_n)>\hat c_{1-\alpha})~.
\]

Our next theorem establishes local properties of our test.

\begin{thm}\label{Thm: test concavity, local}
 Let $\{P_{t}\}$ be a differentiable path in $\mathbf H$ and let Assumptions \ref{Ass: equivalence}, \ref{Ass: local}, \ref{Ass: kn}, and \ref{Ass: quantile regularity} hold with $\tau=0$ and $F\equiv F(P_0)$. Then it follows that
\begin{enumerate}
\item For any $\eta\in\mathbf R$, (i) if $F$ is strictly concave, then $\liminf_{n\to\infty}\pi_n(P_{\eta/\sqrt{n}})=0$, and (ii) if $F$ is non-strictly concave, then
\begin{multline}\label{Thm: test concavity, limit local}
\liminf_{n\to\infty}\pi_n(P_{\eta/\sqrt{n}})\\ \ge P(\|\mathcal M_F' (\mathbb G+\eta\dot F(h))-\{\mathbb G+\eta\dot F(h)\}\|_p>c_{1-\alpha}^*)~,
\end{multline}
where \eqref{Thm: test concavity, limit local} holds with equality if, in addition, the cdf of $\|\mathcal M_F' (\mathbb G+\eta\dot F(h))-\{\mathbb G+\eta\dot F(h)\}\|_p$ is continuous at $c_{1-\alpha}^*$.
\item For any $\eta\le 0$, we have
\begin{align}\label{Thm: test concavity, local size}
\limsup_{n\to\infty}\pi_n(P_{\eta/\sqrt{n}})\le\alpha~.
\end{align}
\item If $\|\mathcal M_F' (\dot F(h))-\dot F(h)\|_p>0$, then
\begin{align}\label{Thm: test concavity, local power}
\liminf_{\eta\uparrow \infty} \liminf_{n\to\infty}\pi_n(P_{\eta/\sqrt{n}}) = 1~.
\end{align}
\end{enumerate}
\end{thm}

The first part of Theorem \ref{Thm: test concavity, local} delivers a lower bound of the local limiting power function. If the true distribution function $F_n\equiv F(P_{\eta/\sqrt n})$ is local to a strictly concave function $F$, then the limiting local power along the path $\{P_{1/\sqrt n}\}$ is zero. This is unfortunate and in fact is an implication of Theorem \ref{Thm: Equivalence, local}. The second part shows that the asymptotic null rejection rate along $\{P_{1/\sqrt n}\}$ is no larger than $\alpha$, establishing the asymptotic local size control.

The first part of Theorem \ref{Thm: test concavity, local} might leave one the impression that the test has poor local power against any sequence of local alternatives $\{P_{\eta/\sqrt n}\}$ (with $\eta>0$). The third part is intended to reconcile such a misconception. Heuristically, it says that our test has nontrivial local power if the sequence $\{P_{\eta/\sqrt n}\}$ does not approach the null too fast. To appreciate the condition $\|\mathcal M_F' (\dot F(h))-\dot F(h)\|_p>0$, we first note that it prevents $F$ from being strictly concave in which case $\mathcal M_F'$ is the identity operator and the limiting local power is zero by Part 1-(i). Second, by \eqref{Eqn: F diff}, we have that, as $n\to\infty$,
\begin{align}
\sqrt n \{F_n- F\}=\sqrt n \{F(P_{h/\sqrt n})- F(P)\}\to \eta\dot F(h)~.
\end{align}
In turn, this implies by Proposition 2.1 in \citet{Beare_Fang2016Grenander} and $\phi(F)=0$ (by the definition of $\mathbf H$) that, as $n\to\infty$,
\begin{align}
\sqrt n \phi(F_n) =\sqrt n \{\phi(F_n)- \phi(F)\} \to \|\mathcal M_F'(\eta\dot F(h))-\eta\dot F(h)\|_p~.
\end{align}
By the positive homogeneity of degree one of $\mathcal M_F'$ (as a Hadamard directional derivative), we thus have: for $\Delta\equiv \|\mathcal M_F'(\dot F(h))-\dot F(h)\|_p$,
\begin{align}
\phi(F_n)= \frac{\eta}{\sqrt n} (\Delta+o(1))~,
\end{align}
for all $\eta>0$. Therefore, $\Delta\equiv \|\mathcal M_F'(\dot F(h))-\dot F(h)\|_p>0$ implies that the third part is concerned with local power along a nontrivial Pitman drift, a canonical device for local power analysis. When $\Delta=0$ (as when $F$ is strictly concave), the speed that $\{P_{\eta/\sqrt n}\}$ approaches the null is too fast in the sense that $\sqrt n \phi(F_n)\to 0$, thereby making the test hard to reject.

\subsection{Comparisons with Existing Tests}\label{Sec: comparison}

We now compare our concavity test with some existing tests. While there is a rich literature on testing monotonicity in settings such as nonparametric regression---see \citet{Chetverikov2018Monotonicity} for a recent study with a brief survey, it is somewhat surprising that results for the same problem in the density context seem rather limited. This is just one piece of evidence consistent with Jon Wellner's view that ``the (shape-constrained) community also needs to do more work to provide inferential methods beyond estimation'' \citep{BanerjeeSamworth2018Interview}.

\citet{Banerjee_Wellner2001} propose likelihood ratio tests for equality at a fixed point, rather than the global shape as we consider, in the setting of nonparametric estimation of a monotone function---see also \citet{Banerjee2005LR}. More related to our setting is the work by \citet{Woodroofe_Sun1999}, who are concerned with testing uniformity versus a monotone (but not uniform) density. Translated to our setup, they study the simple null $F$ being the uniform distribution against the composite alternative that $F$ is convex (but is not a uniform distribution). Therefore, the parameter spaces under the null and the alternative are smaller than the corresponding spaces in our setup. \citet{Woodroofe_Sun1999} propose two tests, namely, the $P$-test and the $D$-test, based on a penalized nonparametric maximum likelihood estimator of the density. The $P$-test statistic is simply the penalized likelihood ratio, while the $D$-test statistic is $\sqrt n$---with $n$ being the sample size---times the uniform distance between the penalized cdf estimator and the uniform cdf.

The simple null hypothesis in \citet{Woodroofe_Sun1999}, though greatly simplifies the asymptotic analysis, may be restrictive from a practical point of view. This is a reflection of the fact that the pointwise asymptotic distributions of the test statistic under a composite null such as ours are highly nonstandard. In a nonparametric regression setting, \citet{Durot2003testing} studies the composite null that the regression function is nonincreasing against the alternative that it is not. The test statistic proposed by Durot is a rescaled version of the uniform distance between a cumulative regression estimator and its least concave majorant. \citet{Durot2003testing} shows that the constant regression functions are asymptotically least favorable, base on which critical values are constructed. \citet{Kulikov_Lopuhaa2004testing} adapt \citet{Durot2003testing}'s test to the density setup based on the same critical values---see their Theorem 3.1, and also Proposition 3.1 in \citet{Kulikov_Lopuhaa2008}. \citet{Kulikov_Lopuhaa2004testing} also propose a test based on the $L_p$-norm (against the empirical distribution) of the empirical cdf and its least concave majorant, also relying on critical values constructed from the least favorable distribution. We note that the test based on the statistic $T_n$ in \citet{Kulikov_Lopuhaa2004testing} is invalid in the sense that it may over-reject under the null, as pointed out by the same authors and confirmed by their simulations.

Tests based on the least favorable distributions, though control the size in a very simple way, may be too conservative and result in power loss. This includes the tests of \citet{Durot2003testing} and \citet{Kulikov_Lopuhaa2004testing}. On the other hand, the $P$-test and the $D$-test explicitly take into account the alternative hypothesis of nondecreasing densities. Therefore, they are powerful in detecting nondecreasing densities, but may perform poorly when the underlying distribution is neither concave nor convex. Our simulations confirm these predictions. Finally, we reiterate that the need of bootstrap for testing concavity as well as the tuning parameters $\kappa_n$ and $t_n$ is in line with the nonstandard nature of the problem, rather than a special attribute of our inferential framework. The implementation of our bootstrap is as simple as calculating the test statistic, or as simple as computing the least favorable majorant.

\iffalse

at some point mention \citet{Sen_Banerjee_Woodroofe2010}???

\begin{enumerate}
  \item \citet{Woodroofe_Sun1999}: $f$ is supported on $[0,1]$, test $\mathrm H_0: f=1$ vs $\mathrm H_1: f$ is nonincreasing; see their second paragraph for quick introduction of their two tests. No need to compute the critical values, which is good.

