Simulation-Based Hypothesis Testing of High Dimensional Means Under Covariance Heterogeneity\\ --- An Alternative Road to High Dimensional Tests
\setcounter{footnote
%Hypothesis testing for mean vectors in high dimensions has gained increasing attentions and stimulated innovative methodologies in statistics. In this paper, we study testing the population mean vector of high dimensional multivariate data for both one-sample and two-sample problems. The proposed simulation-based testing procedures employ maximum-type statistics and use the Gaussian approximation techniques to obtain corresponding critical values. Different from peer tests that heavily rely on the structural conditions on the unknown covariance matrices, the proposed tests allow very general forms of the covariance structures of data and therefore enjoy wide scope of applicability in practice. To enhance powers of the tests against sparse alternatives, we further propose two-step procedures with a preliminary feature screening step. Theoretical properties of the proposed tests are investigated. Extensive numerical experiments on synthetic datasets and empirical applications on identifying diseases-associated gene-sets are provided to support the theoretical results. The proposed tests are easily implemented and computationally efficient in practice. %Simulation studies demonstrate that, comparing to the existing methods, the proposed procedure possesses good performance against strong heterogeneity in the covariance structure of the data. The method is also applied to the analysis of real data sets from leukemia studies in testing the significance of genes.
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\begin{document}


\title{Simulation-Based Hypothesis Testing of High Dimensional Means Under Covariance Heterogeneity\\
--- An Alternative Road to High Dimensional Tests}%\footnote{We are grateful to the Editor and five referees for their very constructive comments and suggestions which have improved the
%presentation of the paper significantly. Jinyuan Chang and Wen-Xin Zhou were supported by the Australian Research Council.  }}

\author{\setcounter{footnote}{1}Jinyuan Chang\footnote{\baselineskip=11pt School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. \newline E-mail: jinyuan.chang@unimelb.edu.au.} ,
\setcounter{footnote}{2} Wen Zhou\footnote{\baselineskip=11pt Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA. \newline E-mail: riczw@stat.colostate.edu.}, \setcounter{footnote}{3} Wen-Xin Zhou\footnote{\baselineskip=11pt School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. \newline E-mail: wenxin.zhou@unimelb.edu.au.} }

%\date{First version: Jan 15th 2014 \qquad Latest version: Jun 1st 2014}
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\maketitle

\begin{abstract}
%Hypothesis testing for mean vectors in high dimensions has gained increasing attentions and stimulated innovative methodologies in statistics.
In this paper, we study testing the population mean vector of high dimensional multivariate data for both one-sample and two-sample problems. The proposed simulation-based testing procedures employ maximum-type statistics and use the Gaussian approximation techniques to obtain corresponding critical values. Different from peer tests that heavily rely on the structural conditions on the unknown covariance matrices, the proposed tests allow very general forms of the covariance structures of data and therefore enjoy wide scope of applicability in practice. To enhance powers of the tests against sparse alternatives, we further propose two-step procedures with a preliminary feature screening step. Theoretical properties of the proposed tests are investigated. Extensive numerical experiments on synthetic datasets and empirical applications on
identifying diseases-associated gene-sets are provided to support the theoretical results. The proposed tests are easily implemented and computationally efficient in practice.

%Simulation studies demonstrate that, comparing to the existing methods, the proposed procedure possesses good performance against strong heterogeneity in the covariance structure of the data. The method is also applied to the analysis of real data sets from leukemia studies in testing the significance of genes.
\end{abstract}
%===============================================================================================%
%===============================================================================================%

\noindent{\bf Keywords}: Feature screening; Gaussian approximation; High dimensional hypothesis test; Simulation-based statistical inference; Testing equality of mean vectors
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\section{Introduction}   \lbl{sec.intro}

High dimensional hypothesis test for mean vectors is in great need in multiple scientific disciplines and practical fields, particularly in modern genomics, image segmentation and quantitative finance. With limited number of subjects being measured, the sample size is relatively small which leads to the so-called ``large $p$, small $n$" problem. Furthermore, the measurements usually possess fairly complex dependence structures. These issues have urged the renovation of traditional multivariate analysis procedures on testing mean vectors. See, for example, \cite{BaiSaranadasa_1996}, \cite{DonohoJin04}, \cite{ChenQin_2010} and \cite{CaiLiuXia_2014}, among others.

In this paper, we consider both the one-sample and two-sample settings. Let $\bX$ and $\bY$ be two $p$-dimensional random vectors with means $\bmu_1$ and $\bmu_2$ and covariance matrices $\bSigma_1$ and $\bSigma_2$, respectively. Consider two independent random samples of independent and identically distributed observations, $\mathcal{X}_n= \{\bX_1, \ldots, \bX_{n}\}$ and $\mathcal{Y}_m=\{\bY_1, \ldots, \bY_m\}$, drawn from the distributions of $\bX$ and $\bY$, respectively. It is of general interest in testing the hypotheses
\begin{enumerate}
\item[(i)] (One-sample problem)
\begin{equation*}\label{eq:onesample0}
H^{\rI}_{0} : \bmu_{1}=\bmu_0 ~~ {versus} ~~ H^{\rI}_{ 1}:\bmu_1\neq\bmu_0
\end{equation*}
for a {\it specified} $p$-dimensional vector $\bmu_0$, which, without loss of generality, is equivalent to
\begin{equation}
H^{\rI}_{0} : \bmu_{1}=\bzero ~~ {versus} ~~ H^{\rI}_{ 1}:\bmu_1\neq\bzero; \label{eq:onesample}
\end{equation}
\item[(ii)] (Two-sample problem)
\begin{equation}\label{eq:twosample}
H^{\rII}_0:\bmu_1=\bmu_2 ~~ versus ~~ H^{\rII}_1:\bmu_1\neq\bmu_2.
\end{equation}
\end{enumerate}
Hypothesis \eqref{eq:onesample} arises when a sample is being compared to a hypothetical population with known prior information, while \eqref{eq:twosample} sparks interest to compare two parallel groups, particularly a control group and a treatment group in scientific experiments. %Both hypotheses consist of $p$ marginal hypotheses regarding the means of the $p$ features/responses.
In the low dimensional setting where $p$ is fixed, traditional tests have been extensively studied for testing both \eqref{eq:onesample} and \eqref{eq:twosample}. For example, the properties for both the one-sample and two-sample Hotelling's $T^2$ tests have been examined under the assumption of normality \citep{Anderson_2003}. We refer to \cite{DemboShao_2006} and \cite{LiuShao_2013} for the large and moderate deviation asymptotics for the Hotelling's $T^2$-statistics without normality.

Generally, the sum of squares-type and the maximum-type statistics are used to test the hypotheses \eq{eq:onesample} and \eq{eq:twosample} in the high dimensional settings. The sum of squares-type statistics aim to mimic the weighted Euclidean norms, $\| \bA^{1/2} \bmu_1 \|_2^2 $ or $\|\bA^{1/2} (\bmu_1-\bmu_2) \|_2^2$ for certain linear transformation $\bA$, and the corresponding tests are powerful for detecting relatively dense signals \citep{BaiSaranadasa_1996, SrivastavaDu_2008, Srivastava_2009, ChenQin_2010}. Statistics of the maximum-type, on the other hand, are preferable for detecting relatively sparse signals \citep{LiuShao_2013,CaiLiuXia_2014} and have been used in a variety of applications including the medical image problem \citep{JamesClymerSchmalbrock_2001}, anomaly detections
\citep{CastagnaSunSiegfried_2003} and gene selections \citep{Martens_2005}. %As discussed in \cite{LiuShao_2013} and \cite{CaiLiuXia_2014}, tests based on the $L_\infty$-type statistics are reasonably more powerful than those based on the $L_2$-type when the alternatives are sparse.

Most existing testing procedures for \eq{eq:onesample} and \eq{eq:twosample} rely on the derivation the pivotal limiting distribution of test statistics, from which the critical value is approximated. In the high dimensional scenarios, various structural assumptions on the unknown covariance matrices have been imposed. For example, the thresholding test statistic proposed by \cite{ZhongChenXu_2013} for the one-sample problem \eqref{eq:onesample} requires weak dependence to guarantee the asymptotic normality, that is, the observation is generated by the model $\bX=\bW+\bmu$ with weakly stationary sequence $\bW=(W_{1},\ldots, W_{p})'$ satisfying $\sum_{k\geq 1}\left|\text{Cov}(W_{1},W_{k+1 })\right|<\infty$. For the two-sample problem \eqref{eq:twosample}, \cite{CaiLiuXia_2014} proposed a testing procedure whose validity requires that, among certain moment conditions, the unknown covariance matrices satisfy $\bSigma_1=\bSigma_2$ and $\max_{1\leq k< \ell \leq p}|\omega_{k \ell}/\s{ \omega_{k k}
\omega_{\ell \ell} }| \leq c$ for precision matrix $\bOmega=(\omega_{k \ell})_{1\leq k,\ell \leq p}$ and some $c\in (0,1)$.

However, in many applications, these assumptions can be very restrictive or difficult to be verified, and therefore limit the scope of applicability for the limiting distribution calibration approach. First, the existence of a pivotal asymptotic distribution relies heavily on the structural assumptions on the unknown covariance/correlation structures, which may not be true in practice. For example, it is very common that the expression levels are highly correlated for genes regulated by the same pathway \citep{WM2012} or associated with the same functionality \citep{K2014}, which results in a complex and non-sparse covariance structure. These empirical evidences indicate that the strong structural assumptions on the covariance matrices may sometimes be unrealistic in real-world applications. Another concern, as pointed out by \cite{CaiLiuXia_2014}, is that the convergence rate to the extreme value distribution of maximum-type statistics is usually slow. Taking the extreme distribution of type I as an example, it was shown in \cite{LiuLinShao_2008} that the convergence rate is of order $O\{  \log(\log n)/ \log(n) \}$. Although the convergence rate may be improved by using suitable intermediate approximations, still its validity relies on the dependence structure of the underlying distribution.

Driven by the above two concerns, we revisit the problem of testing hypotheses \eq{eq:onesample} and \eq{eq:twosample} in high dimensions from a different perspective. Motivated by applications in genomic analysis and image analysis, we are particularly interested in detecting discrepancies when $\bmu_1$ and $\bzero$ or $\bmu_2$ are distinguishable to a certain extent in at least one coordinate. We develop a fully data driven procedure to compute the critical values using the Monte Carlo simulations under appropriate settings. The validity of our procedure is established without enforcing structural assumptions of any kind on the unknown covariances. The main idea is based on the strong approximation of empirical processes by Gaussian processes \citep{VW96,ChernozhukovChetverikovKato_2013}, and to some degree, is similar to that of \cite{LiuShao_2013} that utilizes the intermediate approximation. However, instead of generating independent standard multivariate normal vectors, our approach takes into account correlations among the features and therefore is automatically adapted to the underlying dependence. %Numerical studies provide further evidence that the proposed tests are robust against strong heterogeneity in the covariance structure of the data.

The rest of the paper is organized as follows. In Section~\ref{method.sec}, we describe the simulation-based testing procedures for both hypotheses \eqref{eq:onesample} and \eqref{eq:twosample}. Theoretical properties of the tests are studied in Section~\ref{section:theory}. Numerical studies are reported in Section \ref{section:numerical} to assess the performance of the proposed tests comparing to the peer methods. In Section \ref{real data}, we applied the proposed tests to an acute lymphoblastic leukemia data for identifying disease-associated gene-sets based on the gene expression levels.
The paper is concluded with a brief discussion in Section~\ref{sec.disc}. The underpinning technical details, as well as additional simulation results and empirical data analysis, are relegated to the supplementary material.

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\setcounter{equation}{0}
\section{Methodology}
\label{method.sec} We present our testing procedures for the one-sample and two-sample problems in Sections \ref{section:onesample} and \ref{2s.test}, respectively. Section \ref{section:matrixest} discusses the connections between the proposed tests and covariance estimation, the latter of which is of independent interest. In Section \ref{sec:ps}, we develop a preliminary screening procedure to improve the power of proposed tests against sparse alternatives.

Throughout the paper, we denote by $|\bbeta|_\infty=\max_{1\leq k\leq p}|\beta_k|$ for a $p$-dimensional vector $\bbeta=(\beta_1,\ldots,\beta_p)'$. For a matrix $\bW=(w_{k \ell})_{p\times p}$, its max-norm is defined as $\|\bW\|_\infty=\max_{1\leq k , \ell \leq p}|w_{k  \ell}|$. For two sequences of real numbers $\{a_n\}$ and $\{b_n\}$, we write $a_n \asymp b_n$ if there exist positive constants $c_1$, $c_2$ such that $c_1 \leq a_n/b_n \leq c_2$ for all $n\geq 1$, we write $a_n=O(b_n)$ if there is a constant $C$ such that $|a_n|\leq C|b_n| $ for all sufficiently large $n$, and we write $a_n=o(b_n)$ if $\lim_{n\rightarrow \infty} a_n/b_n =0$. Moreover, let $\bX$ and $\bY$ be two $p$-dimensional random vectors with means $\bmu_1=(\mu_{1 1}, \ldots, \mu_{1 p})'$ and $\bmu_2=(\mu_{2 1}, \ldots, \mu_{2p})'$, covariance matrices $\bSigma_1=(\sigma_{1, k \ell})_{1\leq k, \ell \leq p}$ and $\bSigma_2=(\sigma_{2, k \ell})_{1\leq k, \ell \leq p}$, respectively. Denote by $\bR_1$ and $\bR_2$
the corresponding correlation matrices. Let $\bD_1={\rm diag}\,(\bSigma_1)$ and $\bD_2={\rm diag}\,(\bSigma_2)$ be the diagonal matrices of $\bSigma_1$ and $\bSigma_2$, respectively. Let $\mathcal{X}_n=\{\bX_1, \ldots , \bX_{n} \}$ and $\mathcal{Y}_m=\{\bY_1, \ldots, \bY_{m}\}$ be two independent samples consisting of independent and identically distributed observations drawn from the distributions of $\bX$ and $\bY$, respectively. For each $i=1,\ldots,n$ and $j=1,\ldots,m$, write $\bX_i=(X_{i1},\ldots,X_{ip})'$ and $\bY_j=(Y_{j1},\ldots,Y_{jp})'$. Set $N=n+m$. %and $\lambda=n/m$. Without loss of generality, we assume that $n\leq m$.

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\subsection{Test procedures}

%===========++++++++++++++++++++++++++++++++++++++++++++++++++++++===================%
\subsubsection{One-sample case}
\label{section:onesample}
Consider the maximum-type statistics in the following forms:
\be
 T_{\ns}^{\rI}=\max_{1\leq k\leq p} \s{n} |\bar{X}_k| \qquad \textrm{or}  \qquad
 T^{\rI}_{\rs}=\max_{1\leq k\leq p}\frac{\s{n}|\bar{X}_k|}{\hat{\sigma}_{1 k}},   \label{T12}
\ee where $\bar{X}_k=n^{-1}\sum_{i=1}^{n}X_{i k}$ and $\hat{\sigma}_{1 k}^2=n^{-1}\sum_{i=1}^{n}(X_{i k}-\bar{X}_k)^2$. Throughout, the statistic $T^{\rI}_{\rs}$ is referred as the {\it studentized} statistic, while $T_{\ns}^{\rII}$ is referred as the {\it non-studentized} statistic. Intuitively, large values of $T^{\rI}_{\ns}$ or $T^{\rI}_{\rs}$ provide evidences against $H^{\rI}_0$ in \eqref{eq:onesample} so that the corresponding tests are of the form \beq
    \Psi_{\ns,\alpha}^{\rI}=I\{T_{\ns}^{\rI} >  {\rm cv}_{\ns, \alpha}^{\rI}\}  ~~ \textrm{or} ~~ \Psi_{\rs,\alpha}^{\rI}=I\{T_{\rs}^{\rI} >  {\rm cv}_{\rs , \alpha}^{\rI}\} ,
\eeq where ${\rm cv}_{\ns, \alpha}^{\rI} $ and ${\rm cv}_{\rs , \alpha}^{\rI} $ are the critical/cut-off values. With ${\rm cv}_{\ns, \alpha}^{\rI} $ and ${\rm cv}_{\rs , \alpha}^{\rI} $ properly chosen, we expect the above two tests to have, at least approximately, a prescribed size $\alpha\in (0, 1)$.

For $\nu \in \{\ns, s\}$, ${\rm cv}_{\nu, \alpha}^{\rI} $ is typically specified as the $(1-\alpha)$-quantile of the limiting distribution of $T_{\nu}^{\rI}$ whenever it exists. As discussed in Section \ref{sec.intro}, alternative to the traditional limiting distribution calibration approach, it is possible to compute the critical values via the Monte Carlo simulation in an appropriate setting if we are able to acquire at least partial information on the unknown dependence structure from data. Under the null hypothesis $H^{\rI}_0:\bmu_1=\bzero$, for $\alpha \in (0,1)$, we motivate from the multivariate central limit theorem with fixed $p$ the following procedure to calculate critical values ${\rm cv}_{\ns,\alpha}^{\rI}$ and ${\rm cv}_{\rs ,\alpha}^{\rI}$:
\begin{itemize}
\item[$\bullet$] Let $\wt{\bSigma}_1$ be an estimate of $\bSigma_1$ from the sample $\mathcal{X}_n$, and set $\wt{\bR}_1= \wt{\bD}_1^{-1/2}\wt{\bSigma}_1 \wt{\bD}_1^{-1/2}$ with $\wt{\bD}_1=\textrm{diag}\,(\wt{\bSigma}_1)$. Given $\mathcal{X}_n$, let $\bW^{\rI}_{\ns} \sim \text{N}(\bzero,\wt{\bSigma}_1)$ and $\bW^{\rI}_{\rs} \sim \text{N}(\bzero, \wt{\bR}_1)$ be two multivariate Gaussian random vectors, the critical values ${\rm cv}_{\ns, \alpha}^{\rI}$ and ${\rm cv}_{\rs ,\alpha}^{\rI}$ can be computed by the conditional $(1-\alpha)$-quantiles of $|\bW^{\rI}_{\ns}|_\infty $ and $|\bW^{\rI}_{\rs} |_\infty$; that is,
\begin{align}
\begin{split}
  {\rm cv}_{\ns,\alpha}^{\rI} &= \inf\big\{  t\in \br: \mathbb{P} (  |\bW^{\rI}_{\ns}|_\infty >t \,\big|   \mathcal{X}_n ) \leq \alpha \big\},   \\
  {\rm cv}_{\rs ,\alpha}^{\rI}  &= \inf\big\{ t\in \br: \mathbb{P} (  |\bW^{\rI}_{\rs} |_\infty >t \,  \big| \mathcal{X}_n ) \leq \alpha \big\}.  \label{q12.definition}
\end{split}
\end{align}
\end{itemize}
In practice, we can use the Monte Carlo sample quantiles to approximate ${\rm cv}_{\ns,\alpha}^{\rI}$ and ${\rm cv}_{\rs ,\alpha}^{\rI}$. More specifically, let $ \{\bW_{\ns,1},\ldots,\bW_{\ns,M}\}$ and $\{\bW_{{\rs},1},\ldots,\bW_{{\rs},M}\}$ be random samples independently drawn from $\text{N}(\bzero,\wt{\bSigma}_1)$ and $\text{N}(\bzero, \wt{\bR}_1)$, respectively. Then, ${\rm cv}_{\ns,\alpha}^{\rI}$ and ${\rm cv}_{\rs ,\alpha}^{\rI}$ can be estimated by
\begin{align*}
\begin{split}
\widehat{{\rm cv}}_{\ns,\alpha}^{\, \rI} &= \inf\big\{  t\in \br:   \wh{F}_{\ns,M}^{\rI}(t) \geq 1- \alpha \big\},\\
 \widehat{{\rm cv}}_{\rs ,\alpha}^{\,\rI} &= \inf\big\{  t\in \br:   \wh{F}_{{\rs},M}^{\rI}(t) \geq 1- \alpha \big\},
\end{split}
\end{align*}
where $\wh{F}_{\ns,M}^{\rI}(t) =  M^{-1} \sum_{\ell=1}^M I\{|\bW_{\ns,\ell}|_\infty \leq t \} $ and $\wh{F}_{{\rs},M}^{\rI}(t) = M^{-1}\sum_{\ell=1}^M I\{|\bW_{\rs ,\ell}|_\infty \leq t \} $.
For $\nu\in\{\ns ,  {\rs}\}$, the empirical version of test $\Psi_{\nu,\alpha}^{\rI}$ is therefore defined by \be
     \wh \Psi_{\nu, \alpha}^{\rI}(M) = I\big\{ T_{\nu}^{\rI} > \widehat{{\rm cv}}_{\nu, \alpha}^{\,\rI} \big\},  \label{test.12}
\ee such that the null hypothesis $H^{\rI}_0$ is rejected whenever $\wh \Psi^{\rI}_{\nu, \alpha}(M) =1 $.

In Section \ref{section:matrixest}, we discuss the constructions of $\wt{\bSigma}_1$, and therefore $\wt{\bR}_1$ in details, from which the wide applicability of the test \eqref{test.12} will be explored. The proposed testing procedures are fully data driven and easily computed, the rationale of which are based on Gaussian approximation that characterizes the closeness in distribution of a random process to certain Gaussian process. In Section~\ref{section:theory}, we will show that this simulation-based testing procedure is valid for $p$ growing with sample size $n$; indeed $p$ can be as large as $O\{\exp(n^c)\}$ for some $c>0$.