  \item \citet{Kulikov_Lopuhaa2004testing}: (i) the sup statistic has lfc at uniform density (ii) the weak limit of the sup statistic under strict monotonicity is not available yet; (iii) for Lp with $p\in[1,\infty)$, the lfc property does hold unless the integration is taken against the true cdf, but unlike the sup, its weak limit under strict concavity is available---\citet{Kulikov_Lopuhaa2008}
  \item May also want to compare to \citet{Durot2003testing} even though it is concerned with regression. But this is fine since \citet{Kulikov_Lopuhaa2004testing} did the comparison anyway justified by the stochastic dominance results. This test is simple as it uses lfc.
  \item Mention that lfc is problematic based on \citet{BeareandMoon2015}
  \item Also \citet{LopuhaaMusta2019CLT}: proposes a nonparametric test for monotonicity on the basis of the Lp-distance between the smoothed Grenander-type
estimator and the kernel estimator; basically a bootstrap test; see Theorem 6 in \citet{Durot2007Lp} for verifying Assumption 2.
\end{enumerate}

Ok, tests to compare:

\begin{enumerate}
  \item Dip test + lfc under uniformity: note that \citet{Kulikov_Lopuhaa2004testing} mistakenly uses empirical cdf for this test
  \item Sup distance based on empirical cdf: this is $S_n$ in \citet{Kulikov_Lopuhaa2004testing}, similar to \citet{Durot2003testing} (which is for regression)
  \item The $R_n$ test + lfc in \citet{Kulikov_Lopuhaa2004testing}. Both Rn and Sn have critical values ready.
  \item \citet{LopuhaaMusta2019CLT}
\end{enumerate}

Note: should exclude the $T_n$ test in \citet{Kulikov_Lopuhaa2004testing} since it does not control size; not sure if should look at the P-test in \citet{Woodroofe_Sun1999} which looks at a different test problem.

\fi

\section{Simulation Experiments}\label{Sec: simulation}

We next evaluate the finite sample performance of our test through Monte Carlo simulations. Special attention shall be paid to the tuning parameters $\kappa_n$ (for the KW selection) and $t_n$ (for the rescaled bootstrap). For this, we consider $\kappa_n\in\{n^{-1/7},n^{-1/8}, \log^{-1}n\}$ and $t_n\in\{n^{-1/\varpi}: \varpi=3,4,\ldots,7\}$. We run two sets of simulations in order to compare with the standard bootstrap and some existing monotonicity tests. Throughout, we let the significance level be $5\%$, and all results are based on 1000 Monte Carlo simulations and 500 bootstrap repetitions for each simulation replication (to implement our test).

First, we consider the distributions defined by the following densities supported on $\mathbf R_+$: for any $x\in\mathbf R_+$,
\begin{align}
f_1(x) & =
\frac{1}{\sqrt{\pi/2}}\exp\{-\frac{x^2}{2}\} ~,\label{Eqn: sim1 f1} \\
f_2(x) & =\begin{cases}
3/2 & \text{ if }0\le x<1/2\\
1/2 & \text{ if }1/2\le x<9/10\\
\exp\{9/10-x\}/20 &\text{ if }x\ge 9/10
\end{cases}~.\label{Eqn: sim1 f2}
\end{align}
The function $f_1$ is strictly decreasing, while $f_2$ is weakly decreasing though strictly decreasing on the region $[9/10,\infty)$. Hence, the corresponding distribution functions, denoted $F_1$ and $F_2$, are strictly concave and non-strictly concave respectively. To examine local behaviors of our test, we construct differentiable paths passing through $F_1$ and $F_2$ as follows: for $t\in\mathbf R_+$ and $x\in\mathbf R_+$,
\begin{align}
f_{1,t}(x)&= \frac{\exp\{-tx\}f_1(x)}{\int_{\mathbf R_+} \exp\{-tx\}f_1(x) dx} =
\frac{1}{a(t)}\exp\{-\frac{(x+t)^2}{2}\} ~,\label{Eqn: sim1 f1t} \\
f_{2,t}(x)&=\begin{cases}
e^{-t}3/2 & \text{ if }0\le x<1/2\\
e^t/2 & \text{ if }1/2\le x<9/10\\
b(t)\exp\{9/10-x\}/20 &\text{ if }x\ge 9/10
\end{cases}~,\label{Eqn: sim1 f2t}
\end{align}
where $a(t)\equiv\int_t^\infty \exp\{-u^2/2\}du$ and $b(t)\equiv 20-15e^{-t}-4e^t$. By Example 3.2.1 and Proposition 2.1.1 in \citet{BKRW993Efficient}, these two paths are indeed differentiable (under the null). We then generate i.i.d.\ samples $\{X_i\}_{i=1}^n$ from \eqref{Eqn: sim1 f1}, \eqref{Eqn: sim1 f2}, \eqref{Eqn: sim1 f1t} and \eqref{Eqn: sim1 f2t}, with $n\in\{100,200,300,400,1000\}$. For simplicity, we only consider the sup statistic $\sqrt n\|\hat{\mathbb F}_n-\mathbb F_n\|_\infty$, and report results based on the rescaled and the standard bootstrap.

{\renewcommand{\arraystretch}{1.45}
\begin{table}[!h]
\caption{Pointwise Size Control with Strict Concavity Based on \eqref{Eqn: sim1 f1}}\label{Tab: size pointwise, sc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccccc}
\hline\hline
 &   & \multicolumn{5}{c}{Rescaled} & \multirow{3}{*}[-0.5em]{Standard}\\
\cmidrule{3-7}
 &   & \multicolumn{5}{c}{$t_n$}    & \\
\cmidrule{3-7}
$n$ & $\kappa_n$                     & $n^{-1/3}$ & $n^{-1/4}$ & $n^{-1/5}$ & $n^{-1/6}$ & $n^{-1/7}$ & \\
 \hline
 \multirow{3}{*}{ 100 } & $n^{-1/7}$ & $0.017$    & $0.008$    & $0.006$    & $0.005$    & $0.004$ & \multirow{3}{*}{ $0.053$ }\\
                        & $n^{-1/8}$ & $0.017$    & $0.008$    & $0.006$    & $0.005$    & $0.005$ &   \\
                  & $(\log n)^{-1}$  & $0.016$    & $0.008$    & $0.006$    & $0.005$    & $0.004$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 200 } & $n^{-1/7}$ & $0.022$    & $0.015$    & $0.011$    & $0.009$    & $0.009$ & \multirow{3}{*}{ $0.071$ }\\
                        & $n^{-1/8}$ & $0.021$    & $0.015$    & $0.011$    & $0.009$    & $0.009$ &   \\
                  & $(\log n)^{-1}$  & $0.021$    & $0.015$    & $0.011$    & $0.009$    & $0.009$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 300 } & $n^{-1/7}$ & $0.010$    & $0.002$    & $0.002$    & $0.003$    & $0.003$ & \multirow{3}{*}{ $0.064$ }\\
                        & $n^{-1/8}$ & $0.011$    & $0.002$    & $0.002$    & $0.003$    & $0.002$ &   \\
                  & $(\log n)^{-1}$  & $0.010$    & $0.002$    & $0.002$    & $0.003$    & $0.003$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 400 } & $n^{-1/7}$ & $0.013$    & $0.006$    & $0.000$    & $0.000$    & $0.000$ & \multirow{3}{*}{ $0.068$ }\\
                        & $n^{-1/8}$ & $0.014$    & $0.005$    & $0.000$    & $0.000$    & $0.000$ &   \\
                  & $(\log n)^{-1}$  & $0.014$    & $0.005$    & $0.000$    & $0.000$    & $0.000$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 1000 }& $n^{-1/7}$ & $0.009$    & $0.002$    & $0.003$    & $0.000$    & $0.000$ & \multirow{3}{*}{ $0.078$ }\\
                        & $n^{-1/8}$ & $0.009$    & $0.002$    & $0.003$    & $0.000$    & $0.000$ &   \\
                  & $(\log n)^{-1}$  & $0.009$    & $0.002$    & $0.003$    & $0.000$    & $0.000$ &   \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

Tables \ref{Tab: size pointwise, sc}, \ref{Tab: size pointwise, nsc}, \ref{Tab: size local, sc} and \ref{Tab: size local, nsc} present the numerical results. Inconsistency of the standard bootstrap is prominently evidenced in last columns of Tables \ref{Tab: size pointwise, nsc} and \ref{Tab: size local, nsc} which are based on non-strictly concave functions (so that the operator $\mathcal M$ are only Hadamard directionally differentiable at these distribution functions). On the contrary, our test alleviates the size distortion both pointwise and locally, though the results vary with the choices of the tuning parameters. Overall, the choices $t_n\in\{n^{-1/3},n^{-1/4}\}$ tend to be too small, while $t_n\in\{n^{-1/5},n^{-1/6},n^{-1/7}\}$ are adequate in that they control the size well. Tables \ref{Tab: size pointwise, sc} and \ref{Tab: size local, sc} also show that, when the distribution function is strictly concave, our test by design controls the size both pointwise and locally, though the rejection rates are often close to zero, which is consistent with Theorem \ref{Thm: test concavity, local}. Moreover, as in Tables \ref{Tab: size pointwise, sc} and \ref{Tab: size local, sc}, the standard bootstrap also exhibits size control, due to the facts that the LCM operator is fully (Hadamard) differentiable at strictly concave distribution functions and that the distance between the Grenander distribution estimator and the empirical cdf converges faster than $\sqrt n$ by the Kiefer-Wolfowitz theorems. Finally, we note that the results are quite insensitive to the choice of $\kappa_n$.