%===============================================================================================%
\subsubsection{Two-sample case}
\label{2s.test}

The above testing procedures can be naturally extended to two-sample problem \eqref{eq:twosample}. Consider independent samples $\mathcal{X}_n=\{\bX_1, \ldots , \bX_{n}\}$ and $\mathcal{Y}_m=\{\bY_1, \ldots, \bY_{m}\}$. Analogously to \eq{T12}, we define the two-sample non-studentized and studentized test statistics by
\be
 T_{\ns}^{\rII}=\max_{1\leq k\leq p}  \s{\frac{nm}{n+m}} |\bar{X}_k-\bar{Y}_k| ~~ \textrm{ and } ~~  T_{\rs}^{\rII} = \max_{1\leq k\leq p}   \s{\frac{nm}{m \hat{\sigma}_{1 k}^2   + n \hat{\sigma}_{2 k}^2}}  |\bar{X}_k-\bar{Y}_k|  \, , \label{T34}
\ee
respectively, where $\bar{X}_k=n^{-1}\sum_{i=1}^{n}X_{i k}$, $\bar{Y}_k=m^{-1}\sum_{j=1}^{m}Y_{j k}$, $\hat{\sigma}_{1 k}^2=n^{-1}\sum_{i=1}^{n}(X_{i k}-\bar{X}_k)^2$, and $ \hat{\sigma}_{2 k}^2=m^{-1} \sum_{j=1}^{m}(Y_{j k}-\bar{Y}_k)^2.$ Intuitively, large values of $T_{\ns}^{\rII}$ or $T_{\rs}^{\rII}$ lead to a rejection of the null hypothesis $H^{\rII}_0: \bmu_1 =\bmu_2$. For nominal significance level $\alpha$, we define $\alpha$-level tests of the form
\beq
    \Psi_{\ns,\alpha}^{\rII}=I\{T_{\ns}^{\rII} > {\rm cv}_{\ns,\alpha}^{\rII}\} ~~ \mbox{ or } ~~ \Psi_{\rs,\alpha}^{\rII}=I\{T_{\rs}^{\rII} > {\rm cv}_{\rs ,\alpha}^{\rII}\}\eeq
    with appropriate critical values ${\rm cv}_{\ns,\alpha}^{\rII}$ and ${\rm cv}_{\rs ,\alpha}^{\rII}$.

Let $\wt{\bSigma}_1$ and $\wt{\bSigma}_2$ be estimates of $\bSigma_1$ and $\bSigma_2$ from the samples $\mathcal{X}_n$ and $\mathcal{Y}_m$, respectively. Define
\be
    \wt{\bSigma}_{1,2} = \frac{m}{N}\wt{\bSigma}_1+ \frac{n}{N} \wt{\bSigma}_2,  \quad   \wt{\bD}_{1,2}=\textrm{diag}\,\big( \wt{\bSigma}_{1,2}\big), \quad  \wt{\bR}_{1,2} = \wt{\bD}_{1,2}^{-1/2} \wt{\bSigma}_{1,2} \wt{\bD}_{1,2}^{-1/2}, \label{Sigma12} \ee
and let $\bW_{\ns}^{\rII} \sim  \text{N}(\bzero, \wt{\bSigma}_{1,2}  )$ and $\bW_{\rs}^{\rII}  \sim \text{N}( \bzero,  \wt{\bR}_{1,2})$. As in Section \ref{section:onesample}, the critical values are taken to be the following conditional $(1-\alpha)$-quantiles
\begin{align*}
  {\rm cv}_{\ns,\alpha}^{\rII}  & = \inf\big\{  t\in \br: \mathbb{P}\big(  |\bW_{\ns}^{\rII}|_\infty >t \, |   \mathcal{X}_n, \mathcal{Y}_m \big) \leq \alpha \big\} ,  \\
  {\rm cv}_{\rs ,\alpha}^{\rII} & = \inf\big\{ t\in \br: \mathbb{P}\big(  |\bW_{\rs}^{\rII}|_\infty >t \, | \mathcal{X}_n, \mathcal{Y}_m \big) \leq \alpha \big\},
\end{align*}
which can be computed via the Monte Carlo simulations. Let $\{\bW_{\ns,1},\ldots,\bW_{\ns,M}\}$ and $\{\bW_{{\rs},1},\ldots,\bW_{{\rs},M}\}$ be two random samples independently generated from $\text{N}(\bzero, \wt{\bSigma}_{1,2}  )$ and $\text{N}(\bzero, \wt{\bR}_{1,2}  )$, respectively. Then, ${\rm cv}_{\ns,\alpha}^{\rII}$ and ${\rm cv}_{\rs ,\alpha}^{\rII}$ can be estimated by
\[
\begin{split}
  \widehat{{\rm cv}}_{\ns,\alpha}^{\,\rII} =&~ \inf\big\{  t\in \br: \wh{F}_{\ns,M}^{\rII}(t) \geq 1- \alpha \big\},\\
  \widehat{{\rm cv}}_{\rs ,\alpha}^{\, \rII}  =&~ \inf\big\{  t\in \br:   \wh{F}_{{\rs},M}^{\rII}(t) \geq 1- \alpha \big\},\\
\end{split}
\]
where $\wh{F}_{\ns,M}^{\rII}(t) =  M^{-1} \sum_{\ell=1}^M I\{|\bW_{\ns,\ell}|_\infty \leq t \} $ and $\wh{F}_{{\rs},M}^{\rII}(t) = M^{-1} \sum_{\ell=1}^M I\{|\bW_{\rs ,\ell}|_\infty \leq t \} $. Similarly to \eq{test.12}, for $\nu \in\{\ns, {\rs}\}$, we define the empirical version of $\Psi^{\rII}_{\nu, \alpha}$ by \be
  \wh \Psi^{\rII}_{\nu, \alpha}(M)= I\big\{T^{\rII}_{\nu} > \widehat{{\rm cv}}_{\nu, \alpha}^{\,\rII} \big\},
  \label{test.34}
\ee such that the null hypothesis $H^{\rII}_0$ is rejected as long as $\wh \Psi^{\rII}_{\nu, \alpha}(M)=1$.

%===============================================================================================%

\subsection{Estimation of covariance matrices}
\label{section:matrixest}

As a part of proposed testing procedures, we need estimates of the covariance matrices. Many existing tests rely on the operator-norm consistent estimation of the covariance matrices that requires extra structural assumptions on the unknown covariances such as banding or sparsity. Under these structural assumptions, regularized or rate-optimal estimates of covariance matrices have been developed \citep{BickelLevina_2008a,BickelLevina_2008b,CaiLiu_2011,CaiZhou_2012,CaiYuan_2012}.  and employed for building tests.

In contrast, the proposed tests require much less restrictions on covariance estimates, which grants its wide scope of applicability. In fact, the validity of the simulation-based testing procedures only entails the covariance estimators $\wt{\bSigma}_1$ and $\wt{\bSigma}_2$ to satisfy $\|\wt{\bSigma}_1-\bSigma_1\|_\infty=o_P(1)$ and $\|\wt{\bSigma}_2-\bSigma_2\|_\infty=o_P(1)$, which is shown in Propositions~\ref{proposition.1} and \ref{proposition.2} included in the supplementary material. Denote by $\widehat{\bSigma}_1=(\hat{\sigma}_{1, k\ell})_{1\leq k,l\leq p}$ and $\widehat{\bSigma}_2=(\hat{\sigma}_{2, k\ell})_{1\leq k,l\leq p}$ the sample covariance matrices based on $\mathcal{X}_n $ and $\mathcal{Y}_m$, respectively. The corresponding sample correlation matrices are $\wh{\bR}_q = \wh{\bD}_q^{-1/2}\wh{\bSigma}_q \wh{\bD}_q^{-1/2}$,where $\wh{\bD}_q
=\textrm{diag}\,(\wh{\bSigma}_{q})$ for $q =1, 2$. Matrices $\wh{\bSigma}_{1,2}$ and $\wh{\bR}_{1,2}$ are defined in the same way that leads to $\wt{\bSigma}_{1,2}$ and $\wt{\bR}_{1,2}$ as in \eq{Sigma12}, with $\wt{\bSigma}_1, \wt{\bSigma}_2$ replaced by $\wh{\bSigma}_1, \wh{\bSigma}_2$. Let $\bU=(U_1, \ldots, U_p)' = \bD_1^{-1/2} \, \bX$, where $\bD_1=\textrm{diag}\,(\bSigma_1)$. It will be shown in Lemma \ref{lm2} in the supplementary material that
\begin{itemize}
\item if there exist constants $K_1>0$, $K_2>1$ and $0< \gamma\leq 2$ such
that
$$
    \max_{1\leq k\leq p}  \mathbb{E}   \exp\big(K_1 |U_k|^{\gamma} \big)  \leq K_2 \ \mbox{ and } \  \log(p) = o( n^{ \gamma/2 }),
$$
then $
\|\widehat{\bSigma}_1-\bSigma_1\|_\infty + \|\widehat{\bR}_1-\bR_1\|_\infty  =o_P(1)$; and

\item if there exist constants $r \geq4$, $\de>0$ and $C_1, C_2>0$ such that
$$
    \max_{1\leq k\leq p}\big( \mathbb{E} |U_k|^r \big)^{1/r} \leq C_1  \ \mbox{ and } \
    p \leq C_2 \, n^{r/2-1-\de},
$$
then $ \| \widehat{\bSigma}_1-\bSigma_1\|_\infty + \|\widehat{\bR}_1-\bR_1\|_\infty  =o_P(1).$
\end{itemize} Therefore, the standard and most natural estimator, the sample covariance matrix can be directly employed by proposed tests.



In particular, for the lower order moment case where the $r$th moments of all the coordinates of $\bU$ are uniformly bounded, the convergence rates of $\| \wh \bSigma_1 -\bSigma_1 \|_\infty$ and $\| \wh \bR_1 -\bR_1 \|_\infty$ are  essentially determined by $pn^{1-r/2}$, up to logarithmic factors. Under suitable moment conditions on both $\bX$ and $\bY$, similar results hold for $\widehat{\bSigma}_{1,2}$ and $\widehat{\bR}_{1,2}$. Therefore, the sample covariance and correlation matrices can be directly used in the proposed testing procedures, while the dimension $p$ is allowed to be as large as either $O\{ \exp(n^{c_1}) \}$ or $O(n^{c_2})$ for some $c_1, c_2  >0$, depending on the light or heavy tailedness of the data. In comparison to the existing tests,  we do not enforce any structural assumptions on the covariance matrices $\bSigma_1$ and $\bSigma_2$ for the simulation-based tests. This echoes our motivations in Section \ref{sec.intro} and grants the proposed methods a much wider scope of applicability in practice. As evidenced by extensive numerical studies in Section \ref{section:numerical}, our proposed procedures are fairly robust to various covariance structures with complex forms, even the long range dependence.

Although the proposed tests do not require operator-norm consistent estimates of $\bSigma_1$ and $\bSigma_2$, still one may replace the sample covariance matrix by adaptive and rate-optimal covariance estimators to improve the empirical performance when the underlying covariance satisfies certain structural assumptions. In practice, an additional eigenvalue correction step might be employed to enforce the positive semi-definiteness of the covariance estimator as discussed in \cite{CaiLiu_2011}.

%===============================================================================================%

\subsection{Screening-based testing procedures}
\label{sec:ps}

The proposed testing procedures are valid when the dimension $p$ is much larger than the sample size $n$. However, building tests based on all dimensions may result in large cut-off values which will potentially compromise the power performance. To enhance the power, we propose a two-step procedure that combines the proposed simulation-based tests and a preliminary step on {\it feature screening}, which screens the $p$ measurements before conducting the test. The power of this two-step procedure is expected to improve upon the proposed tests with a large number of irrelevant features excluded. Numerical experiments in Section~\ref{section:numerical} provide evidences that the two-step testing procedures substantially improve the power performance.

We take the non-studentized statistic $T_{\ns}^{\rI}$ in the one-sample problem to illustrate the motivation of the two-step procedure. Under sparse alternatives, assume that irrelevant coordinates with $\mu_{1k}=0$ have been identified and removed from constructing testing statistic. Denote by $\widehat{\mathcal{S}}$ the index set of excluded coordinates upon screening. The analogue of $T_{\ns}^{\rI}$ is given by $T_{\ns}^{f,\rI}=\max_{k\notin\widehat{\mathcal{S}}}\sqrt{n}|\bar{X}_k|$. As $\{1,\ldots,p\}\setminus\widehat{\mathcal{S}}\subset\{1,\ldots,p\}$, the critical value for $T_{\ns}^{f,\rI}$ at a given significance level $\alpha$ is no larger than $\textrm{cv}_{\ns,\alpha}^{\rI}$, the critical value for the original test statistic $T_{\ns}^{\rI}$. For suitably selected $\widehat{\mathcal{S}}$, $T_{\ns}^{f,\rI}$ coincides with $T_{\ns}^{\rI}$ with high probability. Therefore, the two-step procedure will enhance the power in comparison to the original test. As shown in
Section \ref{section:theory}, two-step procedures maintain the nominal significance level and are consistent against sparse alternatives.

%===============================================================================================%

\subsubsection{One-sample case}
\label{ps:sec1}
Given $\bmu_1 = (\mu_{1 1}, \ldots, \mu_{1 p})'$, let $\mathcal{S}_{10}=\{k=1,\ldots, p:\mu_{1 k}=0\}$. The preliminary procedure is aimed at eliminating irrelevant features indexed by $\mathcal{S}_{10}$. Reformulate the original global test of a mean vector to the following $p$ marginal tests:
\[
H^{\rI}_{0k}:\mu_{1 k}=0 ~~ versus ~~ H^{\rI}_{1k}:\mu_{1 k}\neq 0,
\]
for $k=1, \ldots, p$. For the $k$th marginal hypothesis, a standard test statistic is the $t$-statistic $
\textrm{TS}_k^{\rI}={ \s{n} \bar{X}_k}/{\hat{\sigma}_{1 k}}.$
Motivated by the idea of marginal screening \citep{ChangTangWu_2013}, we define the following index set
\begin{equation}
\widehat{\mathcal{S}}_{1 } = \left\{ 1\leq k\leq p:  |\textrm{TS}_k^{\rI} |\leq  \bigg[\sqrt{2}+\frac{\sqrt{2}}{2\log (p)}+\sqrt{\frac{2\log(1/\alpha)}{\log (p)}}\bigg]\sqrt{\log(p)}  \right\}.  \label{ps:sp1}
\end{equation}
We refer to \cite{ChangTangWu_2013} for more discussions on the advantages of the studenized statistics in marginal screening problems. If $|\widehat{\mathcal{S}}_1|<p$, we put $d=p-|\widehat{\mathcal{S}}_{1 }|$ and let $\wt{\bmu}_1 \in \br^d$ be the sub-vector of $\bmu_1 \in \br^p$ containing only the coordinates excluded by $\widehat{\mathcal{S}}_{1}$. We have therefore downsized the original problem and instead, we focus on the reduced null hypothesis $\wt{H}^{\rI}_0: \wt{\bmu}_1=\bzero$ against the alternative $\wt{H}_1^{\rI}: \wt{\bmu}_1\neq\bzero$. The resulting non-studentized and studentized tests are given by
\begin{align}
\Psi_{\ns,\alpha}^{f,\rI} = I\left\{ \max_{k\notin\widehat{\mathcal{S}}_1}\sqrt{n}|\bar{X}_k| > {\rm cv}^{\rI}_{\ns, \alpha}(\widehat{\mathcal{S}}_{1})  \right\}~~ \textrm{and}  ~~ \Psi^{f,\rI}_{\rs,\alpha}=I\left\{ \max_{k\notin\widehat{\mathcal{S}}_1}\frac{\sqrt{n}|\bar{X}_k|}{\hat{\sigma}_{1k}} > {\rm cv}^{\rI}_{\rs, \alpha}(\widehat{\mathcal{S}}_{1})  \right\},  \label{two-step.one-sample.test}
\end{align}
where ${\rm cv}^{\rI}_{\ns, \alpha}(\widehat{\mathcal{S}}_{1 })$ and ${\rm cv}^{\rI}_{\rs, \alpha}(\widehat{\mathcal{S}}_{1 })$ denote the conditional $(1-\alpha)$-quantile of $\max_{k\notin \widehat{\mathcal{S}}_{1 }} | W_{\ns, k}^{\rI}|$ and $\max_{k\notin \widehat{\mathcal{S}}_{1 }} | W_{\rs, k}^{\rI}|$ given $\mathcal{X}_n$, respectively, where $\bW^{\rI}_{\ns}=(W^{\rI}_{\ns,1}, \ldots, W^{\rI}_{\ns,p})'$ and $\bW^{\rI}_{\rs}=(W^{\rI}_{\rs,1}, \ldots, W^{\rI}_{\rs,p})'$ as in \eqref{q12.definition}. If $|\widehat{\mathcal{S}}_1|=p$, we set $\Psi_{\ns,\alpha}^{f,\rI} = \Psi_{\rs,\alpha}^{f,\rI}=0$.


%===============================================================================================%

\subsubsection{Two-sample case}
\label{ps:sec2}
%Recall that $n=\min(n, m)$ and $N=n+m$. Define $\mathcal{S}_{2 0}=\{1\leq k\leq
%p:\mu_{1 k}=\mu_{2 k}\}$, we assume
%\begin{itemize}
%\item[(A1*)] Under the alternative hypothesis $H_1^{\rII}$, there exists a constant $\beta>0$ such that
%\begin{equation}\label{eq:signal2}
%\min_{k\notin \mathcal{S}_{2 0}}\frac{|\mu_{1 k}-\mu_{2 k}|}{\big( \sigma_{1,kk} /n + \sigma_{2,kk}  /m\big)^{1/2}}\geq \sqrt{ 2\beta \log (p) }.
%\end{equation}
%\item[(A2*)] There exists a constant $0\leq \tau \leq 1$ such that $|\mathcal{S}_{2 0}^{{\rm c}}|=O(p^{ \tau} )$.
%\end{itemize}
Similar to the one-sample case, for each $k=1,\ldots,p$, we define $\textrm{TS}_k^{\rII}=  \sqrt{nm}|\bar{X}_k-\bar{Y}_k|/ \big( m\hat{\sigma}_{1 k}^2  + n\hat{\sigma}_{2 k}^2 \big)^{1/2}$ and set
\begin{equation}
\widehat{\mathcal{S}}_{2 }=\left\{1\leq k\leq p: |\textrm{TS}_k^{\rII} | \leq\bigg[\sqrt{2}+\frac{\sqrt{2}}{2\log (p)}+\sqrt{\frac{2\log(1/\alpha)}{\log (p)}}\bigg] \sqrt{\log(p)} \right\}.  \label{ps:sp2}
\end{equation}
If $|\widehat{\mathcal{S}}_2|<p$, we put $d=p-|\widehat{\mathcal{S}}_{2 }|$ and for $q=1,2$, let $\wt{\bmu}_q \in \br^d$ be the sub-vector of $\bmu_q \in \br^p$ indexed by $\{1,\ldots,p\} \setminus \widehat{\mathcal{S}}_{2}$. Then we carry out the testing procedures in Section~\ref{2s.test} to the reduced problem: $\wt{H}^{\rII}_0: \wt{\bmu}_1=\wt{\bmu}_2$ against the alternative $\wt{H}_1^{\rII}: \wt{\bmu}_1\neq\wt{\bmu}_2$. The resulting tests, denoted by $\Psi_{\ns, \alpha}^{f,\rII}$ and $\Psi_{\rs, \alpha}^{f,\rII}$, are therefore defined in the same way as $\Psi_{\ns, \alpha}^{f,\rI}$ and $\Psi_{\rs, \alpha}^{f,\rI}$ in \eqref{two-step.one-sample.test} respectively. If $|\widehat{\mathcal{S}}_2|=p$, we set $\Psi_{\ns,\alpha}^{f,\rII} = \Psi_{\rs,\alpha}^{f,\rII}=0$.


%===============================================================================================%
%===============================================================================================%


%===============================================================================================%
%===============================================================================================%

\setcounter{equation}{0}
\section{Theoretical properties}\label{section:theory}

In this section, we study the properties of the proposed tests including the asymptotic sizes and powers.
%We focus on the framework $M=\infty$ in the Monte Carlo simulation step to approximate the critical values.
In practice, taking $M$ in thousands using numerical devices to increase simulation efficiency is now the rule rather than the exception in the Monte Carlo framework. The difference between such large values of $M$ and using mathematically ideal value $M=\infty$ is particularly small. We therefore focus on the oracle tests $\Psi_{\nu,\alpha}^{\rI}$ and $\Psi_{\nu,\alpha}^{\rII}$ for $\nu\in\{\ns,\rs\}$, and their screening-based analogues $\Psi_{\nu,\alpha}^{f,\rI}$ and $\Psi_{\nu,\alpha}^{f,\rII}$. It is shown that the proposed tests maintain the nominal size asymptotically under very general covariance structures. Moreover, the proposed tests are shown to be consistent against sparse alternatives.