{\renewcommand{\arraystretch}{1.5}
\begin{table}[!h]
\caption{Pointwise Size Control with Non-Strict Concavity Based on \eqref{Eqn: sim1 f2}}\label{Tab: size pointwise, nsc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccccc}
\hline\hline
 &   & \multicolumn{5}{c}{Rescaled} & \multirow{3}{*}[-0.5em]{Standard}\\
\cmidrule{3-7}
 &   & \multicolumn{5}{c}{$t_n$}    & \\
\cmidrule{3-7}
$n$ & $\kappa_n$                     & $n^{-1/3}$ & $n^{-1/4}$ & $n^{-1/5}$ & $n^{-1/6}$ & $n^{-1/7}$ & \\
 \hline
 \multirow{3}{*}{ 100 } & $n^{-1/7}$ & $0.068$    & $0.051$    & $0.046$    & $0.041$    & $0.036$ & \multirow{3}{*}{ $0.132$ }\\
                        & $n^{-1/8}$ & $0.069$    & $0.049$    & $0.044$    & $0.041$    & $0.036$ &   \\
                  & $(\log n)^{-1}$  & $0.070$    & $0.052$    & $0.045$    & $0.041$    & $0.036$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 200 } & $n^{-1/7}$ & $0.082$    & $0.061$    & $0.052$    & $0.046$    & $0.045$ & \multirow{3}{*}{ $0.151$ }\\
                        & $n^{-1/8}$ & $0.084$    & $0.063$    & $0.054$    & $0.045$    & $0.043$ &   \\
                  & $(\log n)^{-1}$  & $0.080$    & $0.063$    & $0.053$    & $0.046$    & $0.044$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 300 } & $n^{-1/7}$ & $0.086$    & $0.066$    & $0.056$    & $0.046$    & $0.040$ & \multirow{3}{*}{ $0.169$ }\\
                        & $n^{-1/8}$ & $0.085$    & $0.068$    & $0.056$    & $0.047$    & $0.040$ &   \\
                  & $(\log n)^{-1}$  & $0.086$    & $0.067$    & $0.056$    & $0.046$    & $0.040$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 400 } & $n^{-1/7}$ & $0.109$    & $0.081$    & $0.064$    & $0.057$    & $0.053$ & \multirow{3}{*}{ $0.191$ }\\
                        & $n^{-1/8}$ & $0.106$    & $0.079$    & $0.064$    & $0.057$    & $0.050$ &   \\
                  & $(\log n)^{-1}$  & $0.108$    & $0.079$    & $0.063$    & $0.058$    & $0.051$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 1000 }& $n^{-1/7}$ & $0.096$    & $0.071$    & $0.063$    & $0.055$    & $0.046$ & \multirow{3}{*}{ $0.205$ }\\
                        & $n^{-1/8}$ & $0.095$    & $0.072$    & $0.063$    & $0.054$    & $0.046$ &   \\
                  & $(\log n)^{-1}$  & $0.093$    & $0.073$    & $0.062$    & $0.056$    & $0.046$ &   \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

{\renewcommand{\arraystretch}{1.5}
\begin{table}[!h]
\caption{Local Size Control with Strict Concavity Based on $f_{1,1/\sqrt n}$ from \eqref{Eqn: sim1 f1t}}\label{Tab: size local, sc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccccc}
\hline\hline
 &   & \multicolumn{5}{c}{Rescaled} & \multirow{3}{*}[-0.5em]{Standard}\\
\cmidrule{3-7}
 &   & \multicolumn{5}{c}{$t_n$}    & \\
\cmidrule{3-7}
$n$ & $\kappa_n$                     & $n^{-1/3}$ & $n^{-1/4}$ & $n^{-1/5}$ & $n^{-1/6}$ & $n^{-1/7}$ & \\
 \hline
 \multirow{3}{*}{ 100 } & $n^{-1/7}$ & $0.017$    & $0.009$    & $0.007$    & $0.008$    & $0.006$ & \multirow{3}{*}{ $0.050$ }\\
                        & $n^{-1/8}$ & $0.016$    & $0.010$    & $0.007$    & $0.007$    & $0.006$ &   \\
                  & $(\log n)^{-1}$  & $0.017$    & $0.008$    & $0.008$    & $0.007$    & $0.006$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 200 } & $n^{-1/7}$ & $0.018$    & $0.006$    & $0.004$    & $0.003$    & $0.003$ & \multirow{3}{*}{ $0.060$ }\\
                        & $n^{-1/8}$ & $0.018$    & $0.006$    & $0.004$    & $0.003$    & $0.003$ &   \\
                  & $(\log n)^{-1}$  & $0.018$    & $0.006$    & $0.004$    & $0.003$    & $0.003$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 300 } & $n^{-1/7}$ & $0.015$    & $0.009$    & $0.006$    & $0.002$    & $0.002$ & \multirow{3}{*}{ $0.059$ }\\
                        & $n^{-1/8}$ & $0.014$    & $0.009$    & $0.006$    & $0.002$    & $0.002$ &   \\
                  & $(\log n)^{-1}$  & $0.016$    & $0.009$    & $0.006$    & $0.002$    & $0.002$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 400 } & $n^{-1/7}$ & $0.015$    & $0.002$    & $0.003$    & $0.000$    & $0.000$ & \multirow{3}{*}{ $0.070$ }\\
                        & $n^{-1/8}$ & $0.015$    & $0.002$    & $0.003$    & $0.000$    & $0.000$ &   \\
                  & $(\log n)^{-1}$  & $0.015$    & $0.002$    & $0.003$    & $0.000$    & $0.000$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 1000 }& $n^{-1/7}$ & $0.009$    & $0.002$    & $0.002$    & $0.002$    & $0.002$ & \multirow{3}{*}{ $0.070$ }\\
                        & $n^{-1/8}$ & $0.010$    & $0.002$    & $0.002$    & $0.002$    & $0.002$ &   \\
                  & $(\log n)^{-1}$  & $0.011$    & $0.002$    & $0.000$    & $0.002$    & $0.002$ &   \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

{\renewcommand{\arraystretch}{1.5}
\begin{table}[!h]
\caption{Local Size Control with Non-Strict Concavity Based on $f_{2,1/\sqrt n}$ from \eqref{Eqn: sim1 f2t}}\label{Tab: size local, nsc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccccc}
\hline\hline
 &   & \multicolumn{5}{c}{Rescaled} & \multirow{3}{*}[-0.5em]{Standard}\\
\cmidrule{3-7}
 &   & \multicolumn{5}{c}{$t_n$}    & \\
\cmidrule{3-7}
$n$ & $\kappa_n$                     & $n^{-1/3}$ & $n^{-1/4}$ & $n^{-1/5}$ & $n^{-1/6}$ & $n^{-1/7}$ & \\
 \hline
 \multirow{3}{*}{ 100 } & $n^{-1/7}$ & $0.070$    & $0.052$    & $0.043$    & $0.041$    & $0.036$ & \multirow{3}{*}{ $0.128$ }\\
                        & $n^{-1/8}$ & $0.070$    & $0.051$    & $0.042$    & $0.038$    & $0.038$ &   \\
                  & $(\log n)^{-1}$  & $0.073$    & $0.051$    & $0.043$    & $0.038$    & $0.034$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 200 } & $n^{-1/7}$ & $0.112$    & $0.074$    & $0.063$    & $0.051$    & $0.045$ & \multirow{3}{*}{ $0.193$ }\\
                        & $n^{-1/8}$ & $0.111$    & $0.076$    & $0.063$    & $0.049$    & $0.045$ &   \\
                  & $(\log n)^{-1}$  & $0.113$    & $0.075$    & $0.062$    & $0.049$    & $0.042$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 300 } & $n^{-1/7}$ & $0.098$    & $0.066$    & $0.055$    & $0.050$    & $0.044$ & \multirow{3}{*}{ $0.182$ }\\
                        & $n^{-1/8}$ & $0.099$    & $0.067$    & $0.058$    & $0.049$    & $0.044$ &   \\
                  & $(\log n)^{-1}$  & $0.100$    & $0.070$    & $0.057$    & $0.050$    & $0.044$ &   \\
                  \cmidrule{3-8}
 \multirow{3}{*}{ 400 } & $n^{-1/7}$ & $0.113$    & $0.084$    & $0.071$    & $0.061$    & $0.055$ & \multirow{3}{*}{ $0.197$ }\\
                        & $n^{-1/8}$ & $0.113$    & $0.085$    & $0.070$    & $0.062$    & $0.055$ &   \\
                  & $(\log n)^{-1}$  & $0.112$    & $0.083$    & $0.069$    & $0.061$    & $0.056$ &   \\
                  \cmidrule{3-8}
\multirow{3}{*}{ 1000 } & $n^{-1/7}$ & $0.092$    & $0.064$    & $0.055$    & $0.046$    & $0.036$ & \multirow{3}{*}{ $0.196$ }\\
                        & $n^{-1/8}$ & $0.093$    & $0.061$    & $0.055$    & $0.044$    & $0.036$ &   \\
                  & $(\log n)^{-1}$  & $0.094$    & $0.066$    & $0.055$    & $0.045$    & $0.037$ &   \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

\iffalse

\begin{table}[!h]
\caption{Pointwise Size Control: Non-Strict Concavity}\label{Tab: size pointwise, nsc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
 &   & \multicolumn{3}{c}{Rescaled} & Standard\\
\cmidrule{3-5}
$n$ & $\kappa_n$ & $t_n=n^{-1/4}$ & $t_n=n^{-1/3}$ & $t_n=n^{-2/5}$ \\
 \hline
 \multirow{3}{*}{ 100 } & $n^{-1/7}$ & 0.0410 & 0.0510 & 0.0650 & \multirow{3}{*}{ 0.0903 }\\
 & $(\log n)^{-1}$ & 0.0450 & 0.0550 & 0.0680 & \\
  & $n^{-1/8}$ & 0.0410 & 0.0530 & 0.0670 & \\
\multirow{3}{*}{ 200 } & $n^{-1/7}$ & 0.0560 & 0.0660 & 0.0770 & \multirow{3}{*}{ 0.0980 }\\
 & $(\log n)^{-1}$ & 0.0510 & 0.0630 & 0.0730 & \\
  & $n^{-1/8}$ & 0.0470 & 0.0590 & 0.0740 & \\
\multirow{3}{*}{ 300 } & $n^{-1/7}$ & 0.0440 & 0.0590 & 0.0770 & \multirow{3}{*}{ 0.1153 }\\
 & $(\log n)^{-1}$ & 0.0570 & 0.0790 & 0.0940 & \\
 & $n^{-1/8}$ &  0.0590 & 0.0680 & 0.0820 & \\
\multirow{3}{*}{ 400 } & $n^{-1/7}$ & 0.0490 & 0.0680 & 0.0830 & \multirow{3}{*}{ 0.1070 }\\
 & $(\log n)^{-1}$ & 0.0550 & 0.0730 & 0.0830 & \\
 & $n^{-1/8}$ &  0.0400 & 0.0520 & 0.0690 & \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}

\begin{table}[!h]
\caption{Local Size Control: Non-Strict Concavity}\label{Tab: size local, lnsc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
  & & \multicolumn{3}{c}{Rescaled} & Standard\\
\cmidrule{3-5}
$n$ & $\kappa_n$ & $t_n=n^{-1/4}$ & $t_n=n^{-1/3}$ & $t_n=n^{-2/5}$ \\
 \hline
 \multirow{3}{*}{ $n=100$ } & $n^{-1/7}$ & 0.0270 & 0.0380 & 0.0490 & \multirow{3}{*}{ 0.0673 }\\
 & $(\log n)^{-1}$ & 0.0310 & 0.0440 & 0.0530 & \\
 & $n^{-1/8}$ & 0.0240 & 0.0310 & 0.0390 & \\
\multirow{3}{*}{ $n=200$ } & $n^{-1/7}$ & 0.0420 & 0.0500 & 0.0660 & \multirow{3}{*}{ 0.0933 }\\
 & $(\log n)^{-1}$ & 0.0350 & 0.0430 & 0.0530 & \\
 & $n^{-1/8}$ & 0.0490 & 0.0600 & 0.0700 & \\
\multirow{3}{*}{ $n=300$ } & $n^{-1/7}$ & 0.0450 & 0.0580 & 0.0710 & \multirow{3}{*}{ 0.1057 }\\
 & $(\log n)^{-1}$ & 0.0490 & 0.0640 & 0.0770 & \\
 & $n^{-1/8}$ & 0.0490 & 0.0600 & 0.0700\\
\multirow{3}{*}{ $n=400$ } & $n^{-1/7}$ & 0.0450 & 0.0540 & 0.0670 & \multirow{3}{*}{ 0.1007 }\\
 & $(\log n)^{-1}$ & 0.0510 & 0.0650 & 0.0820 & \\
 & $n^{-1/8}$ & 0.0420 & 0.0570 & 0.0730\\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}

\begin{table}[!h]
\caption{Pointwise Size Control: Strict Concavity}\label{Tab: size pointwise, sc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
  & & \multicolumn{3}{c}{Rescaled} & Standard\\
\cmidrule{3 -5}
$n$ & $\kappa_n$ & $t_n=n^{-1/4}$ & $t_n=n^{-1/3}$ & $t_n=n^{-2/5}$ \\
 \hline
 \multirow{3}{*}{ 100 } & $n^{-1/7}$ & 0.0070 & 0.0080 & 0.0110 & \multirow{3}{*}{ 0.0190 }\\
  & $(\log n)^{-1}$ & 0.0040 & 0.0080 & 0.0120 & \\
 & $n^{-1/8}$ & 0.0050 & 0.0080 & 0.0120 & \\
\multirow{3}{*}{ 200 } & $n^{-1/7}$ & 0.0001 & 0.0030 & 0.0080 & \multirow{3}{*}{ 0.0126 }\\
  & $(\log n)^{-1}$ & 0.0000 & 0.0040 & 0.0070 & \\
 & $n^{-1/8}$ & 0.0050 & 0.0060 & 0.0070 & \\
\multirow{3}{*}{ 300 } & $n^{-1/7}$ & 0.0050 & 0.0070 & 0.0120 & \multirow{3}{*}{ 0.0117 }\\
 & $(\log n)^{-1}$ & 0.0020 & 0.0020 & 0.0030 & \\
 & $n^{-1/8}$ & 0.0020 & 0.0040 & 0.0050 \\
\multirow{3}{*}{ 400 } & $n^{-1/7}$ & 0.0010 & 0.0010 & 0.0010 & \multirow{3}{*}{ 0.0070 }\\
 & $(\log n)^{-1}$ & 0.0030 & 0.0030 & 0.0050 & \\
 & $n^{-1/8}$ & 0.0030 & 0.0020 & 0.0040 \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}

\begin{table}[!h]
\caption{Local Size Control: Strict Concavity}\label{Tab: size local, lsc}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
 & & \multicolumn{3}{c}{Rescaled} & Standard\\
\cmidrule{3-5}
$n$ & $\kappa_n$ & $t_n=n^{-1/4}$ & $t_n=n^{-1/3}$ & $t_n=n^{-2/5}$ \\
 \hline
 \multirow{3}{*}{ $n=100$ } & $n^{-1/7}$ & 0.0080 & 0.0080 & 0.0110 & \multirow{3}{*}{ 0.0187 }\\
 & $(\log n)^{-1}$ & 0.0070 & 0.0090 & 0.0140 & \\
 & $n^{-1/8}$ & 0.0020 & 0.0050 & 0.0090 & \\
\multirow{3}{*}{ $n=200$ } & $n^{-1/7}$ & 0.0020 & 0.0040 & 0.0060 & \multirow{3}{*}{ 0.0110 }\\
 & $(\log n)^{-1}$ & 0.0040 & 0.0040 & 0.0080 & \\
 & $n^{-1/8}$ & 0.0040 & 0.0070 & 0.0070 & \\
\multirow{3}{*}{ $n=300$ } & $n^{-1/7}$ &  0.0020 & 0.0040 & 0.0050 & \multirow{3}{*}{ 0.0057 } \\
 & $(\log n)^{-1}$ & 0.0020 & 0.0020 & 0.0060 & \\
 & $n^{-1/8}$ & 0.0010 & 0.0010 & 0.0010 & \\
\multirow{3}{*}{ $n=400$ } & $n^{-1/7}$ &  0.0030 & 0.0030 & 0.0060 & \multirow{3}{*}{ 0.0090 } \\
 & $(\log n)^{-1}$ & 0.0020 & 0.0040 & 0.0070 & \\
 & $n^{-1/8}$ & 0.0020 & 0.0030 & 0.0040 & \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}