For $p$-dimensional random vectors $\bX=(X_1,\ldots, X_p)'$ and $\bY=(Y_1, \ldots, Y_p)'$ with means $\bmu_1,\bmu_2$ and covariance matrices $\bSigma_1,\bSigma_2$, their marginally standardized versions are defined by $\bU=(U_{1},\ldots, U_p)'=\bD_1^{-1/2}\bX$ and $\bV=(V_1,\ldots, V_p)'=\bD_2^{-1/2}\bY$, respectively, where $\bD_1=\mbox{diag}\,(\bSigma_1)$ and $\bD_2=\mbox{diag}\,(\bSigma_2)$. As discussed in Section \ref{method.sec}, no structural assumptions on the unknown covariance matrices are enforced. We only impose the following mild moment conditions.

\begin{itemize}
\item[(M1)](Lower order moments) There exist $r \geq 4$ and $K_0>0$ such that the $r$th moments of the components of $\bU$ and $\bV$ are uniformly bounded; that is,
$$
      \max_{1\leq k\leq p} \big( \mathbb{E} |U_k|^r   \big)^{1/r}      \leq K_0 \quad \ \textrm{and}\ \quad
        \max_{1\leq k\leq p} \big( \mathbb{E} |V_k|^r\big)^{1/r}         \leq K_0.
$$
\item[(M2)](Sub-exponential tails) There exist constants $K_1>0$, $K_2>1$ and $0<\gamma\leq 2$ such that
$$
\max_{1\leq k\leq p} \mathbb{E}  \exp\big(K_1|U_k|^{\gamma} \big) \leq K_2\quad \ \textrm{and}\ \quad  \max_{1\leq k\leq p}  \mathbb{E}  \exp\big(K_1 |V_k|^{\gamma} \big) \leq K_2.
$$
\end{itemize}
Throughout this section, we assume that $n, p\geq 2$, $n \asymp m$, $n\leq m$, and that $\{\sigma_{q k}^2 :=\sigma_{q, kk}, q = 1,2\}_{k=1}^n$ is bounded away from $0$ and $\infty$.

%===========++++++++++++++++++++++++++++++++++++++++++++++++++++++===================%
%===========++++++++++++++++++++++++++++++++++++++++++++++++++++++===================%




\begin{theorem}[Asymptotic size of the one-sample tests without screening]  \label{one-sample.size}
Let $\wt{\bSigma}_1=\wh{\bSigma}_1$, the sample covariance matrix, in \eqref{q12.definition} and $\nu \in \{\ns, \rs\}$.
\begin{enumerate}
\item[{\rm (i)}] Assume that {\rm(M1)} holds and $p=O(n^{r/2-1-\de})$ for some $\de>0$. Then as $n ,p  \rightarrow \infty$,
$$
    \mathbb{P}_{H^{\rI}_0}\big\{  \Psi^{\rI}_{\nu,\alpha} =1 \big\} \rightarrow \alpha.
$$
\item[{\rm (ii)}]  Assume that {\rm(M2)} holds for some $\gamma\geq \frac{1}{2}$ and $\log (p)=o(n^{1/7})$. Then as $n ,p \rightarrow \infty$,
$$
 \mathbb{P}_{H^{\rI}_0}\big\{ \Psi^{\rI}_{\nu,\alpha} =1 \big\} \rightarrow \alpha.
$$
\end{enumerate}
\end{theorem}

Theorem \ref{one-sample.size} establishes the validity of the proposed one-sample tests in the sense that the testing procedures in Section \ref{section:onesample} maintain nominal significance level asymptotically. In addition, as evidenced by the numerical experiments in Section~\ref{section:numerical}, the test based on non-studentized statistics outperforms its studentized analogue in terms of maintaining the nominal significance level when the sample size is small. This, however, is not surprising since the inverse operation, say $\wh{\bD}_1^{-1/2}$, usually leads to an augmentation of the estimation error in $\wh{\bD}_1$ and therefore is more sensitive to the sample size. In the following theorem, we summarize the asymptotic power of the proposed one-sample tests under suitable conditions on the lower bound of the signal-to-noise ratios.

\begin{theorem}[Asymptotic power of the one-sample tests without screening] \label{one-sample.power.1}
Let $\wt{\bSigma}_1= \wh{\bSigma}_1$, the sample covariance matrix, in \eqref{q12.definition}. Assume that either condition {\rm(M1)} holds and $p=O(n^{r/2-1-\de})$ for some $\de>0$, or condition {\rm(M2)} holds and $\log(p) = o (n^{\gamma/2} )$. For given $0<\alpha<1$, write \be \la(p,\alpha)=  \s{2\log(p)} + \s{2\log(1/\alpha)}  , \label{threshold} \ee and let $\{ \varepsilon_{n} \}_{n\geq 1}$ be an arbitrary sequence of positive numbers satisfying $\varepsilon_{n} \rightarrow 0$ and $\varepsilon_{n} \s{\log(p)} \rightarrow \infty$ as $n \rightarrow \infty$.
\begin{enumerate}
\item[{\rm (i)}] Under the alternative $H^{\rI}_1$ with
$$
    \f{\max_{1\leq k\leq p} |\mu_{1 k}|}{\max_{1\leq k\leq p} \sigma_{1 k} }   \geq (1+\varepsilon_{n}) \,
    n^{-1/2} \la(p,\alpha) ,
$$
we have as ${n} , p \rightarrow \infty$,
\[
   \mathbb{P}_{H^{\rI}_1}\big\{  \Psi^{\rI}_{\ns ,\alpha} =1 \big\} \rightarrow 1.
\]

\item[{\rm (ii)}] Under the alternative $H^{\rI}_1$ with
$$
    \max_{1\leq k\leq p} \f{|\mu_{1 k} |}{ \sigma_{1 k} }  \geq (1+\varepsilon_{n}) \, n^{-1/2} \la(p,\alpha),
$$
we have as ${n} , p \rightarrow \infty$,
\[
    \mathbb{P}_{H^{\rI}_1} \big\{ \Psi^{\rI}_{\rs ,\alpha} =1 \big\} \rightarrow 1.    \label{power.2}
\]
\end{enumerate}
\end{theorem}

Theorem \ref{one-sample.power.1} shows that, in agreement with intuition, the test based on studentized statistics is consistent in a larger testable region in comparison to the test based on non-studentized statistics. As a complement to Theorem \ref{one-sample.size}, the asymptotic size of the proposed two-sample tests without screening is reported below.


%In addition to the general alternatives as considered in Theorem~\ref{one-sample.power.1}, we are also interested in the power of the test $T^{\rI}_{\rs}$ under the sparse alternative in which $\bmu_1$ only contains a small fraction of non-zero coordinates. Define the subset of $d$-sparse vectors in $\br^p$ by
%\be
%    \mathcal{S}(d) = \bigg\{ \omega =(\omega_1,\ldots,\omega_p)' \in \br^p: \sum_{k=1}^p I\{ \omega_k \neq 0\} = d \bigg\}. \lbl{sparse.set}
%\ee
%We wish to analyze the power of $\Phi^{\rI}_{\rs ,\alpha}$ under the sparse alternative
%\begin{align*}
% H^{\rI}_{s,1} : & \,  \,  \bmu_1 \in \mathcal{S}(d) \, \mbox{ for } d= p^{\tau} \mbox{ and } 0\leq  \tau <1, \, \min_{k: \mu_{1 k} \neq 0} \f{|\mu_{1 k}|}{ \sigma_{1 k} } \geq \sqrt{\frac{ 2\beta \log(p) }{n}} \, \mbox{ for } 0<\beta <1,  \\
% & \mbox{ and the non-zero  locations are randomly uniformly drawn from } \{1,\ldots, p\}.
%\end{align*}
%To do so, we impose the following regularity conditions on the underlying covariance structures.
%\begin{itemize}
%\item[({\rm D-$q$}).] There exist constants $C_1>0$ and $0<\rho<1$ such that $\max_{1\leq k\leq p} \sum_{\ell=1}^p r_{q, k \ell}^2 \leq C_1$ and
%\beq
%    |\Lambda_q(\rho)| := \left\{  1 \leq k\leq p :  |r_{q, k \ell}| \geq \rho \mbox{ for some } 1\leq \ell \leq p,  \, \ell \neq k \right\} =o(p).
%\eeq
%\end{itemize}
%\begin{theorem}[Asymptotic power under sparse alternatives]
%\label{one-sample.power.2}
%Assume that condition (D-1) holds and let $\wt{\bSigma}_1= \wh{\bSigma}_1$. Moreover, assume that either condition (M1) holds with $p=O(n^{r/2-1-\de})$ for some $\de >0$, or condition (M2) holds with $\log(p)=o(n^{ \gamma/2 })$. Then under $H^{\rI}_{s,1}$ with $\beta \geq (1-\s{\tau})^2+\varepsilon$ for some $\varepsilon>0$,
%\beq
%\mathbb{P}_{H^{\rI}_{s,1}}\big\{ \Phi^{\rI}_{\rs ,\alpha} = 1  \big\} \rightarrow 1 \ \ \mbox{ as } n , p \rightarrow \infty.
%\eeq
%\end{theorem}

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\begin{theorem}[Asymptotic size of the two-sample tests without screening]  \label{two-sample.size}
Let $(\wt{\bSigma}_1, \wt{\bSigma}_2)= (\wh{\bSigma}_1, \wh{\bSigma}_2)$, which are the sample covariance matrices, and $\nu \in \{\ns, \rs\}$. Assume that either condition~{\rm(i)} or condition~{\rm(ii)} in Theorem~{\rm\ref{one-sample.size}} holds. Then as $n, p \rightarrow \infty$,
$$
 \mathbb{P}_{H^{\rII}_0}\big\{  \Psi_{\nu,\alpha}^{\rII} =1 \big\} \rightarrow   \alpha.
$$
\end{theorem}

Theorem~\ref{two-sample.size} implies that, under proper moment conditions, the proposed two-sample non-screening tests maintain nominal size $\alpha$ asymptotically, while allowing for either a polynomial or an exponential rate of growth of the dimension $p$ with respect to the sample size $n$. In Theorem~\ref{two-sample.power.1} below, the asymptotic power of the two-sample non-screening tests is analyzed under conditions on the separation distance between $\bmu_1$ and $\bmu_2$.

\begin{theorem}[Asymptotic power of the two-sample tests without screening] \label{two-sample.power.1}
Let $(\wt{\bSigma}_1, \wt{\bSigma}_2)=(\widehat{\bSigma}_1, \widehat{\bSigma}_2)$, which are the sample covariance matrices. Assume that either condition {\rm(M1)} holds and $p=O(n^{r/2-1-\de})$ for some $\de>0$, or condition {\rm(M2)} holds and $\log(p) = o ( n^{\gamma /2  } )$. For given $0<\alpha<1$, let $ \la(p,\alpha)$ be as in \eq{threshold} and let $\{ \varepsilon_n\}_{n\geq 1}$ be an arbitrary sequence of positive numbers satisfying $\varepsilon_n \rightarrow 0$ and $\varepsilon_n \s{\log(p)} \rightarrow \infty$ as $n  \rightarrow \infty$.
\begin{enumerate}
\item[{\rm (i)}] Under the alternative $H^{\rII}_1$ with
$$
   \f{  \max_{1\leq k\leq p} |\mu_{1 k}-\mu_{2 k}| }{ \max_{1\leq k\leq p} ( \sigma_{1 k}^2/n + \sigma_{2 k}^2/m )^{1/2}} \geq (1+ \varepsilon_n ) \,  \la(p,\alpha)  ,
$$
we have as $n, p \rightarrow \infty$,
\[
\mathbb{P}_{H^{\rII}_1}\big\{   \Psi^{\rII}_{\ns,\alpha} = 1 \big\} \rightarrow 1.    \label{power.T3}
\]

\item[{\rm (ii)}] Under the alternative $H^{\rII}_1$ with
$$
    \max_{1\leq k\leq p} \f{|\mu_{1 k}-\mu_{2 k}|}{(  \sigma_{1 k}^2/n + \sigma_{2 k}^2/m  )^{1/2} }  \geq (1+\varepsilon_n) \,  \la(p,\alpha),
$$
we have as $ n, p \rightarrow \infty$,
\[
\mathbb{P}_{H^{\rII}_1}\big\{   \Psi^{\rII}_{\rs ,\alpha} = 1 \big\} \rightarrow 1.   \label{power.T4}
\]
\end{enumerate}
\end{theorem}

%The next result shows the power of the test $\Phi^{\rII}_{\rs ,\alpha}$ under sparse alternatives; that is,
%\begin{align*}
% H^{\rII}_{s,1} : & \,  \,  \bmu_1-\bmu_2 \in \mathcal{S}(d) \, \mbox{ for } d= p^{\tau} \mbox{ and }  0\leq  \tau <1,  \\
% & \, \, \min_{k: \mu_{1k} \neq \mu_{2k}} \f{|\mu_{1k}-\mu_{2 k}|}{( \sigma_{1 k}^2/n + \sigma_{2 k}^2/m )^{1/2}} \geq \s{2\beta \log(p) }  \, \mbox{ for } 0<\beta <1, \mbox{ and the non-zero} \\
% & \, \,  \mbox{ locations of $\bmu_1-\bmu_2$ are randomly uniformly drawn from } \{1,\ldots, p\}.
%\end{align*}
%The proof can be directly adapted from that of Theorem~\ref{one-sample.power.2}, and thus is omitted.
%
%
%
%\begin{theorem}[Asymptotic power under sparse alternatives] \label{sparse.power.2s}
%Suppose that conditions (D-1) and (D-2) are satisfied, and let $(\wt{\bSigma}_1 ,\wt{\bSigma}_2 )=( \wh{\bSigma}_1 ,  \wh{\bSigma}_2)$. Moreover, assume that either condition (M1) holds with $p=O(n^{r/2-1-\de})$ for some $\de>0$, or condition (M2) holds with $\log(p) = o (n^{\gamma/2} )$. Then under $H^{\rII}_{s,1}$ with $\beta \geq (1-\s{\tau})^2+\varepsilon$ for some $\varepsilon>0$,
%\beq
%\mathbb{P}_{H^{\rII}_{s,1}}\big\{ \Phi^{\rII}_{\rs ,\alpha} = 1  \big\} \rightarrow 1 \ \ \mbox{ as } n, p\rightarrow \infty.
%\eeq
%\end{theorem}

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The following theorem establishes asymptotic properties of the proposed two-step testing procedure. Part (i) in Theorem \ref{two-step.one-sample.theory} below shows that the type I error of the proposed screening-based two-step procedures, with or without studentization, can be controlled by the prescribed significance level asymptotically. Similar to the comparison between the studentized and non-studentized tests in Theorem \ref{one-sample.power.1}, parts (ii) and (iii) in Theorem \ref{two-step.one-sample.theory} below also imply that the screening-based two-step studentized test is consistent in a larger region than its non-studentized counterpart.

\begin{theorem}[Asymptotic properties of the one-sample screening-based tests]  \label{two-step.one-sample.theory}
Let $\wt{\bSigma}_1= \wh{\bSigma}_1$, the sample covariance matrix, in \eqref{q12.definition}. Assume that either condition {\rm(M1)} holds and $p=O(n^{r/2-1-\de})$ for some $\de>0$, or condition {\rm(M2)} holds for some $\gamma\geq \frac{1}{2}$ and $\log(p) = o (n^{1/7} )$.
\begin{enumerate}
\item[{\rm (i)}] Under the null $H^{\rI}_0$, we have $\limsup_{n\rightarrow \infty}    \mathbb{P}_{H^{\rI}_0} \big\{  \Psi^{f,\rI}_{\nu ,\alpha} =1  \big\} \leq  \alpha$ for $\nu\in\{\ns,\rs\}$.

\item[{\rm (ii)}] Under the alternative $H^{\rI}_1$ specified in part {\rm(i)} of Theorem {\rm\ref{one-sample.power.1}}, we have $\mathbb{P}_{H^{\rI}_1}  \big\{ \Psi^{f,\rI}_{\ns ,\alpha} =1 \big\} \rightarrow 1$.

\item[{\rm (iii)}] Under the alternative $H^{\rI}_1$ specified in part {\rm(ii)} of Theorem {\rm\ref{one-sample.power.1}},
 we have $\mathbb{P}_{H^{\rI}_1}  \big\{ \Psi^{f,\rI}_{\rs ,\alpha} =1 \big\} \rightarrow 1$.
\end{enumerate}
\end{theorem}

Analogously, the following theorem establishes the limiting null property and the asymptotic power for the proposed two-step procedures with pre-screening in the two-sample settings. The proof of Theorem~\ref{two-step.two-sample.theory} is almost identical to that of Theorem~\ref{two-step.one-sample.theory}, and therefore is omitted in supplementary material.


\begin{theorem}[Asymptotic properties of the two-sample screening-based tests]  \label{two-step.two-sample.theory}
Let $(\wt{\bSigma}_1, \wt{\bSigma}_2)=(\widehat{\bSigma}_1, \widehat{\bSigma}_2)$, which are the sample covariance matrices. Assume that either condition {\rm(M1)} holds and $p=O(n^{r/2-1-\de})$ for some $\de>0$, or condition {\rm(M2)} holds for some $\gamma\geq \frac{1}{2}$ and $\log(p) = o ( n^{1/7  } )$.
\begin{enumerate}
\item[{\rm (i)}] Under the null $H^{\rII}_0$, we have $\limsup_{n\rightarrow \infty}    \mathbb{P}_{H^{\rII}_0} \big\{  \Psi^{f,\rII}_{\nu ,\alpha} =1  \big\} \leq  \alpha$ for $\nu\in\{\ns,\rs\}$.

\item[{\rm (ii)}] Under the alternative $H^{\rII}_1$ specified in part {\rm(i)} of Theorem {\rm\ref{two-sample.power.1}}, we have $\mathbb{P}_{H^{\rII}_1}  \big\{ \Psi^{f,\rII}_{\ns ,\alpha} =1 \big\} \rightarrow 1$.


\item[{\rm (iii)}] Under the alternative $H^{\rII}_1$ specified in part {\rm(ii)} of Theorem {\rm\ref{two-sample.power.1}},
 we have $\mathbb{P}_{H^{\rII}_1}  \big\{ \Psi^{f,\rII}_{\rs ,\alpha} =1 \big\} \rightarrow 1$.
\end{enumerate}
\end{theorem}

%===============================================================================================%
%===============================================================================================%

%===============================================================================================%
%===============================================================================================%

\section{Simulation studies}\label{section:numerical}

%Throughout this section, we denote $T_\nu$ and $T_\nu$-$f$ the testing procedure without and with the pre-screening step, respectively, for $\nu \in \{\ns, \rs\}$. For ease of exposition, we suppress the superscripts $^\rI$ and $^\rII$.

In this section, we report the simulation results from several experiments which were designed to evaluate the performance of the proposed tests, including the non-studentized test without screening $\Psi_{\ns,\alpha}$, the studentized test without screening $\Psi_{\rs,\alpha}$, the non-studentized test with screening $\Psi_{\ns,\alpha}^f$ and the studentized test with screening $\Psi_{\rs,\alpha}^f$, for both one- and two-sample problems. For ease of exposition, we suppress the superscripts $\rm (I)$ and $\rm (II)$. To demonstrate the proposed tests, we also implemented peer testing procedures for comparison. For the one-sample problem, we compared the proposed tests with the test by \cite{ZhongChenXu_2013} (denoted by ZCX hereafter) and the Higher Criticism procedure by \cite{DonohoJin04} (denoted by HC hereafter). For the two-sample problem, we experimented three existing tests: the one by \cite{ChenQin_2010} (denoted by CQ hereafter), the test based on the Higher Criticism procedure \cite{DelaigleHallJin_2011} (denoted by HC2 hereafter), and the test by \cite{CaiLiuXia_2014} (denoted by CLX hereafter). %\red{The test statistic of CQ test is of the sum of square type to utilize relatively non-sparse signals, and the test statistic of CLX test is of the maximum type, which is similar to our proposed tests, that the CLX test maintains powers against sparse alternatives under specific structural assumptions on the unknown covariance matrices.}


The proposed tests are easily implementable via the Monte Carlo method. In the simulation studies, we considered a wide range of covariance structures, including both the sparse and dense settings to investigate the numerical performance of the proposed tests, as well as different values of sample sizes $(n,m)$ and dimension $p$. As discussed in Sections \ref{section:matrixest} and \ref{section:theory}, the validity of the proposed tests is guaranteed as long as the covariance estimators are uniform entrywise consistent. Therefore, we used the sample covariance matrices to generate $M=1500$ Monte Carlo samples to compute the empirical critical values for the tests. The simulation results show that the proposed tests reasonably maintain the nominal significance level, and have substantially better power performance in comparison to peer tests for weak and/or sparse signals.




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%===========++++++++++++++++++++++++++++++++++++++++++++++++++++++===================%

\subsection{One-sample case} \label{subsection:numerical one sample}

Without loss of generality, we took $\bmu_1=\textbf{0}$ under the null hypothesis for the one-sample problem \eqref{eq:onesample}, whereas, under the alternative, we took $\bmu_1=(\mu_{11},\ldots, \mu_{1p})'$ to have $\lfloor \kappa p^{r} \rfloor$ non-zero entries uniformly and randomly drawn from $\{1,\ldots,p\}$, where $\kappa$ was an integer and $\lfloor x \rfloor$ denotes the integer part of $x$. We took $r=0,0.4,0.5,0.7$ and $0.85$, where $\kappa=8$ if $r=0$ and $\kappa=1$ otherwise. The settings $r=0$ and $r=0.7$ or $r=0.85$ mimic the scenarios of sparse and non-sparse signals, respectively. The magnitudes of non-zero entries $\mu_{1\ell}$ were set to be $\{ 2\beta\sigma_{1,\ell\ell}\log(p)/n \}^{1/2}$, where $\sigma_{1,\ell\ell}$ denotes the $\ell$th diagonal entry of $\bSigma_1$. We took $\beta=0.01,0.2,0.4,0.6$ and $\beta=0.01$ to mimic the scenario of weak signals.