\fi

Next, we compare our test with some existing tests. As before, the null hypothesis is that the true cdf is conave (or, equivalently, the true pdf is nonincreasing), while the alternative is that it is not. We shall consider the $D$ and the $P$ tests in \citet{Woodroofe_Sun1999}, the Kolmogorov-Smirnov type test in \citet{Durot2003testing} and \citet{Kulikov_Lopuhaa2004testing}, labelled as the KS test, and the test based on the $R_n$ statistic with $k=1$ in \citet{Kulikov_Lopuhaa2004testing}, labelled as the KL test. We design two families of distributions. The first family consists of distributions defined by: for $\lambda\in\mathbf R$,
\begin{align}\label{Eqn: cdf compare1}
F_\lambda(x) = \begin{cases}
0 & \text{ if }x<0\\
\frac{e^{\lambda x} -1 }{e^{\lambda}- 1} & \text{ if }0\le x\le 1 \\
1 & \text{ if }x>1
\end{cases}~,
\end{align}
where $F_\lambda$ is understood to be the uniform cdf if $\lambda=0$. Clearly, $F_\lambda$ is concave and hence in the null whenever $\lambda\le 0$, but is convex and hence in the alternative if $\lambda>0$---see Figure \ref{Fig: cdfs in simulations}-(a). Thus, this is a setup well suited for the $D$ test and the $P$ test proposed by \citet{Woodroofe_Sun1999}, because the alternative consists of only convex cdfs (or nondecreasing pdfs). As mentioned in Section \ref{Sec: comparison}, the $D$ and the $P$ tests may perform poorly in detecting distributions that are neither convex nor concave. To verify this point numerically, we consider a family of distributions (that belongs to the alternative) defined by: for $\lambda \in (0,1)$,
\begin{align}\label{Eqn: cdf compare2}
F_\lambda(x) = \begin{cases}
0 & \text{ if }x<0\\
-(x-\lambda)^2/\lambda+\lambda & \text{ if }0\le x<\lambda \\
x & \text{ if }\lambda\le x<1\\
1 & \text{ if }\lambda\ge 1
\end{cases}~.
\end{align}
As shown in Figure \ref{Fig: cdfs in simulations}-(b), $F_\lambda$ defined in \eqref{Eqn: cdf compare2} is neither concave nor convex on $[0,1]$ for any $\lambda\in(0,1)$, though it is concave on $[0,\lambda]$ and $[\lambda,1]$.

\begin{figure}
\centering
\subfloat[Starting from the top are cdfs in \eqref{Eqn: cdf compare1} corresponding to $\lambda= -5, -1,-0.2,0.2,1,5$]{
\begin{tikzpicture}[scale=0.75,baseline]
\begin{axis}[domain=0:1, samples=100, restrict y to domain=0:1,
cycle list={%
{NavyBlue,solid},
{Azure4,densely dashdotdotted},
{Green3, densely dotted},
{Green3, densely dotted},
{Azure4,densely dashdotdotted},
{NavyBlue,solid}
}
]
\foreach \theta in {-5,-1, -0.2, 0.2, 1, 5}{%
\addplot+[domain=0:1] {(exp(\theta*\x)-1)/(exp(\theta)-1)}
}
\end{axis}
\end{tikzpicture}}
\quad
\subfloat[Starting from the top are cdfs in \eqref{Eqn: cdf compare2} corresponding to $\lambda= 0.9, 0.6, 0.3$]{
\begin{tikzpicture}[scale=0.75,baseline]
\begin{axis}[domain=0:1, samples=100,restrict y to domain=0:1,
cycle list={%
{Green3, densely dotted},
{Azure4,densely dashdotdotted},
{NavyBlue,solid}
}
]
\foreach \theta in {0.3, 0.6, 0.9}{%
\addplot+[domain=0:1] {(\x<\theta)*(-(\x-\theta)^2/\theta+\theta)+(\x>=\theta)*\x};
}
\end{axis}
\end{tikzpicture}}
\caption{Graphical representations of the cdfs in \eqref{Eqn: cdf compare1} and \eqref{Eqn: cdf compare2}}\label{Fig: cdfs in simulations}
\end{figure}

As suggested by \citet{Woodroofe_Sun1999}, we choose the penalty parameter $c$ for the $D$ and the $P$ tests to be $0.2$, and the corresponding critical values may be found in their Table 1. The critical values for the KS and the KL tests are given in Table 1 in \citet{Kulikov_Lopuhaa2004testing}---see also Table 1 in \citet{Durot2003testing}. For convenience, if $(t_n,\kappa_n)=(n^{-1/3},n^{1-/7})$, then we label our test as R37---``R'' stands for ``the rescaled bootstrap''; if $(t_n,\kappa_n)=(n^{-1/3},\log^{-1}n)$, then the resulting test is labelled as R3L---``L'' stands for ``logarithm''; so on and so forth. Below, we try more sample sizes with $n\in\{100,200,300,400,600,800,1000\}$, but only report results for $n\in\{100,400\}$ in the main text and relegate the rest to Appendix \ref{App: Simulation}.

{\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{5pt}
\begin{table}[!h]
\caption{Comparisons with Existing Tests Based on \eqref{Eqn: cdf compare1}: $n=100$}\label{Tab: comparison, n100}
\begin{footnotesize}
\begin{center}
\begin{tabular}{ccccccccccc}
\hline\hline
  & \multicolumn{10}{c}{$\lambda$}  \\
\cmidrule{2-11}
 & $-1$ & $-0.1$ & $-0.05$ & $0$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ & $1$ \\
\hline
R37    & $0.010$ & $0.084$ & $0.106$ & $0.095$ & $0.124$ &  $0.132$ & $0.181$ & $0.194$ & $0.235$ & $0.819$ \\
R38    & $0.009$ & $0.085$ & $0.107$ & $0.094$ & $0.123$ & $0.133$ & $0.180$ & $0.187$ & $0.235$ & $0.818$ \\
R3L    & $0.009$ & $0.085$ & $0.107$ & $0.096$ & $0.123$ & $0.135$ & $0.180$ & $0.189$ & $0.236$ & $0.818$ \\
R47    & $0.005$ & $0.062$ & $0.084$ & $0.079$ & $0.103$ & $0.109$ & $0.157$ & $0.165$ & $0.206$ & $0.799$ \\
R48    & $0.004$ & $0.064$ & $0.086$ & $0.080$ & $0.104$ & $0.111$ & $0.156$ & $0.165$ & $0.208$ & $0.799$ \\
R4L    & $0.004$ & $0.061$ & $0.088$ & $0.079$ & $0.102$ & $0.110$ & $0.157$ & $0.165$ & $0.209$ & $0.798$ \\
R57    & $0.003$ & $0.049$ & $0.079$ & $0.076$ & $0.094$ & $0.097$ & $0.149$ & $0.152$ & $0.194$ & $0.789$ \\
R58    & $0.003$ & $0.051$ & $0.078$ & $0.075$ & $0.094$ & $0.099$ & $0.146$ & $0.153$ & $0.195$ & $0.789$ \\
R5L    & $0.003$ & $0.050$ & $0.079$ & $0.075$ & $0.095$ & $0.097$ & $0.145$ & $0.154$ & $0.195$ & $0.789$ \\
R67    & $0.002$ & $0.049$ & $0.075$ & $0.075$ & $0.091$ & $0.091$ & $0.137$ & $0.144$ & $0.187$ & $0.784$ \\
R68    & $0.002$ & $0.049$ & $0.076$ & $0.075$ & $0.091$ & $0.091$ & $0.137$ & $0.145$ & $0.187$ & $0.784$ \\
R6L    & $0.002$ & $0.047$ & $0.075$ & $0.073$ & $0.091$ & $0.090$ & $0.144$ & $0.146$ & $0.188$ & $0.787$ \\
R77    & $0.002$ & $0.047$ & $0.073$ & $0.071$ & $0.087$ & $0.088$ & $0.137$ & $0.143$ & $0.180$ & $0.782$ \\
R78    & $0.003$ & $0.045$ & $0.074$ & $0.071$ & $0.088$ & $0.090$ & $0.139$ & $0.145$ & $0.182$ & $0.783$ \\
R7L    & $0.002$ & $0.045$ & $0.074$ & $0.072$ & $0.089$ & $0.088$ & $0.141$ & $0.143$ & $0.182$ & $0.784$ \\ \hline
P-Test & $0.003$ & $0.032$ & $0.030$ & $0.037$ & $0.064$ & $0.068$ & $0.072$ & $0.079$ & $0.119$ & $0.764$ \\
D-Test & $0.000$ & $0.018$ & $0.039$ & $0.036$ & $0.061$ & $0.071$ & $0.090$ & $0.108$ & $0.170$ & $0.809$ \\
KS-Test & $0.000$ & $0.010$ & $0.017$ & $0.018$ & $0.031$ & $0.035$ & $0.048$ & $0.057$ & $0.083$ & $0.642$ \\
KL-Test & $0.000$ & $0.012$ & $0.020$ & $0.024$ & $0.038$ & $0.042$ & $0.052$ & $0.074$ & $0.109$ & $0.765$ \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

{\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{5pt}
\begin{table}[!ht]
\caption{Comparisons with Existing Tests Based on \eqref{Eqn: cdf compare1}: $n=400$}\label{Tab: comparison, n400}
\begin{footnotesize}
\begin{center}
\begin{tabular}{ccccccccccc}
\hline\hline
  & \multicolumn{10}{c}{$\lambda$}  \\
\cmidrule{2-11}
 & $-1$ & $-0.1$ & $-0.05$ & $0$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ & $1$ \\
\hline
R37  & $0.004$ & $0.052$ & $0.077$ & $0.086$ & $0.134$ & $0.189$ & $0.286$ & $0.327$ & $0.420$ & $0.999$ \\
R38  & $0.004$ & $0.053$ & $0.074$ & $0.088$ & $0.134$ & $0.191$ & $0.287$ & $0.326$ & $0.420$ & $0.999$ \\
R3L  & $0.004$ & $0.052$ & $0.075$ & $0.087$ & $0.133$ & $0.191$ & $0.287$ & $0.324$ & $0.422$ & $0.999$ \\
R47  & $0.003$ & $0.033$ & $0.058$ & $0.068$ & $0.107$ & $0.163$ & $0.252$ & $0.296$ & $0.382$ & $0.999$ \\
R48  & $0.003$ & $0.036$ & $0.059$ & $0.065$ & $0.107$ & $0.166$ & $0.252$ & $0.292$ & $0.384$ & $0.999$ \\
R4L  & $0.002$ & $0.035$ & $0.059$ & $0.067$ & $0.107$ & $0.164$ & $0.251$ & $0.296$ & $0.385$ & $0.999$ \\
R57  & $0.003$ & $0.027$ & $0.050$ & $0.061$ & $0.096$ & $0.150$ & $0.237$ & $0.282$ & $0.364$ & $0.999$ \\
R58  & $0.003$ & $0.029$ & $0.053$ & $0.062$ & $0.097$ & $0.150$ & $0.237$ & $0.279$ & $0.365$ & $0.999$ \\
R5L  & $0.003$ & $0.029$ & $0.051$ & $0.061$ & $0.095$ & $0.153$ & $0.237$ & $0.278$ & $0.364$ & $0.999$ \\
R67  & $0.000$ & $0.027$ & $0.048$ & $0.060$ & $0.092$ & $0.144$ & $0.227$ & $0.272$ & $0.349$ & $0.999$ \\
R68  & $0.000$ & $0.024$ & $0.046$ & $0.052$ & $0.080$ & $0.151$ & $0.226$ & $0.276$ & $0.368$ & $0.999$ \\
R6L  & $0.000$ & $0.027$ & $0.050$ & $0.058$ & $0.093$ & $0.146$ & $0.230$ & $0.273$ & $0.354$ & $0.999$ \\
R77  & $0.000$ & $0.023$ & $0.048$ & $0.055$ & $0.087$ & $0.140$ & $0.224$ & $0.268$ & $0.347$ & $0.999$ \\
R78  & $0.000$ & $0.026$ & $0.050$ & $0.053$ & $0.087$ & $0.143$ & $0.223$ & $0.266$ & $0.345$ & $0.999$ \\
R7L  & $0.003$ & $0.024$ & $0.048$ & $0.056$ & $0.088$ & $0.140$ & $0.224$ & $0.265$ & $0.346$ & $0.999$ \\
 \hline
P-Test & $0.000$ & $0.024$ & $0.035$ & $0.041$ & $0.078$ & $0.105$ & $0.189$ & $0.234$ & $0.343$ & $0.999$ \\
D-Test & $0.000$ & $0.013$ & $0.023$ & $0.037$ & $0.065$ & $0.135$ & $0.219$ & $0.294$ & $0.382$ & $0.999$ \\
KS-Test& $0.000$ & $0.010$ & $0.020$ & $0.025$ & $0.046$ & $0.080$ & $0.145$ & $0.198$ & $0.259$ & $0.999$ \\
KL-Test& $0.000$ & $0.009$ & $0.018$ & $0.037$ & $0.060$ & $0.102$ & $0.184$ & $0.258$ & $0.340$ & $1.000$ \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

Tables \ref{Tab: comparison, n100} and \ref{Tab: comparison, n400} record the results for $n\in\{100, 400\}$. Across the choices of $(t_n,\kappa_n)$, when $F_\lambda$ is close to being uniform, our test tends to over-reject (under the null) with smaller $t_n$, especially with $t_n=n^{-1/3}$ and in small samples. All other tests control size well in large samples, but tend to under-reject in small samples. In terms of power, the KS test appears to be the least powerful against local alternatives (i.e., distributions with positive but small $\lambda$). The $D$-test overall appears to be the most powerful one, though the power is comparable to ours and that of the $P$ and the KL tests. For a fair comparison, we also compute the size-adjusted power by subtracting the empirical rejection rates for $\lambda=0$ (the least favorable case) from the corresponding rejections rates for $\lambda>0$. The results are recorded in Table \ref{Tab: comparison}, from which we see that the power patterns more or less remain. Those results for $n\in\{200,300,600,800,1000\}$ share similar patterns---see Appendix \ref{App: Simulation}.

\iffalse
{\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{5pt}
\begin{table}[!h]
\caption{Comparisons with Existing Tests Based on \eqref{Eqn: cdf compare1}: $n=1000$}\label{Tab: comparison}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ccccccccccccccccccc}
\hline\hline
  & \multicolumn{18}{c}{$\lambda$}  \\
\cmidrule{2-19}
 $(t_n,\kappa_n)$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ & $1$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ & $1$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ & $1$ \\
\hline
R37                            & $0.029$ & $0.037$ & $0.086$ & $0.099$ & $0.140$ & $0.724$ & $0.048$ & $0.103$ & $0.200$ & $0.241$ & $0.334$ & $0.913$ & $0.067$ & $0.134$ & $0.299$ & $0.421$ & $0.562$ & $0.904$\\
R38                            & $0.029$ & $0.039$ & $0.086$ & $0.093$ & $0.141$ & $0.724$ & $0.046$ & $0.103$ & $0.199$ & $0.238$ & $0.332$ & $0.911$ & $0.067$ & $0.136$ & $0.301$ & $0.423$ & $0.561$ & $0.905$\\
R3L                            & $0.027$ & $0.039$ & $0.084$ & $0.093$ & $0.140$ & $0.722$ & $0.046$ & $0.104$ & $0.200$ & $0.237$ & $0.335$ & $0.912$ & $0.051$ & $0.133$ & $0.234$ & $0.386$ & $0.491$ & $0.899$\\
R47                            & $0.024$ & $0.030$ & $0.078$ & $0.086$ & $0.127$ & $0.720$ & $0.039$ & $0.095$ & $0.184$ & $0.228$ & $0.314$ & $0.931$ & $0.053$ & $0.122$ & $0.275$ & $0.406$ & $0.553$ & $0.926$\\
R48                            & $0.024$ & $0.031$ & $0.076$ & $0.085$ & $0.128$ & $0.719$ & $0.042$ & $0.101$ & $0.187$ & $0.227$ & $0.319$ & $0.934$ & $0.051$ & $0.120$ & $0.275$ & $0.407$ & $0.553$ & $0.926$\\
R4L                            & $0.023$ & $0.031$ & $0.078$ & $0.086$ & $0.130$ & $0.719$ & $0.040$ & $0.097$ & $0.184$ & $0.229$ & $0.318$ & $0.932$ & $0.052$ & $0.121$ & $0.273$ & $0.405$ & $0.555$ & $0.926$\\
R57                            & $0.018$ & $0.021$ & $0.073$ & $0.076$ & $0.118$ & $0.713$ & $0.035$ & $0.089$ & $0.176$ & $0.221$ & $0.303$ & $0.938$ & $0.048$ & $0.119$ & $0.264$ & $0.402$ & $0.548$ & $0.932$\\
R58                            & $0.019$ & $0.024$ & $0.071$ & $0.078$ & $0.120$ & $0.714$ & $0.035$ & $0.088$ & $0.175$ & $0.217$ & $0.303$ & $0.937$ & $0.047$ & $0.119$ & $0.268$ & $0.403$ & $0.545$ & $0.932$\\
R5L                            & $0.020$ & $0.022$ & $0.070$ & $0.079$ & $0.120$ & $0.714$ & $0.034$ & $0.092$ & $0.176$ & $0.217$ & $0.303$ & $0.938$ & $0.047$ & $0.116$ & $0.264$ & $0.402$ & $0.545$ & $0.932$\\
R67                            & $0.016$ & $0.016$ & $0.062$ & $0.069$ & $0.112$ & $0.709$ & $0.032$ & $0.084$ & $0.167$ & $0.212$ & $0.289$ & $0.939$ & $0.046$ & $0.111$ & $0.265$ & $0.401$ & $0.546$ & $0.937$\\
R68                            & $0.016$ & $0.016$ & $0.062$ & $0.070$ & $0.112$ & $0.709$ & $0.028$ & $0.099$ & $0.174$ & $0.224$ & $0.316$ & $0.947$ & $0.045$ & $0.111$ & $0.264$ & $0.400$ & $0.545$ & $0.937$\\
R6L                            & $0.018$ & $0.017$ & $0.071$ & $0.073$ & $0.115$ & $0.714$ & $0.035$ & $0.088$ & $0.172$ & $0.215$ & $0.296$ & $0.941$ & $0.045$ & $0.110$ & $0.263$ & $0.397$ & $0.544$ & $0.938$\\
R77                            & $0.016$ & $0.017$ & $0.066$ & $0.072$ & $0.109$ & $0.711$ & $0.032$ & $0.085$ & $0.169$ & $0.213$ & $0.292$ & $0.944$ & $0.042$ & $0.110$ & $0.263$ & $0.393$ & $0.542$ & $0.939$\\
R78                            & $0.017$ & $0.019$ & $0.068$ & $0.074$ & $0.111$ & $0.712$ & $0.034$ & $0.090$ & $0.170$ & $0.213$ & $0.292$ & $0.946$ & $0.042$ & $0.111$ & $0.263$ & $0.396$ & $0.542$ & $0.939$\\
R7L                            & $0.017$ & $0.016$ & $0.069$ & $0.071$ & $0.110$ & $0.712$ & $0.032$ & $0.084$ & $0.168$ & $0.209$ & $0.290$ & $0.943$ & $0.042$ & $0.110$ & $0.262$ & $0.392$ & $0.543$ & $0.938$\\\hline
P-Test                         & $0.027$ & $0.031$ & $0.035$ & $0.042$ & $0.082$ & $0.727$ & $0.037$ & $0.064$ & $0.148$ & $0.193$ & $0.302$ & $0.958$ & $0.052$ & $0.118$ & $0.242$ & $0.379$ & $0.544$ & $0.940$\\
D-Test                         & $0.025$ & $0.035$ & $0.054$ & $0.072$ & $0.134$ & $0.773$ & $0.028$ & $0.098$ & $0.182$ & $0.257$ & $0.345$ & $0.962$ & $0.048$ & $0.131$ & $0.304$ & $0.453$ & $0.618$ & $0.944$\\
KS-Test                        & $0.013$ & $0.017$ & $0.030$ & $0.039$ & $0.065$ & $0.624$ & $0.021$ & $0.055$ & $0.120$ & $0.173$ & $0.234$ & $0.974$ & $0.019$ & $0.080$ & $0.198$ & $0.337$ & $0.488$ & $0.957$\\
KL-Test                        & $0.014$ & $0.018$ & $0.028$ & $0.050$ & $0.085$ & $0.741$ & $0.023$ & $0.065$ & $0.147$ & $0.221$ & $0.303$ & $0.963$ & $0.046$ & $0.124$ & $0.283$ & $0.444$ & $0.616$ & $0.957$\\
\hline\hline
\end{tabular}
\end{center}
\end{scriptsize}
\end{table}
}
\fi