The data generation models are described as follows. Let ${\rm Unif}(a,b)$ be the uniform distribution on $(a,b)$ for $b>a$. The following three models were used to generate random samples $\bX_i=\bW_i+\bmu_1$ for $ i=1,\ldots, n$, where $\{\bW_i=(W_{i,1},\ldots, W_{i,p})':i=1,\ldots,n\}$ are independent and identically distributed centered Gaussian random vectors with covariance matrix $\bSigma_1=(\sigma_{1,k \ell})_{1\leq k,\ell \leq p}$.
\begin{enumerate}

\item[(a)] Model 1$^{\rm{(I)}}$ (Bandable $\bSigma_1$): $\sigma_{1,k \ell}=0.4^{|k-\ell |}$ for $1\leq k,\ell \leq p$. This model was employed by \cite{ZhongChenXu_2013}.

\item[(b)] Model 2$^{\rm{(I)}}$ (Block diagonal $\bSigma_1$): $\sigma_{1,k k}$ are independent and identically drawn from ${\rm Unif}(1,2)$, $\sigma_{1,k \ell}=0.7$ for $10(t-1)+1\leq k\neq \ell\leq 10 t$, where $t=1,\ldots, \lfloor p/10\rfloor$, and $\sigma_{1,k \ell}=0$ otherwise.

\item[(c)] Model 3$^{\rm{(I)}}$ (Long range dependence $\bSigma_1$): Let $\theta_1,\ldots, \theta_p$ be independent and identically drawn from ${\rm Unif}(1,2)$, and we took $\sigma_{1,kk}=\theta_k$ and $\sigma_{1,k \ell } =\rho_\alpha(|k -\ell |)$ for $k \neq \ell$, where
$\rho_\alpha(e)=\tfrac12 \{(e+1)^{2H}+(e-1)^{2H}-2e^{2H}\}$ with $H=0.9$.
\end{enumerate}
Model 1$^{\rm{(I)}}$ and Model 2$^{\rm{(I)}}$ have sparse covariance structures while Model 3$^{\rm{(I)}}$ takes long range dependence into account which possesses a non-sparse structure. In addition, we considered the following two models with non-Gaussian data generation mechanisms to study the robustness of the proposed tests against Gaussian assumptions.

\begin{enumerate}
\item[(d)] Model 4$^{\rm{(I)}}$ (Autoregressive process of order one, AR(1), with $t$-distributed innovations): Generate independent and identically distributed $p$-variate random vectors $\bX_i\sim t_\omega(\bmu_1,\bSigma_1) $ for $i=1,\ldots, n$, where $t_\omega(\bmu_1,\bSigma_1)$ is the non-central multivariate $t$-distribution with non-central parameter $\bmu_1$,  $\omega=5$ degrees of freedom, and $\bSigma_1=(\sigma_{1,k\ell})_{1\leq k,\ell\leq p}$ with $\sigma_{1,k \ell}=0.995^{|k-\ell|}$.

\item[(e)] Model 5$^{\rm{(I)}}$ (Moving average process with Beta distributed innovations): For $i=1,\ldots, n$ and $k=1,\ldots, p$, we considered $X_{ik}=\rho_1Z_{i,k}+\rho_2Z_{i,k+1}+\cdots+\rho_pZ_{i,k+p-1}+\mu_k$ where $\rho_{\ell}$ are independent and identically drawn from $0.6{\rm Unif}(-1,1)+0.4\delta_0$ for $\ell=1,\ldots, p$, where $\delta_0$ is the point mass at $0$ and $\{Z_{i,k}\}$ are independent random variables with a common centered $\textrm{Beta}(2,1)$ distribution.
\end{enumerate}
Both the covariance structures in Model 4$^{\rm{(I)}}$ and Model 5$^{\rm{(I)}}$ are non-sparse. Simulation results show that the proposed tests are valid in these non-sparse models with non-Gaussian sampling distributions.

In each model, we generated data with a sample size $n=40$ or $n=80$. The dimension $p$ took values $120,360$ and $1080$. The empirical power and level of significance were computed based on $1500$ simulations. Simulation results for the tests $\Psi_{\ns,\alpha}$, $\Psi_{\rs,\alpha}$, $\Psi_{\ns,\alpha}^f$ and $\Psi_{\rs,\alpha}^f$ and the ZCX and HC tests are summarized in Table \ref{size1} and Figures \ref{f01}-\ref{f05}. Table \ref{size1} displays the empirical sizes of all the tests. It can be seen that in all the models, the empirical sizes of the non-studentized tests $\Psi_{\ns,\alpha}$ and $\Psi_{\ns,\alpha}^f$ are reasonably close to the nominal level $0.05$ for both $n=40$ and $n=80$. Comparing the results for $n=40$ and $n=80$, the proposed studentized tests $\Psi_{\rs,\alpha}$ and $\Psi_{\rs,\alpha}^f$ have slightly inflated size when $n$ is relatively small but improve with larger sample sizes. The ZCX test maintains
the nominal size for Model 1$^{\rm{(I)}}$ but fails in the presence of long range dependence or non-sparse covariance structures. The HC procedure also fails in maintaining the nominal significance when the sample size $n$ is small or the dependency is strong and complex.



\begin{sidewaystable}
    \centering
{\renewcommand{\arraystretch}{1.1}
 \begin{tabular}{cccccccccccccccc}\toprule
  &  \multicolumn{3}{c}{Model 1$^{\rm{(I)}}$}      &      \multicolumn{3}{c}{Model 2$^{\rm{(I)}}$}    &    \multicolumn{3}{c}{Model 3$^{\rm{(I)}}$}       &  \multicolumn{3}{c}{Model 4$^{\rm{(I)}}$}    &    \multicolumn{3}{c}{Model 5$^{\rm{(I)}}$}                                  \\ \toprule
$\text{tests}~$/$~p$ &  120&360&1080&120&360&1080&120&360&1080&  120&360&1080&  120&360&1080\\\toprule
& \multicolumn{15}{c}{$n=40$}     \\ \midrule
 $\Psi_{\ns,\alpha}$  & 0.037&0.027&0.021&0.035&0.033&0.020& 0.025&0.028&0.023 &0.054&0.044&0.033& 0.044&0.023&0.023\\ [0.5ex]
  $\Psi_{\rs,\alpha}$  & 0.133 & 0.126 & 0.168& 0.094&0.114&0.216&0.093&0.113&0.202 & 0.065&0.080&0.096& 0.094&0.190&0.252  \\ [0.5ex]
   $\Psi^f_{\ns,\alpha}$  & 0.044& 0.045& 0.043& 0.041&0.045&0.030& 0.039&0.027&0.039 & 0.054&0.046&0.033&0.055&0.030&0.032  \\ [0.5ex]
    $\Psi^f_{\rs,\alpha}$  &  0.150 & 0.154& 0.194& 0.093&0.152&0.226 &0.095&0.170&0.218 & 0.060&0.058&0.093& 0.095&0.205&0.305  \\ [0.5ex]
ZCX  & 0.064&0.078 & 0.089& 1&1&1& 1&1&1 & 0.382&0.487&0.673& 0.389&1&1  \\ [0.5ex]
HC & 0.165& 0.296& 0.545& 0.195&0.284&0.510& 0.145&0.278&0.418&  0.370&0.450&0.535&0.179&0.280&0.518 \\
 \midrule

& \multicolumn{15}{c}{$n=80$}     \\ \midrule

 $\Psi_{\ns,\alpha}$ & 0.037&0.036&0.029& 0.043&0.032&0.025&0.040&0.032&0.042 & 0.049&0.047&0.040& 0.034&0.039&0.036 \\ [0.5ex]
  $\Psi_{\rs,\alpha}$  & 0.060& 0.082& 0.092& 0.073&0.092&0.091&0.082&0.083&0.094 &0.058&0.058&0.067& 0.083&0.090&0.125  \\ [0.5ex]
   $\Psi^f_{\ns,\alpha}$  & 0.048& 0.045& 0.043& 0.053&0.043&0.034&0.051&0.045&0.040 &  0.049&0.048&0.044&0.046&0.054&0.045  \\ [0.5ex]
    $\Psi^f_{\rs,\alpha}$  & 0.086& 0.097& 0.094&0.088&0.091&0.106&0.095&0.091&0.110 &0.060&0.058&0.069& 0.091&0.098&0.157 \\ [0.5ex]
ZCX  & 0.080&0.072&0.071&1&1&1&1&1&1&  0.404&  0.506&0.702& 0.593&1&1 \\ [0.5ex]
HC & 0.129&0.170&0.247& 0.132&0.148&0.250&0.126&0.175&0.223& 0.380&0.420&0.506&0.124&0.138&0.283
  \\ \bottomrule
  \end{tabular}
  }
  \caption{Empirical sizes of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) for the one-sample problem \eqref{eq:onesample}, along with those of the tests by \cite{ZhongChenXu_2013} (ZCX) and by \cite{DonohoJin04} (HC)  at 5\%  nominal significance. Models with Gaussian data and bandable, block diagonal or long range dependence covariance matrices, the autoregressive model with $t$-distributed innovations, and the moving average model with Beta distributed innovations are considered when $n=40, 80$ and $p=120,360, 1080$.  }\label{size1}
\end{sidewaystable}



\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S11p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S11p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S11p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S11p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample}, along with the power of the test by \cite{ZhongChenXu_2013} (ZCX) at 5\% nominal significance for the Gaussian data and bandable covariance matrices in Model 1$^{\rm{(I)}}$ when $n=80$ and $p=1080$.}
   \label{f01}
\end{figure}

\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S12p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S12p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S12p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S12p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample} at 5\% nominal significance for the Gaussian data and block diagonal covariance matrices in Model 2$^{\rm{(I)}}$ when $n=80$ and $p=1080$.}
   \label{f02}
\end{figure}


To compare the empirical powers, we took $n=80$, $p=1080$. For Model 1$^{\rm{(I)}}$, we compared the proposed tests with the ZCX test (Figure \ref{f01}), whereas, for the other four models, we focused on comparing the proposed tests as they maintain the nominal size reasonably well (Figures \ref{f02}-\ref{f05}). Figure \ref{f01} shows that the proposed studentized tests and non-studentized test with screening, $\Psi_{\rs,\alpha}$, $\Psi_{\rs,\alpha}^f$ and $\Psi_{\ns,\alpha}^f$, provide non-trivial powers against alternatives with sparse signals ($r=0$) even under the weak signal settings ($\beta=0.01$); in contrast, the ZCX test improves its power as the signal getting dense, which is expected for sum of squares-type statistics. As the signal strength increases, all tests under consideration gain powers. The proposed tests with screening, $\Psi_{\ns,\alpha}^f$ and $\Psi_{\rs,\alpha}^f$, outperform the ZXC test under sparse alternatives ($r=0,0.4$), and their powers are close to that of the ZCX test for dense signals ($r\geq 0.7$). From Figures \ref{f02} to \ref{f05}, we observe that the screening procedure substantially improves the power performance of the tests for all settings, which reflects the heuristic discussions and motivations in Section \ref{sec:ps}. The proposed non-studentized test with screening $\Psi_{\ns,\alpha}^f$ performs comparably to, or better than, the studentized test without screening $\Psi_{\rs,\alpha}$ under sparse alternatives ($r\leq 0.5$). This suggests that $\Psi_{\ns,\alpha}^f$ is more preferable in practice given its capability in maintaining the nominal significance for small sample size. More extensive simulations were carried out for dimensions $p=120$ and $360$, from which the comparisons are in line with the cases reported here. It can be further observed that the empirical powers of all the tests increase in $p$.  These numerical results support our theoretical analysis. The additional simulation results are placed in the supplementary material.


\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S13p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S13p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S13p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S13p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample} at 5\% nominal significance for the Gaussian data and long range dependence covariance matrices in Model 3$^{\rm{(I)}}$ when $n=80$ and $p=1080$.}
   \label{f03}
\end{figure}

\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S14p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S14p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S14p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S14p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample} at 5\% nominal significance for the autoregressive process model, Model 4$^{\rm{(I)}}$, with $t$-distributed innovations when $n=80$ and $p=1080$.}
   \label{f04}
\end{figure}

\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S15p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S15p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S15p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S15p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample} at 5\% nominal significance for the moving average process with Beta distributed innovations in Model 5$^{\rm{(I)}}$ when $n=80$ and $p=1080$.}
   \label{f05}
\end{figure}


In summary, the numerical results show that the proposed tests, particularly the studentized tests and the non-studentized test with screening, $\Psi_{\rs,\alpha}$, $\Psi_{\rs,\alpha}^f$ and $\Psi_{\ns,\alpha}^f$, outperform the peer tests under sparse alternatives when the covariance structure is non-sparse, in other words, when the dependency is spread. The proposed tests are robust against unknown covariance structures and Gaussianity. The non-studentized test with screening $\Psi_{\ns,\alpha}^f$ maintains the nominal significance for small sample size and has good powers against sparse alternatives, which is recommended for practical applications with relatively small sample size. The studentized tests with screening $\Psi_{\rs,\alpha}^f$ is more powerful and thus is preferable in applications with relatively large samples, such as biomedical research with a large cohort.


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\subsection{Two-sample case}
\label{subsection:numerical two sample}

Similar settings to those in Section \ref{subsection:numerical one sample} were used to examine the proposed tests for the two-sample problem \eqref{eq:twosample}. Without loss of generality, we took $\bmu_1=\bmu_2=\textbf{0}$ under the null hypothesis, whereas, under the alternative, we let $\bmu_1=(\mu_{11},\ldots, \mu_{1p})'$ to have $\lfloor \kappa p^{r} \rfloor$ non-zero entries uniformly and randomly drawn from $\{1,\ldots,p\}$, where $\kappa$ is an integer. As before, we considered $r=0,0.4,0.5,0.7$ and $0.85$, where $\kappa=8$ if $r=0$ and $\kappa=1$ otherwise. The settings $r=0$ and $r=0.7$ or $r=0.85$ mimic sparse and non-sparse scenarios, respectively. The magnitudes of non-zero entries $\mu_{1\ell}$ were set to be $\{2\beta\sigma_{\ell\ell}\log(p)(1/n+1/m)\}^{1/2}$, where $\sigma_{\ell\ell}$ is the $\ell$th diagonal entry of the pooled covariance matrix $\bSigma_{1,2}$ as in \eqref{Sigma12}. We took $\beta=0.01,0.2,0.4,0.6$ and $\beta=0.01$ to mimic the scenario of weak signals.

The following three models were used to generate random samples $\bX_i=\bW_i+\bmu_1, \bY_j=\bV_j+\bmu_2$ for $ i=1,\ldots, n$ and $j=1,\ldots, m$, where $\bW_i=(W_{i,1},\ldots, W_{i,p})'$ and $\bV_j=(\bV_{j1},\ldots, \bV_{jp})'$ are independent and identically distributed centered Gaussian random vectors with covariance matrices $\bSigma_1=(\sigma_{1,k \ell})_{1\leq k,\ell \leq p}$ and $\bSigma_2=(\sigma_{2,k \ell})_{1\leq k,\ell \leq p}$, respectively.
\begin{enumerate}
\item[(a)] Model 1$^{\rm{(II)}}$ (Block diagonal covariance matrices): For $k=1,\ldots, p$ and $q=1,2$, $\sigma_{q,kk}$ are independent and identically drawn from ${\rm Unif}(1,2)$, $\sigma_{q,k\ell}=0.7$ for $10(t-1)+1\leq k\neq \ell\leq 10t$, where $t=1,\ldots, \lfloor p/10\rfloor$, and $\sigma_{q,k\ell}=0$ otherwise. This model was studied in \cite{CaiLiuXia_2014}.


\item[(b)] Model 2$^{\rm{(II)}}$ (Non-sparse covariance matrices): Let $\bF=(f_{k \ell})_{1\leq k,\ell\leq p}$ with $f_{k k}=1,f_{k,k+1}=f_{k+1,k}=0.5$; $\bU_q \sim \mathcal{U}(\mathcal{V}_{p,k_0})$, the uniform distribution on the Stiefel manifold for $q=1,2$; and $\bTheta=(\theta_{k \ell})_{1\leq k, \ell\leq p}$ be a diagonal matrix with $\theta_{kk}$ independent and identically  drawn from ${\rm Unif}(1,6)$. Set $k_0=10$ and put $\bSigma_q=\bTheta^{1/2}(\bF+\bU_q \bU'_q)\bTheta^{1/2}$ for $q=1,2$. A similar model was studied in \cite{CaiLiuXia_2014}.

\item[(c)] Model 3$^{\rm{(II)}}$ (Long range dependence): Let $\theta_{11},\ldots, \theta_{1p},\theta_{21},\ldots,\theta_{2p}$ be independent and identically drawn from ${\rm Unif}(1,2)$; for $q=1,2$, we took $\sigma_{q,kk}=\theta_{qk}$ and $\sigma_{q,k \ell } =\rho_\alpha(|k -\ell |)$ for $k\neq \ell$, where
$\rho_\alpha(e)=\tfrac12 \{(e+1)^{2H}+(e-1)^{2H}-2e^{2H}\}$ with $H=0.9$.
\end{enumerate}
Model 1$^{\rm{(II)}}$ imposes sparse covariance structures while Model 2$^{\rm{(II)}}$ and Model 3 $^{\rm{(II)}}$ account for non-sparse dependence structures. In addition, we considered the following two models with non-Gaussian data generation mechanisms to study the robustness of the proposed tests against Gaussianity.

\begin{enumerate}
\item[(d)] Model 4$^{\rm{(II)}}$ (Autoregressive process of order one, AR(1), with $t$-distributed innovations): Generate independent and identically distributed $p$-variate random vectors $\bX_i\sim t_{\omega_1}(\bmu_1,\bSigma_1)$, $\bY_j\sim t_{\omega_2}(\bmu_2,\bSigma_2)$ for $i=1,\ldots, n$ and $j=1,\ldots, m$, where $t_{\omega_1}(\bmu_1,\bSigma_1)$ and $t_{\omega_2}(\bmu_2,\bSigma_2)$ are the non-central multivariate $t$-distributions with non-central parameters $\bmu_1,\bmu_2$, degrees of freedoms $\omega_1=5, \omega_2=7$, and $\bSigma_1=(\sigma_{1,k\ell})_{1\leq k,\ell\leq p}$ with $\sigma_{1,k \ell}=0.995^{|k-\ell|}$, $\bSigma_2=(\sigma_{2,k\ell})_{1\leq k,\ell\leq p}$ with $\sigma_{2,k \ell}=0.7^{|k-\ell|}$.

\item[(e)] Model 5$^{\rm{(II)}}$ (Moving average process with Gamma distributed innovations): For $i=1,\ldots, n$, $j=1,\ldots, m$, and $k=1,\ldots, p$, set $X_{i k}  =\rho_{1,1}Z_{i,k}+\rho_{1,2}Z_{i,k+1}+\cdots+\rho_{1,p}Z_{i,k+p-1}+\mu_{1k}$ and $ Y_{j k}   =\rho_{2,1}\wt Z_{j,k}+\rho_{2,2}\wt Z_{j,k+1}+\cdots+\rho_{2,p} \wt Z_{j,k+p-1}+\mu_{2k}$, where $\rho_{1,\ell}$ are independent and identically drawn from $0.6{\rm Unif}(-1,1)+0.4\delta_0$ and $\rho_{2,\ell}$ are independent and identically drawn from $0.8{\rm Unif}(-1,1)+0.2\delta_0$ for $\ell=1,\ldots, p$, where $\delta_0$ is the point mass at $0$, $\{Z_{i,k}\}$ and $\{ \wt Z_{j,k}\}$ are independent random variables with a common centered $\textrm{Gamma}(1,4)$ and $\textrm{Gamma}(4,1)$ distributions, respectively.
\end{enumerate}
Model 4$^{\rm{(I)}}$ and Model 5$^{\rm{(I)}}$ impose non-sparse covariance structures. Simulation results show that the proposed tests are valid in these non-sparse models with non-Gaussian sampling distributions.