{\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{4pt}
\begin{table}[!h]
\caption{Size-Adjusted Power Comparisons with Existing Tests Based on \eqref{Eqn: cdf compare1}}\label{Tab: comparison}
\begin{scriptsize}
\begin{center}
\begin{tabular}{cccccccccccccc}
\hline\hline
  & \multicolumn{6}{c}{$\lambda$ for $n=100$} & & \multicolumn{6}{c}{$\lambda$ for $n=400$}\\
\cmidrule{2-7}  \cmidrule{9-14}
 & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ & $1$ & & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ & $1$  \\
\hline
R37                            & $0.029$ & $0.037$ & $0.086$ & $0.099$ & $0.140$ & $0.724$ && $0.048$ & $0.103$ & $0.200$ & $0.241$ & $0.334$ & $0.913$ \\
R38                            & $0.029$ & $0.039$ & $0.086$ & $0.093$ & $0.141$ & $0.724$ && $0.046$ & $0.103$ & $0.199$ & $0.238$ & $0.332$ & $0.911$ \\
R3L                            & $0.027$ & $0.039$ & $0.084$ & $0.093$ & $0.140$ & $0.722$ && $0.046$ & $0.104$ & $0.200$ & $0.237$ & $0.335$ & $0.912$ \\
R47                            & $0.024$ & $0.030$ & $0.078$ & $0.086$ & $0.127$ & $0.720$ && $0.039$ & $0.095$ & $0.184$ & $0.228$ & $0.314$ & $0.931$ \\
R48                            & $0.024$ & $0.031$ & $0.076$ & $0.085$ & $0.128$ & $0.719$ && $0.042$ & $0.101$ & $0.187$ & $0.227$ & $0.319$ & $0.934$ \\
R4L                            & $0.023$ & $0.031$ & $0.078$ & $0.086$ & $0.130$ & $0.719$ && $0.040$ & $0.097$ & $0.184$ & $0.229$ & $0.318$ & $0.932$ \\
R57                            & $0.018$ & $0.021$ & $0.073$ & $0.076$ & $0.118$ & $0.713$ && $0.035$ & $0.089$ & $0.176$ & $0.221$ & $0.303$ & $0.938$ \\
R58                            & $0.019$ & $0.024$ & $0.071$ & $0.078$ & $0.120$ & $0.714$ && $0.035$ & $0.088$ & $0.175$ & $0.217$ & $0.303$ & $0.937$ \\
R5L                            & $0.020$ & $0.022$ & $0.070$ & $0.079$ & $0.120$ & $0.714$ && $0.034$ & $0.092$ & $0.176$ & $0.217$ & $0.303$ & $0.938$ \\
R67                            & $0.016$ & $0.016$ & $0.062$ & $0.069$ & $0.112$ & $0.709$ && $0.032$ & $0.084$ & $0.167$ & $0.212$ & $0.289$ & $0.939$ \\
R68                            & $0.016$ & $0.016$ & $0.062$ & $0.070$ & $0.112$ & $0.709$ && $0.028$ & $0.099$ & $0.174$ & $0.224$ & $0.316$ & $0.947$ \\
R6L                            & $0.018$ & $0.017$ & $0.071$ & $0.073$ & $0.115$ & $0.714$ && $0.035$ & $0.088$ & $0.172$ & $0.215$ & $0.296$ & $0.941$ \\
R77                            & $0.016$ & $0.017$ & $0.066$ & $0.072$ & $0.109$ & $0.711$ && $0.032$ & $0.085$ & $0.169$ & $0.213$ & $0.292$ & $0.944$ \\
R78                            & $0.017$ & $0.019$ & $0.068$ & $0.074$ & $0.111$ & $0.712$ && $0.034$ & $0.090$ & $0.170$ & $0.213$ & $0.292$ & $0.946$ \\
R7L                            & $0.017$ & $0.016$ & $0.069$ & $0.071$ & $0.110$ & $0.712$ && $0.032$ & $0.084$ & $0.168$ & $0.209$ & $0.290$ & $0.943$ \\\hline
P-Test                         & $0.027$ & $0.031$ & $0.035$ & $0.042$ & $0.082$ & $0.727$ && $0.037$ & $0.064$ & $0.148$ & $0.193$ & $0.302$ & $0.958$ \\
D-Test                         & $0.025$ & $0.035$ & $0.054$ & $0.072$ & $0.134$ & $0.773$ && $0.028$ & $0.098$ & $0.182$ & $0.257$ & $0.345$ & $0.962$ \\
KS-Test                        & $0.013$ & $0.017$ & $0.030$ & $0.039$ & $0.065$ & $0.624$ && $0.021$ & $0.055$ & $0.120$ & $0.173$ & $0.234$ & $0.974$ \\
KL-Test                        & $0.014$ & $0.018$ & $0.028$ & $0.050$ & $0.085$ & $0.741$ && $0.023$ & $0.065$ & $0.147$ & $0.221$ & $0.303$ & $0.963$ \\
\hline\hline
\end{tabular}
\end{center}
\end{scriptsize}
\end{table}
}

Tables \ref{Tab: comparison2, n100} and \ref{Tab: comparison2, n400} record the rejection rates for $n\in\{100,400\}$ based on the design \eqref{Eqn: cdf compare2}. As expected, the $D$ and the $P$ tests, which are designed to be powerful against convex alternatives, now perform poorly and in most cases have power less than $5\%$---this is the case even with large sample sizes. The KS and the KL tests appear more powerful than the $D$ and the $P$ tests, but their performance is strictly dominated by our tests across all sample sizes and pairs $(t_n,\kappa_n)$, with substantial power discrepancies in many cases. We remind the reader that in practice, one can rarely rule out a priori distributions such as those in \eqref{Eqn: cdf compare2}. Thus, the simulations suggest that our test is more robust in the sense that they are powerful against a larger class of alternatives. Those results for $n\in\{200,300,600,800,1000\}$ share similar patterns---see Appendix \ref{App: Simulation}.

Admittedly, the simulation results reinforce the importance of the choice of $\kappa_n$ for our test---the choice of $t_n$ does not appear as important. Overall, they provide comforting numerical evidence that our test is a useful addition to the literature. The associated computation cost is reasonable: with an Intel Xeon CPU E5-1650 v4 3.60GHz, a single replication based on 1000 samples from the design in \eqref{Eqn: cdf compare1} and 500 bootstrap repetitions is completed in 15 seconds.