In each model, we generated data with sample sizes $n=m=40$ and $n=m=80$. The dimension $p$ took values $ 120,360$ and $1080$. The empirical power and level of significance were computed based on $1500$ simulations.




\begin{sidewaystable}
    \centering
    {\renewcommand{\arraystretch}{1.1}
   \begin{tabular}{cccccccccccccccc}\toprule

  &  \multicolumn{3}{c}{Model 1$^{(\rm{II})}$}      &      \multicolumn{3}{c}{Model 2$^{(\rm{II})}$}    &    \multicolumn{3}{c}{Model 3$^{(\rm{II})}$}     &    \multicolumn{3}{c}{Model 4$^{(\rm{II})}$} &    \multicolumn{3}{c}{Model 5$^{(\rm{II})}$}                   \\ \toprule

$\text{tests}~$/$~p$ &  120&360&1080&120&360&1080&120&360&1080&120&360&1080&120&360&1080\\\midrule
     & \multicolumn{15}{c}{$(n,m)=(40,40)$}     \\ \midrule

 $\Psi_{\ns,\alpha}$ & 0.039&0.041&0.041&0.042&0.044&0.039&0.051&0.036&0.030& 0.052&0.036&0.042& 0.034&0.030&0.032\\ [0.5ex]
  $\Psi_{\rs,\alpha}$  &  0.094&0.112&0.125 &0.092&0.097&0.116&0.096&0.095&0.130&0.086&0.090&0.092&0.094&0.141&0.158  \\ [0.5ex]
   $\Psi^f_{\ns,\alpha}$  & 0.055&0.048&0.057& 0.049&0.055&0.054& 0.063&0.045&0.044 &  0.055&0.039&0.052&0.044&0.038&0.045  \\ [0.5ex]
    $\Psi^f_{\rs,\alpha}$  & 0.092&0.120&0.152& 0.098&0.131&0.053&0.092&0.093&0.160 & 0.090&0.094&0.094 &0.092&0.174&0.180\\ [0.5ex]
HC2  & 0.140&0.202&0.343& 0.127&0.156&0.308& 0.138& 0.221&0.315 & 0.280&0.344& 0.391&0.155&0.210&0.447 \\ [0.5ex]
CQ & 0.044&0.049&0.034& 0.046&0.049&0.051&0.046&0.059&0.051& 0.064&0.066&0.054 &0.046&0.047&0.051 \\[0.5ex]
CLX & 0.101&0.103&0.138&0.081&0.087&0.098&0.085&0.093&0.116 &  0.204&0.181&0.137& 0.081&0.127&0.157\\
 \midrule

& \multicolumn{15}{c}{$(n,m)=(80,80)$}     \\ \midrule

 $\Psi_{\ns,\alpha}$ & 0.054&0.039&0.046& 0.053&0.040&0.040& 0.042&0.038&0.049& 0.046&0.045&0.047&0.041&0.036&0.035\\ [0.5ex]
  $\Psi_{\rs,\alpha}$  & 0.074&0.062&0.086& 0.058&0.064&0.090&0.064&0.071&0.090 & 0.059&0.065&0.074& 0.070&0.090&0.097\\ [0.5ex]
   $\Psi^f_{\ns,\alpha}$  & 0.065&0.052&0.060& 0.063&0.050&0.058& 0.052&0.044&0.063 &  0.047&0.048&0.056 &0.046&0.056&0.045 \\ [0.5ex]
    $\Psi^f_{\rs,\alpha}$  & 0.088&0.076&0.098&0.070&0.080&0.093&0.073&0.084&0.093   & 0.062&0.069&0.086&0.086&0.092&0.099 \\ [0.5ex]
HC2  & 0.112&0.132&0.170&0.079&0.076&0.133&0.092&0.147&0.186 & 0.237&0.292&0.377& 0.099&0.150&0.207\\ [0.5ex]
CQ & 0.046&0.039&0.048&0.048&0.038&0.048&0.038&0.044&0.049 &   0.044&0.054&0.056& 0.051&0.053&0.053\\[0.5ex]
CLX & 0.107&0.090&0.104&0.057&0.057&0.089&0.058&0.060&0.083 & 0.289&0.352& 0.297&0.059&0.087&0.098
  \\ \bottomrule
  \end{tabular}
  }
  \caption{Empirical sizes of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) for the two-sample problem \eqref{eq:twosample}, along with those of the tests by  \cite{DelaigleHallJin_2011} (HC2), \cite{ChenQin_2010} (CQ), and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance. Models with Gaussian data and block diagonal, non-sparse or long range dependence covariance matrices, the autoregressive process model with $t$-distributed innovations, and the moving average processes model with Gamma distributed innovations are considered when $n=m=40, 80$ and $p=120,360,1080$.  }\label{size2}

\end{sidewaystable}






\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S21p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S21p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S21p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S21p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ) and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance for the Gaussian data and block diagonal covariance matrices in Model 1$^{(\rm{II})}$ when $n=m=80$ and $p=1080$.}
   \label{f11}
\end{figure}

\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S22p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S22p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S22p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S22p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ) and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance for the Gaussian data and non-sparse covariance matrices in Model 2$^{(\rm{II})}$ when $n=m=80$ and $p=1080$.}
   \label{f12}
\end{figure}

\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S23p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S23p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S23p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S23p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ) and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance for the Gaussian data and long range dependence covariance matrices in Model 3$^{(\rm{II})}$ when $n=m=80$ and $p=1080$.}
   \label{f13}
\end{figure}



The numerical results on the proposed tests $\Psi_{\ns,\alpha}$, $\Psi_{\rs,\alpha}$, $\Psi_{\ns,\alpha}^f$ and $\Psi_{\rs,\alpha}^f$ and the HC2, CQ and CLX tests are summarized in Table \ref{size2} and Figures \ref{f11}-\ref{f15}. Table \ref{size2} displays the empirical sizes. It can be seen that in all the models, the empirical size for the proposed non-studentized tests $\Psi_{\ns,\alpha}$ and $\Psi_{\ns,\alpha}^f$ are reasonably close to the nominal level $0.05$ for both $n=m=40$ and $n=m=80$. Comparing the results for $n=m=40$ and $n=m=80$, the studentized tests, $\Psi_{\rs,\alpha}$ and $\Psi_{\rs,\alpha}^f$, have slightly inflated significance when the sample size is relatively small but improve when the sample size increases. Additionally, the CLX test fails to maintain the nominal size for Model 4$^{{\rm (II)}}$ due to the strong dependency in the covariance structures. Analogous to the observation in Section \ref{subsection:numerical one sample}, it is difficult for the HC2 procedure to maintain the nominal significance when the sample size is small or the dependency is strong and complex. The CQ test maintains the nominal significance reasonably well in all the models.


\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S24p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S24p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S24p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S24p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the test by \cite{ChenQin_2010} (CQ) at 5\% nominal significance for the autoregressive process model, Model 4$^{\rm{(II)}}$, with $t$-distributed innovations when $n=m=80$ and $p=1080$.}
   \label{f14}
\end{figure}

\begin{figure}[h!]
    \centering
    \includegraphics[height=7cm,width=7cm]{figures/S25p1080f1.png}
    \includegraphics[height=7cm,width=7cm]{figures/S25p1080f2.png}
 \includegraphics[height=7cm,width=7cm]{figures/S25p1080f3.png}
 \includegraphics[height=7cm,width=7cm]{figures/S25p1080f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ) and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance for the moving average process with Gamma distributed innovations in Model 5$^{\rm{(II)}}$ when $n=m=80$ and $p=1080$.}
   \label{f15}
\end{figure}



To evaluate the power, we compared the proposed tests with the CQ and CLX tests for $n=m=80$ and $p=1080$. It can be seen that the tests with screening, $\Psi_{\ns,\alpha}^f$ and $\Psi_{\rs,\alpha}^f$, outperform both the CQ and CLX tests against alternatives with sparse signals $(r=0)$ for different signal strength $\beta$. On the other hand, all the tests perform similarly when the signals become less sparse and strong (see the right bottom panels in Figures \ref{f11} to \ref{f13} and Figure \ref{f15}). The CQ test gains more powers when signals become less sparse, as expected for sum of squares-type statistics. Its power approaches to those of the proposed tests with screening $\Psi_{\ns,\alpha}^f$ and $\Psi_{\rs,\alpha}^f$ when the signals become less sparse and stronger ($r\geq 0.5, \beta\geq 0.4$) in all the models except Model 4$^{\rm{(II)}}$. In Model 4$^{\rm{(II)}}$, all the proposed tests outperform the CQ test substantially (Figure \ref{f14}) as the sum of squares-type test statistics may lose power for heavy tailed sampling distributions. The CLX test performs similarly to the proposed tests without screening $\Psi_{\ns,\alpha}$ and $\Psi_{\rs,\alpha}$, but is outperformed by the proposed tests with screening for all settings. The simulation results agree with the heuristic discussion and the theoretical justification that the screening step substantially improves the power of proposed tests. Similar to the observations in Section \ref{section:onesample}, the non-studentized test with screening $\Psi_{\ns,\alpha}^f$ is preferable in practice whenever the sample size is relatively small. More extensive simulations were carried out for dimensions $p=120$ and $360$, from which the comparisons are consistent with the cases that are reported here. The empirical powers of all the tests also increase in $p$. The additional simulation results are placed in the online supplementary material.


To sum up, the numerical results show that the proposed tests, particularly the studentized tests  and the non-studentized test with screening, $\Psi_{\rs,\alpha}$, $\Psi_{\rs,\alpha}^f$ and  $\Psi_{\ns,\alpha}^f$, outperform the existing methods when the covariance structure is non-sparse and complex. The proposed tests are robust against both unknown covariance structures and Gaussianity. The non-studentized test with screening $\Psi_{\ns,\alpha}^f$ maintains the nominal significance for small sample sizes and has good powers against sparse alternatives, which is recommended for practical applications whenever the sample sizes are limited.

%===============================================================================================%
%===============================================================================================%

\setcounter{equation}{0}
\section{Empirical study} \label{real data}

To achieve certain biological functions, genes tend to work in groups which are known as gene-sets. Analysis and interpretation based on gene-sets derive more power than focusing on individual gene in extracting biological insights \citep{S05}. It has drawn increasing attentions to identify gene-sets associated with biological states of interest, such as gene-sets associated with disease developments or evolutionary mutations, in biomedical research and general genome analysis \citep{S05,ET_2007,RNR_2008,T11}. The gene-sets are technically defined in gene ontology (GO) system which provides structured vocabularies producing the names of the sets or known as GO terms. Three categories of gene ontologies of interest have been identified: biological processes (BP), cellular components (CC) and molecular functions (MF). That is, a particular gene-set will belong to one of these three groups of GO terms.

Statistically, identifying interesting gene-sets out of $G$ candidate gene-sets $\mathcal{S}_1,\ldots, \mathcal{S}_G$ based on independent samples from two biological states is equivalent to test hypotheses
$$H_{0s}: \bmu_{1,s}=\bmu_{2,s}\ \ \textit{versus}\ \ H_{1s}: \bmu_{1,s}\neq \bmu_{2,s}$$ for $s=1,\ldots,G$, where the $p_s$ dimensional vector $\bmu_{i,s}$ models the mean expression levels of $p_s$ genes in the gene-set $\mathcal{S}_s$ under the biological condition $i$ for $i=1,2$, which may correspond to control and mutation groups or health and diseased groups in practice. It is common that gene-sets overlap with each other as one particular gene may belong to several functional groups, and the size of a gene-set $p_s$ usually range from a small to a very large number. The selection of gene-sets therefore encounters both multiplicity and high dimensionality. As suggested by \cite{ChenQin_2010}, we applied the proposed tests to each gene-set. With $p$-values obtained for all $G$ gene-sets, similar to \cite{ChenQin_2010}, we further employed multiple testing procedures such as the Bonferroni method for controlling the family-wise error rate or procedures such as the Benjamini-Hochberg procedure \citep{BH95} for controlling the false discovery rate (FDR) to identify significant gene-sets. In practice, we compute the empirical $p$-values of the proposed tests by $\widehat{{\rm pv}} = M^{-1}\sum_{\ell=1}^M I\big\{|\bW_{\ns,\ell}|_\infty>T_{\ns}^{\rII}\big\}$ or $\widehat{{\rm pv}}=M^{-1}\sum_{\ell=1}^M I\big\{ |\bW_{\rs ,\ell}|_\infty>T_{\rs}^{\rII}\big\}$. Here $\{\bW_{\ns,1},\ldots,\bW_{\ns,M}\}$ and $\{\bW_{\rs,1},\ldots,\bW_{\rs,M}\}$ are two random samples independently generated from $\textrm{N}(\bzero,\widehat{\bSigma}_{1,2})$ and $\textrm{N}(\bzero,\widehat{\bR}_{1,2})$, where $\widehat{\bSigma}_{1,2}$ and $\widehat{\bR}_{1,2}$ are defined via replacing $\widetilde{\bSigma}_1$ and $\widetilde{\bSigma}_2$ in (\ref{Sigma12}) by sample covariance matrices $\widehat{\bSigma}_1$ and $\widehat{\bSigma}_2$, respectively. The empirical $p$-values for the proposed two-step procedure are computed similarly.

We applied the above procedure to a human acute lymphoblastic leukemia (ALL) dataset which is available at \url{http://www.ncbi.nlm.nih.gov}. The data contains gene expression levels from microarray experiments for patients suffering from ALL of either T-lymphocyte type or B-lymphocyte type leukemia. This dataset was originally analyzed by \cite{C04} to provide insight into the genetic mechanism on ALL development. The data was also analyzed by \cite{DKV08} and \cite{ChenQin_2010}, among others using different methodologies. To illustrate the proposed tests, we focus on the 75 patients of B-lymphocyte type leukemia, who were classified into two groups: 35 patients with BCR/ABL fusion and 40 patients with cytogenetically normal NEG, i.e. $n=35$ and $m=40$. We employed the approach in \cite{G2005} to conduct preliminary data processing. To focus on high dimensional scenarios, we also excluded gene-sets with small numbers of genes that only gene-sets with $p_s\geq 19$ were retained. It remained $G=1853,262$ and $284$ unique GO terms in the BP, CC and MF categories, respectively. The largest gene-set contained $p_s=3050$ genes in the BP category, $p_s=3145$ genes in the CC category, and $p_s=3040$ genes in the MF category. Identifications of gene-sets associated to the BCR/ABL fusion display biological insights on the development of B-lymphocyte type leukemia and provide lists of functional groups for potential clinical treatments. Our aim is therefore to identify gene-sets with significantly different expression levels between the BCR/ABL and NEG groups for each of the three categories.

%\begin{figure}[h!]
%  \centering
%  \includegraphics[height=9cm,width=12cm]{image/real/BP.png}
%  \caption{Histograms of $p$-values for BP gene category using the proposed two-sample tests without and with pre-screening , $T_\nu$ and $T_\nu$-$f$ for $\nu=1,2$.}
%  \label{f2a}
%\end{figure}

The sample size of the ALL data is relatively small comparing to the maximum $p_s$, we therefore employed the proposed two-sample non-studentized tests $\Psi_{\ns,\alpha}$ and $\Psi_{\ns,\alpha}^f$ in analysis as suggested by simulation studies in Section \ref{section:numerical}. Based on empirical $p$-values, we further employed the Benjamini-Hochberg (BH) procedure for controlling the FDR at $0.015$ and identify significant gene-sets. For the proposed tests, we let $M=50000$ and used the sample covariance matrices to generate samples. Simulation studies in Section \ref{section:numerical} have shown that the test by \cite{CaiLiuXia_2014} may inflate type I error rate for small sample size, we therefore only consider the test by \cite{ChenQin_2010} (CQ) as a comparison. For each category, the numbers of gene-sets being identified are summarized in Table \ref{t2}. The proposed two-stage test, $\Psi_{\ns,\alpha}^f$, identified more gene-sets than other
methods. Particularly, $\Psi_{\ns,\alpha}^f$ found more disease associated gene-sets than $\Psi_{\ns,\alpha}$ which reflects the power improvement of the proposed two-stage testing procedure as discussed before. The overlap among gene-sets increases in the number of genes within each gene-set $p_s$ \citep{ChenQin_2010}, which explains the relatively large number of identified disease-associated gene-sets in Table \ref{t2}.


\begin{table}[h]
\centering
\noindent{
\begin{tabular}{c|cc|ccc|ccc}\toprule
GO    & \multirow{2}{*}{Total} &
\multirow{2}{*}{$\Psi_{\ns,\alpha}$} &  \multicolumn{3}{c|}{$\Psi_{\ns,\alpha}^f$ and CQ}   &  \multirow{2}{*}{$\max_{s} p_s$} & \multirow{2}{*}{$\min_s p_s$} &
\multirow{2}{*}{$\lfloor \bar{p}_s\rfloor$} \\[.75ex]
  Category    &  &   & $\Psi_{\ns,\alpha}^f$ only & Both & CQ only  \\
 \midrule

BP & 1853 & 1082 & 302 & 1065 & 136&   3050 & 20 & 150\\[0.5ex]
CC &  262 & 93 & 33 & 117 & 28& 3145 &  19 & 280\\[0.5ex]
MF &  284 & 136 & 63 & 133 & 14 & 3040 & 19 & 157\\
\bottomrule
\end{tabular}}
\caption{Numbers of identified BCR/ABL assoicated gene-sets for each GO category using different tests in conjunction with the BH procedure for controlling FDR at $0.015$. Columns labeled by the name of tests records the number of identified gene-sets by the corresponding testing procedures, where $\Psi_{\ns,\alpha}$ and $\Psi_{\ns,\alpha}^f$ are the proposed non-studentized tests without and with screening, and CQ stands for the test by \cite{ChenQin_2010}.}\label{t2}
\end{table}

By carefully investigating the gene-sets identified by both the proposed tests $\Psi_{\ns,\alpha}$ and $\Psi_{\ns,\alpha}^f$, we found that gene-sets GO:0005758 (mitochondrial intermembrane space) and GO:0004860 (protein kinase inhibitor activity) were identified as diseases-associated in the CC and MF categories. The functions of these two interesting gene-sets were recently studied and recognized associated with the development of ALL \citep{BK14,CW09}. Particularly, the protein kinase inhibition has been considered to be essential for the mechanism of T-lymphocyte type ALL \citep{CW09} and our finding suggests its connection with B-lymphocyte type ALL as well. In the supplementary materials, we list the top 15 gene-sets that were identified to be diseases-associated by the proposed non-studentized test with screening $\Psi_{\ns,\alpha}^f$ but missed by the CQ test with the FDR controlled at 0.015. The association of these gene-sets with the ALL may deserve further biological validations using the polymerase chain reaction.




%===============================================================================================%
%===============================================================================================%

\section{Discussion}  \label{sec.disc}

In this paper, we study hypothesis testing of high dimensional mean vectors in both the one-sample and two-sample settings. Unlike the existing tests, the proposed procedures enforce no structural assumptions on the unknown covariance matrices, and are powerful against sparse alternatives.

The test statistics are taken as the extreme values of partial sums or self-normalized partial sums, whose distributions can be approximated by the extreme values of centred Gaussian vectors with the same covariance structure as the $p$-vector of all marginal statistics. The uniform convergence of the approximated distributions to the target distributions of the test statistics is guaranteed whenever $\|\widehat{\bSigma}-\bSigma\|_\infty+\|\widehat{\bR}-\bR\|_\infty=o_{P}(1)$; that is, the estimates of the covariance and correlation matrices are max-norm consistent, which can be easily met using the sample covariance matrices under mild moment conditions. Therefore, the proposed tests allow for very general form of the unknown covariance matrices even for non-invertible covariances that model perfect correlated variables. This important feature of the proposed tests is supported by simulation studies, whereas most existing methods may encounter difficulties in maintaining the nominal sizes for complex and/or strong dependency. Additional numerical experiments regarding perfectly correlated variables are also included in the supplementary material.

To further improve the power of proposed tests, we developed a preliminary feature screening step to construct the two-step testing procedures $\Psi_{\ns,\alpha}^f$ and $\Psi_{\rs,\alpha}^f$. In principle, the pre-screening is based on marginal signal-to-noise ratios and improves the numerical performance of the proposed tests against sparse alternatives. Theoretical justifications of the two-step tests are provided by Theorems \ref{two-step.one-sample.theory} and \ref{two-step.two-sample.theory}. Numerical studies demonstrate their superior performance over the others. For example, when the sample size is relatively small, the non-studentized test with screening $\Psi_{\ns,\alpha}^f$ is more robust than others yet maintains satisfactory powers. Particularly, the data analysis for gene-set selections shows that the two-step testing procedures are capable of identifying diseases-associated gene-sets that were missed by the other peer tests. It is appealing to notice the connection between the preliminary feature screening and augment classifications in high dimensional settings \citep{FanFan08}. For a binary classification problem, which is essentially a two-sample problem, the preliminary feature screening step might be employed, in conjunction with a broader class of statistics such as the Kolmogorov-Smirnov test, to select features with discriminant powers and therefore improve the prediction performance.


\section*{Supplementary Material}

Supplementary material available online includes technical proofs, derivation of the main theorems, more extensive simulation results, and details regarding real data analysis.

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The Supplementary Materials including technical proofs and more numerical studies are organized as follows. Section A introduces notations and important auxiliary lemmas. In Section B, we report the derivations of theoretical justifications for the proposed tests for one-sample problems. The derivations of theoretical justification for the proposed two-sample testing procedures are given in Section C. The theoretical justification of the proposed two-step procedures, the proposed tests with screening, are displayed in Section D. In Section E, more simulations studies are reported. Section F presents more results for the analysis of the acute lymphoblastic leukemia data recorded by \cite{C04}.

\section{Preliminaries}


Throughout, we use $C$ and $c$ to denote positive constants that are independent of $(n,m,p)$, which may take different values at each occurrence.


The proposed simulation-based procedures for testing the equality of means are founded on the idea of Gaussian approximation. In this section, we state some useful results on high dimensional Gaussian approximations recently established by \cite{ChernozhukovChetverikovKato_2013} with some discussions. In line with Section~\ref{method.sec}, let $\mX, \mX_1, \ldots, \mX_{n} \in \br^p$ be independent and identically distributed random vectors with mean $\bmu_1$ and covariance matrix $\bSigma_1=(\sigma_{1, k\ell})$. Write $\bX=(X_1, \ldots, X_p)'$, $\bX_i=(X_{i 1}, \ldots, X_{i p})'$ and define
$$
    T_0 = \max_{1\leq k \leq p} \s{n} \bar{X}_k \qquad  \mbox{ with } \quad  \bar{X}_k= n^{-1}\sum_{i=1}^{n} X_{i k}.
$$
Moreover, let $\{ \bG_i =(G_{i 1}, \ldots , G_{i p})'\}_{i=1}^{n}$ be a sequence of independent centred Gaussian random vectors in $\br^p$ with covariance matrix $\bSigma_1$. The Gaussian analogue of $T_0$ can be defined as
$$
    Z_0 = \max_{1\leq k\leq p}  n^{-1/2} \sum_{i=1}^{n} G_{i k}.
$$

For $k=1,\ldots, p$, let $\sigma^2_{1k}=\var(X_k)=\sigma_{1,kk}$ and for $r\geq 1$, define the moments
\be
 \mu_{r}(\bX)  = \big( \mathbb{E} | \bU |_{\infty}^r \big)^{1/r}, \ \ \nu_{r}(\bX) = \max_{1\leq k\leq p} \big( \mathbb{E} |  U_k |^r  \big)^{1/r}, \label{maximum.moment}
\ee
where $ \bU =\bU(\bX)= ( U_1,\ldots, U_p)' =\bD_1^{-1/2}\bX$ and $\bD_1=\textrm{diag}(\bSigma_1)$. These quantities will play an important role in our theoretical analysis. For every $0<t<1$, set $u(t) = \max \{  u_X(t), u_G(t)  \}$, where $u_X(t)$ and $u_G(t)$ are the $(1-t)$-quantiles of $\max_{1\leq i\leq n}| \bU_i|_{\infty}$ for $\bU_i=\bD_1^{-1/2}\bX_i$ and $\max_{1\leq i\leq n} | \bD_1^{-1/2} \bG_i|_\infty $, respectively. Further, for $r\geq 4$ as in condition (M1), we write
\be
    \vartheta_{n,p}  = \vartheta_{n,p}(r) = p  n^{1-r/2}.  \label{alpha.np}
\ee



Lemmas~\ref{CCK_2013} and \ref{CCK_2013_31} below are Theorem 2.2 and Lemma~3.1 in \cite{ChernozhukovChetverikovKato_2013}.


\begin{lemma}   \label{CCK_2013}
{\rm Assume that $\sigma_{1,kk}$ is bounded away from $0$ and $\infty$. Then for any $0<t<1$, \beqn
    \lefteqn{ \sup_{ x \in \br} \left|  \mathbb{P}(T_0\leq x) - \mathbb{P}(Z_0 \leq x) \right|  } \nn \\
    && \leq C  \Big[  \big( \nu_{3}^{3/4} \vee \nu_{4}^{1/2} \big) \, n^{-1/8}  \{ \log(p n/t)\}^{7/8} +
    u(t) \, n^{-1/2} \{ \log(p n/t) \}^{3/2} +  t  \, \Big],  \label{HD.CLT}
\eeqn where $C>0$ is a constant independent of $n$, $p$ and $t$. }
\end{lemma}

\begin{lemma}  \label{CCK_2013_31}
{\rm Assume that there exist constants $C>c>0$ such that for every $1\leq k\leq p$, $c\leq \sigma_{1,kk},\sigma_{2,kk}\leq C$. Then
$$\sup\limits_{x \in \mathbb{R}}\left|\mathbb{P}\left(\max\limits_{1\leq k\leq p}X_k\leq x\right)-
\mathbb{P}\left(\max\limits_{1\leq k\leq p}Y_k\leq x\right)\right| \leq C' \Delta^{1/3}\left\{ 1\vee \log\left(p/\Delta\right)\right\}^{2/3},$$ where $C'>0$ is a constant depending only on $c$ and $C$ and $\Delta:=\|\bSigma_1-\bSigma_2\|_\infty$. }
\end{lemma}

\noindent{\bf Remark:} The upper bound given in \eqref{HD.CLT} can be simplified under conditions (M1) and (M2), respectively. By Markov's inequality, we have for all $u>0$,
\beq
 \mathbb{P}\left( \max_{1\leq i\leq n }  | \bD^{-1/2} \bG_i | > u  \right)
   \leq 2 p n \{1-\Phi(u)\}  \leq    u^{-1}\exp\{\log (p n)- u^2/2\}.
\eeq
This implies that for every $0<t<1$, $u_G(t) \leq \s{  2 \log ( p n/t)  }$. Under condition (M1), once again using Markov's inequality yields, for every $u>0$,
\beq
 \mathbb{P} \left( \max_{1\leq i\leq n } |\bU_{i} |_{\infty} > u  \right) \leq u^{-r} \, \mathbb{E} \left( \max_{1\leq i\leq n} |\bU_{i} |_{\infty}^r  \right)  = u^{-r} \mu_{n,r}^r,
\eeq
where $\mu_{n,r} :=\{ \mathbb{E} (  \max_{1\leq i\leq n} | \bU_i |_{\infty}^r  ) \}^{1/r}$. For $0<t<1$, it follows directly from the definition that $u_X(t) \leq t^{-1/r}   \mu_{n,r}$. In particular, taking
$$
t=\min\bigg\{ 1, \,     \left[  \mu_{n,r} n^{-1/2}\{\log(pn)\}^{3/2}  \right]^{r/(r+1)}  \bigg\}
$$
in \eqref{HD.CLT} implies by the inequality $\mu_{n,r}= \{ E (\max_{1\leq i\leq n} |\bU_{i}|_\infty^r ) \}^{1/r}\leq (pn)^{1/r} \nu_r$ that the upper bound on the right side of \eqref{HD.CLT} is of order
\be
     n^{-1/8} \{ \log(p n) \}^{7/8} +\vartheta_{n,p}^{1/(r+1)} \{\log(pn)\}^{3/2}, \label{poly.rate}
\ee where $\vartheta_{n,p}=pn^{1-r/2}$ is as in \eqref{alpha.np}.

On the other hand, if condition (M2) holds, then it is easy to see that $u_X(t)$ can be bounded by some multiple of $\{ \log ( pn/t)\}^{1/\gamma}$. Taking $t=n^{-1/2}$ gives the optimal rate of convergence in \eqref{HD.CLT} which is of order \be
     n^{-1/8}  \{ \log(p n)\}^{7/8} + n^{-1/2} \{\log(p n)\}^{3/2+1/ \gamma }.    \label{exp.rate}
\ee
In the proofs below, \eqref{poly.rate} and \eqref{exp.rate} will be applied directly.

\medskip

Recall that $\wh{\bSigma}_1=(\hat{\sigma}_{1, k\ell})$ with $\hat{\sigma}_{1, k\ell}  =  n^{-1} \sum_{i=1}^{n} (X_{i k}-\bar{X}_k) (X_{i \ell}-\bar{X}_{\ell})$ and $\hat{\sigma}_{1 k}^2= \hat{\sigma}_{1,kk }$. Furthermore, denote by
$\bR_1 =(r_{1, k\ell})$ the correlation matrix of $\bX$ and its sample analogue is given by $\wh{\bR}_1 =(\hat{r}_{1, k\ell})$ with
$$
    \hat{r}_{1, k\ell} = \f{\sum_{i=1}^{n} (X_{i k}- \bar{X}_k)(X_{i \ell} - \bar{X}_{\ell})}{ \sqrt{ \sum_{i=1}^{n}  (X_{i k}-\bar{X}_k)^2  \sum_{i=1}^{n} (X_{i \ell}-\bar{X}_{\ell})^2  }} .
$$
The following lemma provides non-asymptotic bounds on the differences $\wh{\bSigma}_1 - \bSigma_1$ and $\wh{\bR}_1 - \bR_1$ in the elementwise $\ell_{\infty}$-norm.



%Throughout this section, we use $c, c_1, c_2, \ldots$ and $C, C_1,
%C_2, \ldots$ to denote absolute constants that are independent of
%$n$, $m$ and $p$. Their values, however, may change from line to
%line. Without loss of generality, assume throughout the following
%that $2 \leq p =p_n \leq \exp(n)$. For two sets $A$ and $B$, we use
%$A \bigtriangleup B$ to denote their symmetric difference; that is,
%$A \bigtriangleup B= (A \setminus B) \cup (B \setminus A)$.


\begin{lemma}[Probabilistic estimates] \label{lm2}
{\rm  Suppose that $n, p\geq 2$ and $\log(p) \leq n$.
\begin{enumerate}
\item[(i)] Assume that condition (M1) holds. Then there exist constants $C_1, C_2>0$ independent of $n$ and $p$ such that, with probability at least $1- C_1 \{ n^{-1} + \vartheta_{n,p}^{2/(r+2)}\}$,
\begin{align}
        &   \max\left(  \big\| \bD_1^{-1/2} \widehat{\bSigma}_1  \bD_1^{-1/2} - \bR_1   \big\|_\infty,  \big\| \widehat{\bR}_1 - \bR_1 \big\|_\infty  \right)  \nn \\
 & \qquad \qquad \qquad       \leq C_2  \left[ \nu_4^2 \,  n^{-1/2} \{ \log(p n) \}^{1/2} +  \nu_r^2  \, \vartheta_{n,p}^{2/(r+2)} + \nu_r^2 \, \vartheta_{n,p}^{2/r} \log(p)  \right].  \label{max.bound.1}
\end{align}


\item[(ii)] Assume that condition (M2) holds. Then there exist constants $C_3, C_4>0$ independent of $n$ and $p$ such that, with probability at least $1- C_3  n^{-1}$,
\begin{align}
  &  \max\left( \big\| \bD_1^{-1/2} \widehat{\bSigma}_1  \bD_1^{-1/2} - \bR_1   \big\|_\infty,  \big\| \widehat{\bR}_1 - \bR_1 \big\|_\infty  \right) \nn \\
    &  \qquad \qquad \qquad \qquad \qquad\qquad   \leq C_4  \left[
   n^{-1/2} \{\log (pn) \}^{1/2}  + n^{-1} \{ \log(p n)\}^{2/\gamma}   \right]. \label{max.bound.2}
\end{align}

\item[(iii)] Assume that $\nu_{4}$ in \eqref{maximum.moment} is uniformly bounded. Then for
$0<t\leq \s{n}$,
\be
    \mathbb{P}\left\{ \sqrt{n} \big| \widehat{\bD}_1^{-1/2} ( \widehat{\bmu}_1 - \bmu_1 ) \big|_\infty  \geq  t  \right\} \leq C   p \exp(-c t^2 ) + n^{-1}, \label{max.bound.3}
\ee
where $\widehat{\bD}_1=\mbox{diag}\,(\hat{\sigma}_{11}^2, \ldots, \hat{\sigma}^2_{1p})$, $\widehat{\bmu}_1=(\bar{X}_1, \ldots, \bar{X}_p)'$ and $C, c>0$ are constants independent of $n$ and $p$.


\end{enumerate}

}
\end{lemma}

\noindent {\bf Proof}: Recall that for $i=1,\ldots, n$, $\bU_i=(U_{i1},\ldots, U_{ip})'$ with $U_{ik}=(X_{i k}-\mu_{1 k})/ \sigma_{1k}$ and define $S_{kn}=\sum_{i=1}^{n} U_{ik}$ and $V_{kn}=( \sum_{i=1}^{n} U_{ik}^2)^{1/2}$. First, observe that
\be
    \f{\hat{\sigma}_{1, k\ell }}{\sigma_{1k} \sigma_{1\ell}} = n^{-1}\sn ( U_{ik}- n^{-1}S_{kn}) (U_{i\ell}- n^{-1}S_{\ell n}) =   n^{-1}\sn  U_{ik} U_{i \ell}  - n^{-2} S_{kn} S_{\ell n}. \label{sample.cov.dec}
\ee
Moreover, put $t_{kn}=n^{-1/2}S_{kn}/V_{kn}$ for $k=1, \ldots, p$, such that the sample correlations can be expressed as
\beqn
     \hat{r}_{1,k\ell}
    % \f{\sum_{i=1}^{n} (X_{i k}- \bar{X}_k)(X_{i \ell} - \bar{X}_{\ell} )}{\{\sum_{i=1}^{n}  (X_{i k}-\bar{X}_k)^2  \sum_{i=1}^{n} (X_{i \ell}-\bar{X}_{\ell})^2\}^{1/2}}  } \nn \\
     = \f{\sum_{i=1}^{n} U_{i k} U_{i \ell}}{V_{kn} V_{ \ell n}} \big\{(1-t^2_{kn})(1-t^2_{ \ell n}) \big\} ^{-1/2} - t_{kn} t_{ \ell n} \big\{(1-t^2_{kn})(1-t^2_{ \ell n}) \big\} ^{-1/2}. ~~~~~ \label{corr.dec}
\eeqn
By \eqref{sample.cov.dec} and \eqref{corr.dec}, lying in the heart of the proof is a careful analysis of the following quantity:
\be
    \De_1  = \max_{1\leq k \leq  \ell \leq p} \left| n^{-1} \sum_{i=1}^{n} U_{i k} U_{i\ell} - r_{1, k\ell} \right|.   \label{De1.def}
\ee

\medskip
\noindent
\textbf{\textit{Case 1.}} Assume that condition (M1) holds. We follow a standard procedure: first show $\De_1 $ is concentrated around its expectation $\mathbb{E} \De_1 $, and then upper bound the expectation. Note that, $\bU_1, \ldots, \bU_n$ are independent and identical distributed random vectors in $\br^p$ with mean zero and covariance matrix $\bR_1$. Applying Theorem~3.1 in \cite{EinmahlLi_2008} with $s=\frac{1}{2}r $ and $\| \cdot \|= \| \cdot \|_\infty$ yields, for every $t>0$,
\begin{align}
\mathbb{P}\big( \De_1  \geq 2  \mathbb{E} \De_1  + t \big)  & \leq \exp\{-(nt)^2/(3 \nu_4^4  n)\} + C\,  t^{-r/2}  n^{1-r/2}  \max_{1\leq i\leq n} \mathbb{E} \bigg(  \max_{1\leq k \leq \ell \leq p} |U_{ik} U_{i\ell}|^{r/2} \bigg) \nn \\
& \leq \exp\{-(nt)^2/(3 \nu_4^4   n)\} + C\,  t^{-r/2}   n^{1-r/2}  \max_{1\leq i\leq n} \mathbb{E} \bigg(  \max_{1\leq k \leq p} |U_{ik}|^{r} \bigg) \nn \\
& \leq  \exp\{- n  t^2/(3 \nu_4^4 )\} + C\,     t^{-r/2}\nu_r^r  \,  p   n^{1-r/2}, \label{Fuk.Nagaev.ineq}
\end{align}
where by Lemma~A.1 in \cite{ChernozhukovChetverikovKato_2013},
\begin{align}
      \mathbb{E}  \De_1  & \leq C \bigg[  n^{-1/2}\{\log(p)\}^{1/2} \left\{ \max_{1\leq k\leq \ell \leq p} \sn \mathbb{E}  (U_{ik} U_{i\ell})^2  \right\}^{1/2}  \nn \\
   & \qquad  \qquad\qquad\qquad \qquad\qquad +   n^{-1} \log(p) \left\{ E \left( \max_{1\leq i\leq n}\max_{1\leq k\leq \ell \leq p}  U_{ik}^2 U_{i\ell}^2 \right) \right\}^{1/2} \bigg] \nn \\
    & \leq  C \left[    \nu_4^2 n^{-1/2}\{ \log(p) \}^{1/2} + \nu_r^2 \,  p^{2/r}   n^{-1+2/r} \log(p) \right]. \label{mean.ubd}
\end{align}
Together, \eqref{mean.ubd} and \eqref{Fuk.Nagaev.ineq} imply by taking $t=\max[ \nu_4^2 \, n^{-1/2} \{\log(n)\}^{1/2},   \nu_r^2 \,  \vartheta_{n,p}^{2/(r+2)} ]$ that
\be
    \mathbb{P} \left[ \De_1 > C \left\{ \nu_4^2  \,  n^{-1/2} \{ \log(pn)\}^{1/2} + \nu_r^2  \,  \vartheta_{n,p}^{2/(r+2)} + \nu_r^2  \,   \vartheta_{n,p}^{2/r} \log(p) \right\} \right] \leq C \left\{ n^{-1} + \vartheta_{n,p}^{2/(r+2)} \right\}. \label{De1.tail.prob}
\ee

Next we focus on the self-normalized sums $t_{kn}$ in the second term on the right side of \eqref{corr.dec}. Using Theorem 2.19 in \cite{PenaLaiShao_2009} to the non-negative random variables $U_{ik}^2$ gives, for every $0<\varepsilon<1$,
\be
    \mathbb{P} \left\{   V_{kn}^2  \leq  ( 1-\varepsilon)n \right\} \leq \exp\left\{ -  n \varepsilon^2 /( 2 \mathbb{E} U_{ik}^4 ) \right\} \leq \exp\left\{ - n \varepsilon^2 /( 2  \nu_{4}^4 ) \right\}.  \label{tail.1}
\ee
Moreover, it follows from Lemma~6.4 in \cite{JingShaoWang_2003} that for $t>0$,
\be
   \mathbb{P} \left\{  |S_{kn}| \geq t (4  \s{n}  + V_{kn}) \right\} \leq 4\exp(-t^2/2). \label{sn.tail}
\ee
Then it is concluded from \eqref{tail.1} and \eqref{sn.tail} that, with probability at least $1-4\exp(-t^2/2)-\exp( - c  n )$, $|t_{kn}|  \leq t(  n^{-1/2}+ 4V_{n,k}^{-1} ) \leq  7 t  n^{-1/2} $. Taking $t = \{ 2 \log(p n) \}^{1/2}$ we obtain that
$$
    \mathbb{P} \left\{    \max_{1\leq k\leq p}|t_{kn}| > C n^{-1/2}\{ \log(pn) \}^{1/2} \right\} \leq C  n^{-1}.
$$
The last display, joint with \eqref{corr.dec} and \eqref{De1.tail.prob} proves the bound for $\| \widehat{\bR}_1- {\bR}_1 \|_\infty$ in \eqref{max.bound.1}.


Let $\De_2 = \max_{1\leq k\leq p}| n^{-1} S_{kn}|$. The arguments leading to \eqref{mean.ubd} and \eqref{De1.tail.prob} can be used to prove that, respectively, $\mathbb{E} \De_2 \leq C[ n^{-1/2} \{\log(p)\}^{1/2} + \nu_r   \,  p^{1/r}  n^{-1+1/r} ]$ and for $t>0$,
$$
    \mathbb{P}\big( \De_2 \geq 2\mathbb{E} \De_2 + t  \big)   \leq \exp(- n t^2/3) + C \, t^{-r} \nu_r^r \,  p  n^{1-r}.
$$
This completes the proof of \eqref{max.bound.1} in view of \eqref{sample.cov.dec} and \eqref{De1.tail.prob}.

\medskip
\noindent
\textbf{\textit{Case 2.}} Under condition (M2), it follows from Theorem 6 in \cite{DelaigleHallJin_2011} that for all $y>0$ and $1\leq k\leq \ell \leq p$,
\beq
       \mathbb{P} \left\{ \left| \sum_{i=1}^{n} \big( U_{ik} U_{i\ell} - \mathbb{E} U_{ik} U_{i\ell} \big) \right|  > y \right\} \leq 2 \exp\left\{ -  y^2/(4n)\right\} + C \exp( -c y^{\gamma/2} ),
\eeq
where $C, c>0$ are constants depending only on $K_1$, $K_2$ and $\gamma$ in (M2). This, combined with a union bound yields
\[
   \mathbb{P}\left( \De_1  > C  \left[ n^{-1/2}\{\log(pn)\}^{1/2} + n^{-1} \{\log (p n)\}^{2/\gamma} \right] \right) \leq C n^{-1},  \label{var.ubd.exp}
\]
where $\De_1$ is as in \eqref{De1.def}. A completely analogous argument leads to
$$
 \mathbb{P}\left( \De_2  > C   \left[ n^{-1/2} \{\log(pn)\}^{1/2} + n^{-1} \{\log (p n)\}^{1/\gamma} \right] \right) \leq C n^{-1}.
$$
Assembling the above calculations completes the proof of \eqref{max.bound.2}.

\medskip

Finally we prove \eqref{max.bound.3}. It is well-known the $t$-statistic can be expressed as
\be
    \f{\s{n}(\bar{X}_k-\mu_{1  k})}{\hat{\sigma}_{1 k}} = \f{S_{kn}/V_{kn}}{\{ 1- n^{-1}(S_{kn}/V_{kn})^2 \}^{1/2}}.  \label{equiv}
\ee
Let $ \varepsilon_{n} = \min[ \frac{1}{2}, \nu_{4}^2 \, n^{-1/2} \{ 2\log(pn) \}^{1/2} ]$. Inequalities (\ref{tail.1}) and (\ref{sn.tail}) imply, for $0<t\leq \sqrt{n}$,
\begin{align}
    & \mathbb{P}\left\{ \sqrt{n}   \, |\widehat{\bD}_1^{-1/2} ( \widehat{\bmu}_1 - \bmu_1 )|_\infty   \geq  t  \right\}  \nn \\
    & = \mathbb{P}\left\{  \max_{1\leq k\leq p}  \f{ \s{n}(\bar{X}_k - \mu_{1 k} )}{\hat{\sigma}_{1 k}}   \geq t  \right\}    \nn \\
    &   \leq p \max_{1\leq k\leq p} \mathbb{P} \left\{ S_{kn}/V_{kn} \geq t(1+t^2n^{-1})^{-1/2} \right\} \nn \\
    &  \leq p \max_{1\leq k\leq p} \left[  \mathbb{P}\left\{  S_{kn} \geq  \f{t (4 \s{n}+ V_{kn})}{(1+4\s{2})(1+t^2 n^{-1})^{1/2}} \right\} +  \mathbb{P}\left\{ V_{kn}^2  \leq ( 1- \varepsilon_{n} )n \right\} \right] \nn \\
      & \leq 3 p \exp(- c  t^2 ) + n^{-1}. \nn
\end{align}
A completely analogous argument will lead to the same bound for $ \mathbb{P} \big\{ \sqrt{n}   \, |\widehat{\bD}_1^{-1/2} ( \widehat{\bmu}_1 - \bmu_1 )|_\infty  \leq -t   \big\} , $ and hence completes the proof of Lemma \ref{lm2}. \qed
%===================================================================================================
\section{Proof of the asymptotic null properties: one-sample case}


\subsection{Proof of Proposition~\ref{proposition.1}}

We first introduce a proposition that gives non-asymptotic bounds on the distributional error between the one-sample test statistic $T^{\rI}_{\nu}$ and its Gaussian analogue under the null hypothesis for $\nu \in \{ \ns, \rs \}$. We then turn to a proof of Theorem~\ref{one-sample.size}.

\begin{proposition} \label{proposition.1} Let $\nu \in \{\ns, s\}$, $\Theta^{\rI}_{\ns}= \|\wt{\bSigma}_1-\bSigma_1 \|_{\infty}$ and $\Theta^{\rI}_{\rs} = \|\wt{\bR}_1-\bR_1 \|_{\infty}$. The estimator $\wt{\bSigma}_1$ of $\bSigma_1$ is such that $\mbox{diag}\,(\wt{\bSigma}_1)= \mbox{diag}\,(\widehat{\bSigma}_1)$.
\begin{enumerate}
\item[{\rm (i)}] Assume that {\rm(M1)} holds. Then under $H^{\rI}_0$,
\begin{align*}
  & \sup_{x \geq 0} \big| \mathbb{P}\big\{  T_{\nu}^{\rI} >x \big\} - \mathbb{P}\big\{ |\bW^{\rI}_{\nu }  |_{\infty} >x  |  \mathcal{X}_n  \big\} \big|  \\
  & \leq  C_1 \left[  \{\Theta_{\nu}^{\rI} \}^{1/3}\{ \log(p  n) \}^{2/3}  + n^{-1/8} \{ \log(p n) \}^{7/8} +\vartheta_{n,p}^{1/(r+1)} \{\log(pn)\}^{3/2} \right],
\end{align*}
where $\vartheta_{n,p} = pn^{1-r/2}$.