{\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{5pt}
\begin{table}[!h]
\caption{Comparisons with Existing Tests Based on \eqref{Eqn: cdf compare2}: $n=100$}\label{Tab: comparison2, n100}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccccccc}
\hline\hline
  & \multicolumn{9}{c}{$\lambda$}  \\
\cmidrule{2-10}
 & $0.1$ & $0.2$ & $0.3$ & $0.4$ & $0.5$ & $0.6$ & $0.7$ & $0.8$ & $0.9$ \\
\hline
R37    & $0.127$ & $0.221$ & $0.339$ & $0.482$ & $0.652$ & $0.664$ & $0.607$ & $0.386$ & $0.056$  \\
R38    & $0.125$ & $0.223$ & $0.336$ & $0.483$ & $0.656$ & $0.669$ & $0.607$ & $0.387$ & $0.053$     \\
R3L    & $0.127$ & $0.222$ & $0.338$ & $0.483$ & $0.654$ & $0.669$ & $0.608$ & $0.386$ & $0.055$  \\
R47    & $0.102$ & $0.184$ & $0.273$ & $0.406$ & $0.566$ & $0.586$ & $0.503$ & $0.296$ & $0.030$     \\
R48    & $0.103$ & $0.182$ & $0.273$ & $0.407$ & $0.569$ & $0.584$ & $0.499$ & $0.299$ & $0.029$     \\
R4L    & $0.102$ & $0.182$ & $0.275$ & $0.406$ & $0.563$ & $0.582$ & $0.504$ & $0.297$ & $0.031$  \\
R57    & $0.097$ & $0.164$ & $0.255$ & $0.374$ & $0.531$ & $0.537$ & $0.447$ & $0.248$ & $0.024$     \\
R58    & $0.095$ & $0.160$ & $0.252$ & $0.377$ & $0.529$ & $0.537$ & $0.448$ & $0.250$ & $0.024$     \\
R5L    & $0.096$ & $0.162$ & $0.256$ & $0.374$ & $0.527$ & $0.539$ & $0.449$ & $0.247$ & $0.024$  \\
R67    & $0.086$ & $0.153$ & $0.238$ & $0.358$ & $0.494$ & $0.508$ & $0.411$ & $0.213$ & $0.020$     \\
R68    & $0.084$ & $0.152$ & $0.239$ & $0.358$ & $0.497$ & $0.506$ & $0.412$ & $0.212$ & $0.020$     \\
R6L    & $0.086$ & $0.153$ & $0.237$ & $0.357$ & $0.499$ & $0.512$ & $0.412$ & $0.214$ & $0.020$     \\
R77    & $0.087$ & $0.147$ & $0.234$ & $0.346$ & $0.479$ & $0.484$ & $0.389$ & $0.193$ & $0.019$     \\
R78    & $0.085$ & $0.148$ & $0.233$ & $0.344$ & $0.475$ & $0.486$ & $0.389$ & $0.196$ & $0.019$     \\
R7L    & $0.085$ & $0.146$ & $0.234$ & $0.342$ & $0.480$ & $0.483$ & $0.394$ & $0.196$ & $0.019$     \\ \hline
P-Test & $0.015$ & $0.018$ & $0.008$ & $0.009$ & $0.010$ & $0.004$ & $0.003$ & $0.006$ & $0.003$  \\
D-Test & $0.030$ & $0.048$ & $0.039$ & $0.042$ & $0.032$ & $0.017$ & $0.006$ & $0.004$ & $0.000$  \\
KS-Test& $0.017$ & $0.055$ & $0.090$ & $0.121$ & $0.154$ & $0.126$ & $0.086$ & $0.021$ & $0.000$ \\
KL-Test& $0.025$ & $0.050$ & $0.043$ & $0.039$ & $0.015$ & $0.003$ & $0.000$ & $0.000$ & $0.000$ \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

{\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{5pt}
\begin{table}[!h]
\caption{Comparisons with Existing Tests Based on \eqref{Eqn: cdf compare2}: $n=400$}\label{Tab: comparison2, n400}
\begin{footnotesize}
\begin{center}
\begin{tabular}{cccccccccc}
\hline\hline
  & \multicolumn{9}{c}{$\lambda$}  \\
\cmidrule{2-10}
 & $0.1$ & $0.2$ & $0.3$ & $0.4$ & $0.5$ & $0.6$ & $0.7$ & $0.8$ & $0.9$ \\
\hline
R37    & $0.207$ & $0.595$ & $0.938$ & $0.995$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $0.718$  \\
R38    & $0.207$ & $0.598$ & $0.935$ & $0.995$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $0.721$  \\
R3L    & $0.207$ & $0.596$ & $0.937$ & $0.995$ & $1.000$ & $1.000$ & $1.000$ & $1.000$ & $0.723$  \\
R47    & $0.173$ & $0.516$ & $0.908$ & $0.991$ & $1.000$ & $1.000$ & $1.000$ & $0.995$ & $0.563$     \\
R48    & $0.176$ & $0.517$ & $0.908$ & $0.991$ & $1.000$ & $1.000$ & $1.000$ & $0.995$ & $0.564$     \\
R4L    & $0.175$ & $0.514$ & $0.908$ & $0.991$ & $1.000$ & $1.000$ & $1.000$ & $0.995$ & $0.561$     \\
R57    & $0.157$ & $0.485$ & $0.886$ & $0.987$ & $0.998$ & $1.000$ & $1.000$ & $0.985$ & $0.463$     \\
R58    & $0.155$ & $0.487$ & $0.886$ & $0.989$ & $0.998$ & $0.999$ & $1.000$ & $0.986$ & $0.470$     \\
R5L    & $0.158$ & $0.485$ & $0.887$ & $0.988$ & $0.998$ & $1.000$ & $1.000$ & $0.985$ & $0.467$     \\
R67    & $0.143$ & $0.469$ & $0.876$ & $0.985$ & $0.998$ & $0.999$ & $1.000$ & $0.975$ & $0.394$     \\
R68    & $0.143$ & $0.471$ & $0.875$ & $0.985$ & $0.998$ & $0.999$ & $1.000$ & $0.976$ & $0.393$     \\
R6L    & $0.144$ & $0.468$ & $0.875$ & $0.986$ & $0.998$ & $0.999$ & $1.000$ & $0.977$ & $0.391$     \\
R77    & $0.137$ & $0.462$ & $0.864$ & $0.980$ & $0.997$ & $0.999$ & $1.000$ & $0.966$ & $0.346$     \\
R78    & $0.134$ & $0.461$ & $0.864$ & $0.979$ & $0.997$ & $0.999$ & $1.000$ & $0.967$ & $0.350$     \\
R7L    & $0.136$ & $0.456$ & $0.866$ & $0.981$ & $0.997$ & $0.999$ & $1.000$ & $0.967$ & $0.348$     \\ \hline
P-Test & $0.020$ & $0.014$ & $0.010$ & $0.015$ & $0.010$ & $0.007$ & $0.007$ & $0.005$ & $0.003$  \\
D-Test & $0.050$ & $0.054$ & $0.042$ & $0.038$ & $0.032$ & $0.012$ & $0.011$ & $0.004$ & $0.000$  \\
KS-Test& $0.076$ & $0.295$ & $0.689$ & $0.907$ & $0.966$ & $0.977$ & $0.962$ & $0.750$ & $0.056$ \\
KL-Test& $0.107$ & $0.232$ & $0.356$ & $0.362$ & $0.298$ & $0.083$ & $0.003$ & $0.000$ & $0.000$ \\
\hline\hline
\end{tabular}
\end{center}
\end{footnotesize}
\end{table}
}

\section{Conclusion}\label{Sec: conclusion}

This paper studies estimation of and inference on a cumulative distribution function with concavity constraint. The estimation results generalize the asymptotic order results in \citet{Kiefer_Wolfowitz1976minimax} to settings with unbounded support and contiguous distributions. These results are not only of interest in their own right, but also useful for conducting inference on the concavity. In particular, in conjunction with the rescaled bootstrap of \citet{Dumbgen1993} and the recent work of \citet{FangSantos2018HDD}, they allow us to build up a test that controls size, pointwise and locally, even when the distribution function is strictly concave (in which case the test statistic is asymptotically degenerate). Through simulation studies, we find that our test is powerful against a larger class of alternatives, compared to some existing tests such as those in \citet{Woodroofe_Sun1999} that are designed to work against a specific class of alternatives, or those in \citet{Durot2003testing} and \citet{Kulikov_Lopuhaa2004testing} that rely on critical values from the least favorable asymptotic distributions. In Appendix \ref{App: extension}, we show how the testing results may be extended to a general setup that includes regression and hazard rate problems as special cases.

In our limited simulation studies, we also find that the choice of $\kappa_n$ for the KW selection does not appear as important as the choice of $t_n$ for implementing the rescaled bootstrap. A formal investigation of both choices is, however, beyond the scope of this paper, and therefore left for future study.

\clearpage

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