\item[{\rm (ii)}] Assume that {\rm(M2)} holds. Then under $H^{\rI}_0$,
\begin{align*}
  & \sup_{x \geq 0}  \big|\mathbb{P}\big\{ T_{\nu}^{\rI} >x \big\} -\mathbb{P}\big\{  |  \bW^{\rI}_{\nu } |_{\infty} >x | \mathcal{X}_n   \big\} \big|  \\
 &  \leq  C_2  \Big[  \{ \Theta_{\nu}^{\rI}\}^{1/3}  \{ \log(p  n) \}^{2/3}   + n^{-1/8}   \{\log(p n ) \}^{7/8} + n^{-1/2} \{\log(p  n)\}^{3/2+1/\gamma} \Big].
\end{align*}
%In addition, if we take $\wt{\bSigma}_1=\wh{\bSigma}_1$ and $\wt{\bR}_1=\wh{\bR}_1$, then $b_{\nu}\leq C_4 \, n^{-1/2} \log^{1/2} (p  \vee n)$ with probability greater than $1-c_5 \, n^{-1}$.
\end{enumerate}
The constants $C_1, C_2>0$ are independent of $n, p, r$ and $\gamma$.
\end{proposition}

\begin{proof}

For the sake of clarity we give the proof only for the studentized statistic $T_{\rs} := T_{\rs}^{\rI}$, as the non-studentized statistic $T_{\ns}^{\rI}$ can be dealt with in a similar way. To begin with, observe that for every $x\in \br$, $|x|=\max(x, -x)$. Therefore, for $t>0$,
\beqn
 \mathbb{P}\big(  T_s > t \big)  = \mathbb{P}\left\{    \max_{1\leq k\leq p } \max\left( \f{  \s{n} \bar{X}_k}{\hat{\sigma}_{1 k}} , -  \f{ \s{n}\bar{X}_k}{\hat{\sigma}_{1 k}} \right) > t   \right\}.  \label{prob.dec}
\eeqn
Define a new sequence of dilated random vectors $\bX^{{\rm ext}}_1, \ldots, \bX^{{\rm ext}}_{n}$ taking values in $\br^{2p}$, given by $\bX^{{\rm ext}}_i=(X^{{\rm ext}}_{i, 1}, \ldots, X^{{\rm ext}}_{i ,2p})'=(\mX'_i, - \mX'_i)'$. In this notation, we have $\mathbb{P}(   T_s > t ) =\mathbb{P}(  T_s^{{\rm ext}}  > t  )$, where
$$
      T_{\rs}^{{\rm ext}} = \s{n}\max_{1\leq k\leq 2 p} \f{\bar{X}^{{\rm ext}}_k}{\hat{\sigma}^{{\rm ext}}_{k}} \quad \mbox{ with } \ \
    \bar{X}^{{\rm ext}}_k = n^{-1} \sum_{i=1}^{n} X^{{\rm ext}}_{i,k}, \ \ (\hat{\sigma}_{k}^{{\rm ext}})^2 = n^{-1} \sum_{i=1}^{n} (X^{{\rm ext}}_{i,k} -\bar{X}^{{\rm ext}}_k)^2.
$$
Hence, we only need to focus on $  T_{\rs}^+ := \s{n} \max_{1\leq k\leq p } \bar{X}_k /\hat{\sigma}_{1 k} $ without loss of generality.

Recall that, conditional on $\{\bX_1, \ldots, \bX_{n} \}$, $\bW^{\rI}_{\rs}=(W_{\rs,1}^{\rI}, \ldots, W_{\rs, p}^{\rI})'$ is a centred Gaussian random vector with covariance matrix $\wt{\bR}_1$. Put \be
 Z_{\rs}^+=\max_{1\leq k\leq p} W_{\rs , k}^{\rI}. \label{tilde.L2+}
\ee
We shall show that, with high probability, the distribution of $T_{{\rs}}^+$ and the conditional distribution of $ Z_{{\rs}}^+$ given $\mathcal{X}_n$ are close enough so that the conditional quantiles of $Z_s^+$ given $\mathcal{X}_n$ provide reasonable estimates of the quantiles of $ T_{{\rs}}^+$. For this purpose, we introduce the following random variables that serve as intermediate approximations.



Let $\{\bG_{i}=\left( G_{i1}, \ldots,  G_{ip}\right)'\}_{i=1}^n$ be independent $p$-variate centered Gaussian random vectors with covariance matrix $\bR_1$, and define
\be
  Z^*_{\rs} = \max_{1\leq k\leq p}  n^{-1/2}\sn G_{ik}, \qquad
     T^*_{\rs}  =  \max_{1\leq k\leq p}  n^{-1/2} \sum_{i=1}^{n}U_{ik},  \label{T2*}
\ee
where $U_{ik}= (X_{i k}-\mu_{1 k})/\sigma_{1k}$. For the random variables $Z_{\rs}^*$ and $T_{\rs}^*$, by Lemma~\ref{CCK_2013} we have the following Berry-Esseen type bound:
\be
    d_n := \sup_{x\in \br} \big| \mathbb{P}\big( T_{\rs}^* \leq x \big) - \mathbb{P}\big( Z_{\rs}^* \leq x\big) \big| ,  \label{dif.1}
\ee
where the order of $d_n$ depends on the moment conditions imposed on $\mX_i$ as described in \eqref{poly.rate} and \eqref{exp.rate}. Further, applying Theorem~3 in \cite{ChernozhukovChetverikovKato_2014} to $Z_{\rs}^*$ gives, for any $\varepsilon>0$,
\be
    \sup_{x\in \br }\mathbb{P} \big(  |  Z_{\rs}^*  -x | \leq \varepsilon  \big) \leq 4 \varepsilon \left[  1 + \{ 2\log(p)   \}^{1/2}  \right].  \label{con.dif}
\ee

On the other hand, we see that under $H^{\rI}_0$, $T_{\rs}^*=\s{n} \max_{1\leq k\leq p} \bar{X}_k/  \sigma _{1k}$ and hence
\be
   \left|  T_{\rs}^+  -  T_{\rs}^*  \right| \leq    \max_{1\leq k\leq p} \left| \f{\hat{\sigma}_{1 k}}{ \sigma_{1k}} -1 \right|    \cdot \max_{1\leq k \leq p}  \f{\s{n}|\bar{X}_k|}{\hat{\sigma}_{1 k}} \leq    \max_{1\leq k\leq p} \left| \f{\hat{\sigma}_{1 k}}{ \sigma_{1k}} -1 \right|  \cdot    \sqrt{n} \big| \widehat{\bD}_1^{-1/2}  \widehat{\bmu}_1 \big|_\infty .  \label{dif.2}
\ee
For $t>0$, define the event
\be
   \mathcal{E}_0(t) = \left\{  \max_{1\leq k\leq p} \left|  \hat{\sigma}_{1 k}/  \sigma_{1k}  -1 \right| \leq t  \right\}.  \label{E0}
\ee
On this event, we have $ |  T_{\rs}^+  -  T_{\rs}^* |  \leq t \sqrt{n}  | \widehat{\bD}_1^{-1/2}  \widehat{\bmu}_1  |_\infty$. Together with (iii) in Lemma \ref{lm2}, this yields for any $\varepsilon>0$ that
\begin{align}
      \mathbb{P}\big(  |  T_{\rs}^+  -  T_{\rs}^* | >\varepsilon \big)   \leq C  p \exp\{ -c (\varepsilon/t)^2 \} + \mathbb{P}\{ \mathcal{E}_0(t)^{{\rm c}} \} ,  \label{difference.probability.bound}
\end{align}
where $\mathcal{E}_0^{{\rm c}}$ denotes the complimentary set of $\mathcal{E}_0$.


For the two Gaussian maxima $Z_{\rs}^+$ and $Z_{\rs}^*$ given in \eqref{tilde.L2+} and \eqref{T2*}, respectively, it follows from Lemma \ref{CCK_2013_31} that
\be
    \hat{d}_n :=  \sup_{x \in \br} \big| \mathbb{P}\big( Z_{\rs}^* \leq x \big) -  \mathbb{P} \big(   Z_{\rs}^+ \leq x | \mathcal{X}_n \big) \big| \leq     C  \Theta_{\rs}^{1/3}  \{ 1 + \log(p/\Theta_s )\}^{2/3},  \label{dif.3}
\ee
where $\Theta_{\rs} =\Theta_{\rs}^{\rI} = \|\wt{\bR}_1- \bR_1  \|_{\infty}$.

Consequently, combination of inequalities \eqref{dif.1}, \eqref{con.dif}, \eqref{difference.probability.bound} and \eqref{dif.3} gives, for every $x\in \br$ and $t, \varepsilon >0$,
\begin{align*}
 & \mathbb{P}\big(  T_{\rs}^+ > x \big)  \\
 & \leq  \mathbb{P}\big( T_{\rs}^* > x - \varepsilon \big) + \mathbb{P}\big(  |   T_{\rs}^+  - T_{\rs}^*  | > \varepsilon \big)  \\
 &\leq  \mathbb{P}\big(  Z_{\rs}^* > x-\varepsilon \big) + d_n  + \mathbb{P}\big(  |  T_{\rs}^+  - T_{\rs}^*  | > \varepsilon \big) \\
 & \leq  \mathbb{P}\big( Z_{\rs}^* > x  \big) + C  \varepsilon \{\log(p) \}^{1/2}  + d_n + \mathbb{P}\big(   |  T_{\rs}^+  -  T_{\rs}^*  | > \varepsilon \big) \\
 & \leq  \mathbb{P}\big(    Z_{\rs}^+ > x | \mathcal{X}_n \big)  +  C   \varepsilon \{\log(p) \}^{1/2} + d_n + \hat{d}_n + C  p \exp\{ -c (\varepsilon/t)^2 \} + \mathbb{P}\{ \mathcal{E}_0(t)^{{\rm c}} \} .
\end{align*}
A similar argument leads to the reverse inequality, which together the previous display implies by taking $\varepsilon=C t \{\log(pn)\}^{1/2}$ that
\begin{align}
     \sup_{x \in \br }\big| \mathbb{P}\big(  T_{\rs}^+ \leq  x  \big) -  \mathbb{P}\big(  Z_{\rs}^+ \leq  x |  \mathcal{X}_n \big)  \big|     \leq d_n + \hat{d}_n +  C \left\{  t  \log(pn) + n^{-1} \right\}    + \mathbb{P}\{ \mathcal{E}_0(t)^{{\rm c}} \}, \label{BE.bound.1}
\end{align}
where $d_n$ and $\hat{d}_n$ are as in \eqref{dif.1} and \eqref{dif.3}, respectively.

In the case where condition (M1) holds, by (i) in Lemma \ref{lm2}, taking $ t \asymp  \nu_4^2\,  n^{-1/2}\{ \log(pn) \}^{1/2} + \nu_r^2  \,   \vartheta_{n,p}^{2/(r+2)} + \nu_r^2 \,  \vartheta_{n,p}^{2/r}   \log(p)$ in \eqref{BE.bound.1} yields that $\mathbb{P}\{ \mathcal{E}_0(t)^{{\rm c}} \}  \leq C  \{ n^{-1} + \vartheta^{2/(r+2)}_{n,p}  \}$. Substituting this into \eqref{BE.bound.1} proves the first assertion of the proposition in view of \eqref{poly.rate}.

Finally, when condition (M2) holds, (ii) in Lemma \ref{lm2} implies by taking $t  \asymp  n^{-1/2} \{\log (p n) \}^{1/2} + n^{-1} \{ \log (p n)\}^{2/\gamma}$ in \eqref{BE.bound.1} that $\mathbb{P}\{ \mathcal{E}_0(t)^{{\rm c}} \} \leq C n^{-1}$. Together with \eqref{exp.rate} and \eqref{BE.bound.1}, this completes the proof of the second assertion of the proposition.

\end{proof}

%============================================================================

\subsection{Proof of Theorem~\ref{one-sample.size}}
We now prove Theorem~\ref{one-sample.size}. As before, we only give the proof for the test that is based on the studentized statistic. For $\alpha\in (0,1)$ given, recall that ${\rm cv}_{\rs ,\alpha}^{\rI}$ is the conditional $(1-\alpha)$-quantile of $\bW_{\rs}^{\rI} \sim N(0, \widehat{\bR}_1)$ given $\mathcal{X}_n=\{\bX_i\}_{i=1}^n$. Then it follows from (i), Proposition~\ref{proposition.1} and Lemma~\ref{lm2} that, under condition (M1),
\begin{align*}
    & \left| P_{H_0^{\rI}}\big\{ \Psi_{\rs ,\alpha}^{\rI}=1  \big\} - P\big( |\bW_{\rs}^{\rI} |_\infty >{\rm cv}_{\rs ,\alpha}^{\rI} | \mathcal{X}_n \big)  \right|  \xrightarrow P 0,
\end{align*}
as $n\rightarrow \infty$. Applying Theorem~3 in \cite{ChernozhukovChetverikovKato_2014} and Comment 5 after it to $|\bW_{\rs}^{\rI} |_\infty$ implies by \eqref{q12.definition} that
\begin{align*}
      \alpha & \geq  P\big( |\bW_{\rs}^{\rI} |_\infty >{\rm cv}_{\rs ,\alpha}^{\rI} | \mathcal{X}_n \big)  \\
    & \geq P\big( |\bW_{\rs}^{\rI} |_\infty >{\rm cv}_{\rs ,\alpha}^{\rI} - n^{-1} | \mathcal{X}_n \big) - P\big( {\rm cv}_{\rs ,\alpha}^{\rI} - n^{-1} \leq |\bW_{\rs}^{\rI} |_\infty \leq {\rm cv}_{\rs ,\alpha}^{\rI} | \mathcal{X}_n \big)  \\
    & \geq \alpha - C n^{-1}E\big( |\bW_{\rs}^{\rI} |_\infty | \mathcal{X}_n \big)  \geq \alpha - C n^{-1}\{\log(p)\}^{1/2}.
\end{align*}
The last two displays jointly complete the proof of the theorem. \qed

%============================================================================
\subsection{Proof of Theorem~\ref{one-sample.power.1}}

As in the proof of Proposition~\ref{proposition.1}, we start with $T_{\rs}=T_{\rs}^{\rI}$. A standard result on Gaussian maximum yields
\be
 \mathbb{E} \big(  |  \bW_{\rs}^{\rI}  |_{\infty} |   \mathcal{X}_n \big) \leq  \{  2 \log(p) \}^{1/2} + \{ 2 \log(p) \}^{-1/2} \leq [ 1+ \{ 2\log(p)\}^{-1}] \{  2 \log(p) \}^{1/2},
    \label{mean.gm}
\ee where $ \mathcal{X}_n =\{\bX_i \}_{i=1}^n$. The following fundamental result \citep{Borell_1975} shows the concentration of $ |\bW_{\rs}^{\rI} |_{\infty}$ around its mean $ \mathbb{E} ( | \bW_{\rs}^{\rI} |_{\infty} |   \mathcal{X}_n )$ that for every $u>0$,
$$
    \mathbb{P}\big\{ | \bW_{\rs}^{\rI}  |_{\infty} \geq \mathbb{E}\big( |  \bW_{\rs}^{\rI} |_{\infty} |    \mathcal{X}_n \big) + u  \big|   \mathcal{X}_n \big\} \leq  \exp(-{u^2}/{2})
$$
Together with \eqref{mean.gm}, this implies
\be
    {\rm cv}_{\rs ,\alpha}^{\rI} \leq [1+ \{2\log(p)\}^{-1}]\s{   2 \log(p)   } + \s{ 2\log(1/\alpha) }. \label{cv.ubd}
\ee

Next, recall that $T_{\rs}=\max_{1\leq k\leq p}\max\big( \s{n}\bar{X}_k/\hat{\sigma}_{1k}, -\s{n}\bar{X}_k/\hat{\sigma}_{1k} \big)$, and let $\mathcal{E}_0(t)$ be as in \eqref{E0} for some $0<t \leq \frac{1}{2}$. Set $k_0=\arg\max_{1\leq k\leq p}   |\mu_{1 k}|/ \sigma_{1k}$, and assume without loss of generality that $\mu_{1 k_0}>0$. Then, on the event $\mathcal{E}_0(t)$, \beq
        T_{\rs}  \geq \f{\s{n} \bar{X}_{k_0}  }{\hat{\sigma}_{1  k_0}}  =  \f{\s{n}(\bar{X}_{k_0} - \mu_{1  k_0})}{\hat{\sigma}_{1 k_0}}  + \f{\s{n}\mu_{1 k_0}}{\hat{\sigma}_{1  k_0}}  \geq         \f{\s{n}(\bar{X}_{k_0} - \mu_{1  k_0})}{\hat{\sigma}_{1  k_0}}  + (1+t)^{-1} \f{ \sqrt{n} \mu_{1 k_0}}{\sigma_{1  k_0}}.
\eeq

By condition (M1) with $p=O(n^{r/2-1-\de})$, we have $\vartheta_{n,p}=O(n^{-\de})$. This, together wiht Lemma~\ref{lm2} implies by taking $t \asymp  n^{-2\de/(r+2)}$ that $ \mathbb{P}\{\mathcal{E}_0(t)^{{\rm c}} \} \leq C \{ n^{-1} + n^{-2\de/(r+2)}  \}$. Further, choose $u=u_{n,p}$ in such a way that $(1+t)\big[ 1+ \{\log(p)\}^{-1}+u \big]=1+\varepsilon_{n}$, for $\varepsilon_{n}>0$ satisfying that $\varepsilon_{n}\rightarrow0$ and $\varepsilon_{n} \s{\log(p) } \rightarrow\infty$. Consequently,
$$
    \max_{1\leq k\leq p} \f{\s{n}|\mu_{1 k}|}{\sigma_{1 k}} \geq (1+t) \big[ 1+ \{\log(p)\}^{-1}+u \big]  \la(p,\alpha),
$$
where $\la(p,\alpha)$ is as in \eq{threshold}. The last display together with Lemma~\ref{lm2} yields, as $n, p\rightarrow \infty$,
\begin{equation}
\begin{split}
\mathbb{P}\big(  T_{\rs } > {\rm cv}_{\rs ,\alpha}^{\rI} \big) \geq&~  \mathbb{P} \big(  T_{\rs}^+ >  [ 1+ \{ \log(p) \}^{-1}  ] \, \la(p,\alpha)  \big)   \\
     \geq&~ \mathbb{P}\big( T_{\rs}^+ >  [ 1+ \{ \log(p) \}^{-1}  ] \, \la(p,\alpha) , \, \mathcal{E}_0(t) \, \big)  \\
     \geq&~ 1- \mathbb{P}\big\{  \s{n} (\bar{X}_{k_0}-\mu_{1 k_0}) < - u   \la(p,\alpha)\hat{\sigma}_{1 k_0} \big\}
    - \mathbb{P}\big\{ \mathcal{E}_0(t)^{{\rm c}} \big\}   \\
     \geq&~ 1- n^{-1} -  C \exp\{ -c  u^2 \la^2(p,\alpha) \}- \mathbb{P}\big\{ \mathcal{E}_0(t)^{{\rm c}} \big\}  \\
      \geq&~ 1-  n^{-1} -  C \exp\{-c   u^2 \log(p) \} - \mathbb{P}\big\{\mathcal{E}_0(t)^{{\rm c}} \}  \\
      \rightarrow & 1 . \label{power.lbd}
\end{split}
\end{equation}


Under condition (M2), taking $t \asymp  n^{-1/2} \{ \log (pn)\}^{1/2}+ n^{-1} \{  \log(p n)\}^{2/\gamma} $ instead gives $\mathbb{P}\{ \mathcal{E}_0(t)^{{\rm c}}  \} \leq C n^{-1}$. Then the conclusion follows directly from \eqref{cv.ubd} and \eqref{power.lbd}.

Next we consider the statistic $T_{\ns} = T_{\ns}^{\rI}$ without studentization. Analogously to \eqref{mean.gm} and the concentration inequality after it, now we have for $\bW_{\ns}^{\rI}$, \beq \mathbb{E} \big(|\bW_{\ns}^{\rI} |_{\infty}  \big| \mathcal{X}_n \big) \leq   \max_{1\leq k\leq p}\hat{\sigma}_{1 k} \, \cdot   [1+ \{2\log(p)\}^{-1} ]\{ 2\log(p)  \}^{1/2} \eeq and for every $u>0$,
$$
\mathbb{P}\big\{  | \bW_{\ns}^{\rI}  |_{\infty} \geq \mathbb{E} \big( | \bW_{\ns}^{\rI}  |_{\infty} | \mathcal{X}_n \big) + u \, \big| \mathcal{X}_n \big\} \leq   \exp\bigg( -\f{u^2}{2\max_{1\leq k\leq p}\hat{\sigma}_{1 k}^2 } \bigg),
$$
The last two displays joint imply that on the event $\mathcal{E}_0(t)$,
\be
    {\rm cv}_{\ns, \alpha}^{\rI} \leq (1+t)   \max\limits_{1\leq k\leq p}  \sigma_{1 k}    \left( [1+ \{2 \log(p)\}^{-1}]\{ 2\log(p)  \}^{1/2}  +\{  2\log(1/\alpha)  \}^{1/2} \right). \label{cv.ubd'}
\ee

Following the same argument that leads to \eqref{power.lbd}, it suffices to estimate the probability that $\mathbb{P}\{ \s{n}  (\bar{X}_k-\mu_{1 k}) > y\}$ for $y>0$. Assuming the uniform boundedness of fourth moments, put $\wh{X}_{i k}=(X_{i k} -\mu_{1 k})I\{|X_{i k} -\mu_{1 k}|\leq  \sigma_{1 k} \s{n}  \}$, such that $|\mathbb{E}  \wh{X}_{i k} -\mu_{1 k}| \leq  \sigma_{1 k}  \nu_4^4  \, n^{-3/2} $. Then it follows from Bernstein's inequality that for $y>2 \sigma_{1k}\nu_4^4 \,  n^{-1}$,
\begin{align}
   &  \mathbb{P}\left\{ \f{1}{\s{n}}\sn (X_{i k}-\mu_{1 k}) > y  \right\}    \nn \\
    &  \leq  \mathbb{P}\left\{ \f{1}{\s{n}}\sn \big( \wh{X}_{i k}-\mathbb{E} \wh{X}_{i k} \big) >\f{y}{2}  \right\} +  n \max\limits_{1\leq i\leq n} \mathbb{P}\left( |X_{ik}-\mu_{1 k} | > \sigma_{1k} \s{n} \,  \right) \nn  \\
    &  \leq \exp\left\{ -\f{1}{C} \min\left(  \f{y^2 }{\sigma^2_{1k}}  , \f{y}{ \sigma_{1 k }} \right) \right\} + \nu_4^4 \, n^{-1} .  \nn
\end{align}

The rest of the proof is similar to that for $T_{\rs}^{\rI}$ and thus is omitted. The proof of Theorem~\ref{one-sample.power.1} is then complete. \qed

%\subsubsection{Proof of Theorem~\ref{one-sample.power.2}.}
%
%Under the regularity condition (D-1), first we derive the asymptotic expression for ${\rm cv}_{\rs ,\alpha}={\rm cv}_{\rs ,\alpha}^{\rI}$ given in \eqref{q12.definition} as $p\rightarrow \infty $. To this end, we need the following result on the asymptotic distribution of Gaussian maxima under dependence which is Lemma 6 in \cite{CaiLiuXia_2014} with slight modification.  Let $(G_1, \ldots, G_p)'$ be a $p$-variate centred Gaussian random vector with covariance matrix $\bR_1$.
%
%\begin{lemma} \lbl{lm.CLX}
%Suppose that condition (D-1) holds. Then for every $t\in \br$, as $p\rightarrow \infty$
%\beq
%\mathbb{P}\left( \max_{1\leq k \leq p} |G_k|  \leq \s{  2\log (p)- \log(\log p ) + t   } \,  \right) \rightarrow \exp\left\{ -\f{1}{ \s{\pi}} \exp(-t/2) \right\}  .
%\eeq
%\end{lemma}
%
%For $0<\alpha<1$, put ${\rm cv}_{ \alpha}=\s{   2\log (p) -\log(\log p ) +c_{\alpha} }$, where $c_{\alpha}:=-\log \pi -  2\log[\log\{(1-\alpha)^{-1}\}]$. A direct consequence of Lemma \ref{lm.CLX} is that $\mathbb{P} (\max_{1\leq k\leq p}|G_k| >{\rm cv}_{\alpha}  ) \rightarrow \alpha$ as $p\rightarrow \infty$. This, together with Lemmas~\ref{CCK_2013_31} and~\ref{lm2} implies ${\rm cv}_{\rs ,\alpha}  = {\rm cv}_{\alpha}\{ 1+o(1)\}$ as $ p \rightarrow \infty$.
%
%With the above preparations, we are ready to prove the asymptotic power. First, we consider the case where $\beta\geq (1-\s{\tau} )^2 + \varepsilon$ for some $\varepsilon>0$. Under $H^{\rI}_{s,1}$, set $\mathcal{S}_{10}=\{1\leq k\leq p: \mu_{1 k} = 0\}$ such that
%\begin{align*}
% \mathbb{P}\big\{  T^{\rI}_{\rs}   > {\rm cv}_{\rs ,\alpha} \big\}
%    & \geq \mathbb{P}\left\{ \max_{k\in \mathcal{S}_{10}^{{\rm c}} } \f{\s{n} (\bar{X}_k-\mu_{1 k})}{\hat{\sigma}_{1 k}} {\rm sign}(\mu_{1 k})  > {\rm cv}_{\rs ,\alpha} - \min\limits_{k\in \mathcal{S}_{10}^{{\rm c}}  } \f{\s{n} |\mu_{1k}|}{\hat{\sigma}_{1 k}} \right\}  \\
%   & \geq \mathbb{P}\left\{ \max_{k\in \mathcal{S}_{10}^{{\rm c}} } \f{\s{n} (\bar{X}_k-\mu_{1 k})}{\hat{\sigma}_{1 k}} {\rm sign}(\mu_{1 k})  > {\rm cv}_{\rs ,\alpha} - \f{\s{2\beta \log (p)}}{1+t} \right\}
%   - \mathbb{P}\big\{\mathcal{E}_{0}(t)^{{\rm c}} \big\}
%\end{align*}
%for $\mathcal{E}_{0}(t)$ as in (\ref{E0}), and $t=t(n,p)=o(1)$ is chosen in the same way as in the end of the proof of Proposition~\ref{proposition.1} such that $\mathbb{P}\{\mathcal{E}_{0}(t)^{{\rm c}}\} =o(1)$. For $\beta \geq (1-\s{\tau})^2+\varepsilon$ and $M>0$ large enough, it is easy to see that ${\rm cv}_{\rs ,\alpha}-(1+t)^{-1}\s{2 \beta \log (p)}< \s{2\log(p^{\tau}) -\log\{\log (p^{\tau})\} -M}$ for all sufficiently large $p$.
%
%
%Applying Proposition~\ref{proposition.1} to $\max_{k\in \mathcal{S}_{10}^{{\rm c}}}  \{  \s{n} \, {\rm sign}(\mu_{1 k})(\bar{X}_k-\mu_{1 k})/\hat{\sigma}_{1 k} \}$ yields, as $ p \rightarrow \infty$,
%\begin{align*}
%   & \mathbb{P}\big\{  T^{\rI}_{\rs}  > {\rm cv}_{\rs ,\alpha} \big\} \\
%    & \geq
%\mathbb{P}\left\{ \max_{k\in \mathcal{S}_{10}^{{\rm c}} }  \f{\s{n}(\bar{X}_k-\mu_{1 k})}{\hat{\sigma}_{1 k}}  {\rm sign}(\mu_{1 k})  > \s{2\log (p^{\tau}) -\log \{\log(p^{\tau})\} -M} \right\}
%   -\mathbb{P}\big\{ \mathcal{E}_{0}(t)^{{\rm c}} \big\}  \\
%   & \rightarrow   \mathbb{P}\left\{ \max_{1\leq k\leq p^{\tau}  } G_{k} > \s{2\log (p^{\tau}) -\log \{\log(p^{\tau})\} -M} \right\}  \\
%   & \rightarrow 1 - \exp\left\{ -\f{1}{2\s{\pi}}\exp(M/2) \right\}.
%\end{align*}
%Finally letting $M \rightarrow \infty$ completes the proof of the theorem.  \qed



%===================================================================================================
\section{Proof of the asymptotic null properties: two-sample case}

\subsection{Proof of Theorem~\ref{two-sample.size}}


Recall that $n=\min(n,m)$ and $\lambda = n/m$. The following result extends Proposition~\ref{proposition.1} and provides a non-asymptotic error bound between the distribution of the two-sample test statistic and that of its Gaussian analogue under the null hypothesis. Throughout the following, we write $ \mathcal{X}_n = \{ \bX_i \}_{i=1}^n$ and $\mathcal{Y}_m = \{ \bY_j \}_{j=1}^m$.


\begin{proposition} \label{proposition.2} Let $\nu \in \{\ns,{\rs}\}$, $\Theta_{\ns}^{\rII}=  \|\wt{\bSigma}_{1,2}-\bSigma_{1,2}\|_{\infty}$ and $\Theta^{\rII}_{\rs} = \| \wt{\bR}_{1,2} -\bR_{1,2} \|_{\infty}$. The estimators $\wt{\bSigma}_1$ and $\wt{\bSigma}_2$ of $\bSigma_1$ and $\bSigma_2$, respectively, are such that $\mbox{diag}\,(\wt{\bSigma}_1)= \mbox{diag}\,(\widehat{\bSigma}_1)$ and $\mbox{diag}\,(\wt{\bSigma}_2)= \mbox{diag}\,(\widehat{\bSigma}_2)$.


\begin{enumerate}
\item[{\rm (i)}] Assume that {\rm(M1)} holds. Then under $H^{\rII}_0$,
\begin{align*}
  & \sup_{x\geq 0}  \big| \mathbb{P}\big\{  T_{\nu}^{\rII} >x \big\} -\mathbb{P}\big\{ |  \bW^{\rII}_{\nu}  |_{\infty} >x  | \mathcal{X}_n, \mathcal{Y}_m  \big\} \big| \\
  &\qquad  \leq    C_1 \left[   ( \Theta^{\rII}_{\nu} )^{1/3} \{ \log(pn) \}^{2/3}  + n^{- 1/8 } \{ \log(pn) \}^{7/8} + \vartheta_{n,p}^{1/(r+1)}  \{\log(p n)\}^{3/2}  \right]
\end{align*}
for $\vartheta_{n,p}$ as in \eqref{alpha.np}.


\item[{\rm (ii)}] Assume that {\rm(M2)} holds. Then under $H^{\rII}_0$,
\begin{align*}
   & \sup_{x \geq 0 }  \big| \mathbb{P}\big\{  T_{\nu}^{\rII} > x \big\}
 -\mathbb{P}\big\{  |  \bW^{\rII}_{\nu } |_{\infty} >x  |   \mathcal{X}_n, \mathcal{Y}_m  \big\} \big|  \\
 & \quad  \leq C_2  \Big[  ( \Theta_{\nu}^{\rII})^{1/3}  \{ \log(p   n) \}^{2/3}   + n^{-1/8}   \{\log(p n ) \}^{7/8} + n^{-1/2} \{\log(p n)\}^{3/2+1/\gamma}   \Big].
\end{align*}
\end{enumerate}
The constants $C_1, C_2>0$ and independent of $n, m , p , r$ and $\gamma$.
\end{proposition}


As a direct consequence of Proposition~\ref{proposition.2}, the proof for the validity of the two-sample tests based on the statistics $ T_{\ns}^{\rII}$ and $ T_{\rs}^{\rII}$ is almost identical to that of Theorem~\ref{one-sample.size}, and hence is omitted.

\begin{proof}[Proof of Proposition~\ref{proposition.2}]
The basic idea is essentially in line with that for proving Proposition~\ref{proposition.1}, and we shall only focus on the studentized test statistic $T_{\rs}=T_{\rs}^{\rII}$ with slight abuse of notation. First, define a pooled sequence of random vectors $\{\bxi_i=(\xi_{i 1}, \ldots, \xi_{i p})' \}_{i=1}^N$ as follows:
\beq
    \xi_{i k}  = \left\{
           \begin{array}{ll}
         X_{i k}-\mu_{1 k},      \qquad  &   1\leq i \leq n,  \\
                                       %     \vspace{-0.2cm}   \\
               - \lambda  (Y_{i-n, k}-\mu_{2 k}), \qquad  &  n+1 \leq  i \leq N ,             \end{array}   \right.
\eeq
for $N=n+m$ and $\lambda=n/m$ as specified in Section \ref{method.sec}. It is easy to see that $\mathbb{E} \bxi_i =\bzero$ and
\beq
\mathbb{E}( \bxi_i \bxi_i' )  = \left\{
\begin{array}{ll}
   \bSigma_1,      \qquad  &   1\leq i \leq n  ,  \\
   \lambda^2   \bSigma_2,  \qquad &     n+1 \leq  i \leq N.  \end{array}   \right.
\eeq
Moreover, define weighted versions of $\bxi_i$:
$$
\tilde \bxi_i= (\tilde \xi_{i1}, \ldots, \tilde \xi_{ip})' =\bD^{-1/2}\bxi_i  \qquad   \mbox{ and } \qquad
\hat{\bxi}_i = ( \hat \xi_{i1}, \ldots, \hat \xi_{ip})' =\wh{\bD}^{-1/2}\bxi_i,
$$
where $\bD={\rm diag}(s_1^2, \ldots, s_p^2)$ and $\wh{\bD}={\rm diag}(\hat{s}_1^2, \ldots, \hat{s}_p^2)$ with
\be
s_k^2 = \f{n}{N}\left(   \sigma^2_{1 k} + \lambda    \sigma_{2k}^2 \right)~~~\textrm{and}~~~
 \hat{s}_k^2 = \f{n}{N} \left(   \hat{\sigma}_{1 k}^2 + \lambda   \hat{\sigma}_{2 k}^2  \right).  \label{sk2}
\ee In the above notation, we have under $H_0^{\rII}$, $ T_{\rs} = T_{\rs}^{\rII} = N^{-1/2}\max_{1\leq k\leq p}  |  \sum_{i=1}^N \hat{\xi}_{i k}|$. According to the discussions below \eqref{prob.dec}, it is sufficient to focus on on the following statistic: \beq
        T_{\rs}^+ = \max_{1\leq k\leq p}  N^{-1/2}\sum_{i=1}^N \hat{\xi}_{i k}.
\eeq In the definition of $ T_{\rs}^+$, replacing the variance estimates $\hat{s}_k^2$ with their population analogues leads to $T^*_{\rs}  ={N}^{-1/2} \max_{1\leq k\leq p} \sum_{i=1}^N \tilde \xi_{i k}$, satisfying \beq
   \big|  T_{\rs}^{+}  - T_{\rs}^*  \big| \leq  \max_{1\leq k\leq p} \left| \f{\hat{s}_k}{s_k} -1 \right|  \sqrt{n} \left| \widehat{\bD}_{1,2}^{-1/2}(\widehat{\bmu}_1 -\widehat{\bmu}_2) \right|_\infty  ,
\eeq
where $\widehat{\bD}_{1,2}=\mbox{diag}\,(\widehat{\bSigma}_1+\lambda \widehat{\bSigma}_2)$. For $0<t\leq \frac{1}{2}$, define the subset
\be
    \mathcal{E}_{n,m}(t) = \left\{  \max_{1\leq k\leq p} \left| \f{\hat{\sigma}_{\nu k}^2}{\sigma^2_{\nu k}} -1 \right|  \leq t , \ \  \nu =1, 2 \right\}.  \label{E12}
\ee
On this event, we have $1- t \leq ( \hat{s}_k/s_k)^2 \leq 1+t $ for all $k=1,\ldots,p$. By \eqref{sk2}, we apply the inequality $1+ \frac{1}{2} u - \frac{1}{2} u^2  \leq (1+u)^{1/2} \leq 1+ \frac{1}{2} u$ that holds for all $u\geq -1$ to obtain
$$
    \mathcal{E}_{n,m}(t) \subseteq \left\{  \max_{1\leq k\leq p} \left| \f{\hat{s}_k}{s_k} -1 \right| \leq \f{t(1+t)}{2} \right\}.
$$


Next, let $\{ ( G_{i1}, \ldots, G_{ip})' \}_{i=1}^N$ be independent centred Gaussian random vectors with the same covariance matrix as $\tilde \bxi_i$, and define
\be
    Z^*_{\rs} = \max_{1\leq k\leq p}  N^{-1/2} \sum_{i=1}^N G_{ik} .  \label{two.sample.L2}
\ee
Observe that for each $k$, $N^{-1}\sum_{i=1}^N \mathbb{E}  \tilde \xi_{i k}^2 =1$. Hence we use Lemma~\ref{CCK_2013} to bound the Kolmogorov distance between $T^*_{\rs}$ and $Z^*_{\rs}$, i.e.
\beq
 d_{n,m} =   \sup_{x \in \br} \big| \mathbb{P}\big( T^*_{\rs} \leq x \big) - \mathbb{P}\big(  Z^*_{\rs} \leq x \big) \big|,
\eeq
and the anti-concentration inequality \eqref{con.dif} holds for $Z^*_{\rs}$ given in \eqref{two.sample.L2}.


Conditional on $\mathcal{X}_n$ and $\mathcal{Y}_m$, the $p$-vector $\bW_{\rs}^{\rII}=(W_{\rs,1}^{\rII},\ldots, W_{\rs,p}^{\rII})'$ is centred Gaussian with covariance matrix $\wt{\bR}_{1,2} =\wt{\bD}_{1,2}^{-1/2}  ( \wt{\bSigma}_1+  \lambda \wt{\bSigma}_2  ) \wt{\bD}_{1,2}^{-1/2}$, where $\wt{\bD}_{1,2}  = {\rm diag}\,( \wt{\bSigma}_1+ \lambda \wt{\bSigma}_2  )={\rm diag}\,( \wh{\bSigma}_1+ \lambda \wh{\bSigma}_2 )=\wh{\bD}_{1,2} $. Note that $Z_s^*$ given in \eqref{two.sample.L2} is the maximum of the $p$ coordinates of a centred Gaussian random vector with covariance matrix $\bR_{1,2}$. Once again, using Lemma \ref{CCK_2013_31} gives \beq
   \hat{d}_{n,m} = \sup_{x \in \br} \big| \mathbb{P}\big(  Z_{\rs}^* \leq x \big) - \mathbb{P}\big( Z_{\rs}^+ \leq x
    |  \mathcal{X}_n, \mathcal{Y}_m \big) \big| \leq C   \Theta_{\rs}^{1/3}  \{ 1+ \log(p/\Theta_{\rs} ) \}^{2/3},
\eeq where $Z_{\rs}^+=\max_{1\leq k\leq p} W_{{\rs},k}^{\rII}$ and $\Theta_{\rs}=\Theta_{\rs}^{\rII} = \| \wt{\bR}_{1,2} -\bR_{1,2}\|_{\infty} $.

The rest of the proof is similar to the one-sample case and thus is omitted.
\end{proof}



\subsection{Proof of Theorem~\ref{two-sample.power.1}}


To begin with, we note that inequality \eqref{cv.ubd} holds for ${\rm cv}_{\rs, \alpha}^{\rII}$, the conditional $(1-\alpha)$-quantile of $\mW_{\rs}^{\rII}$. Define $  k_0  = \arg \max_{1\leq k\leq p} \sigma^{-1}_k |\mu_{1 k} -\mu_{2 k}| $ with $\sigma_{k}^2 =\sigma^2_{1 k}/n + \sigma^2_{2 k}/m$. Put $ \hat{\sigma}_{ k}^2 = \hat{\sigma}_{1 k}^2 /n + \hat{\sigma}_{2 k}^2/m $, and assume without loss of generality that $\mu_{1 k_0} > \mu_{2 k_0}$. Then on the event $\mathcal{E}_{n,m}(t)$ given in \eqref{E12} for $0<t\leq \frac{1}{2}$,
\[
\begin{split}
      T_{\rs}^{\rII}& \geq \f{(\bar{X}_{k_0}-\mu_{1 k_0}) -(\bar{Y}_{k_0}-\mu_{2 k_0})  }{\hat{\sigma}_{ k_0}} +  \f{\mu_{1 k_0}-\mu_{2 k_0}}{\hat{\sigma}_{ k_0}}   \\
     &\geq \f{(\bar{X}_{k_0}-\mu_{1 k_0}) -(\bar{Y}_{k_0}-\mu_{2 k_0})  }{\hat{\sigma}_{ k_0}} + (1+t/2)^{-1} \f{\mu_{1 k_0}-\mu_{2 k_0}}{ \sigma_{ k_0}}.
\end{split}
\]
A chain of inequalities that are similar to those in \eqref{power.lbd} hold for $\mathbb{P}\big\{ T_{\rs}^{\rII}>{\rm cv}^{\rII}_{\rs,\alpha} \big\}$, and thus the conclusion in (ii) follows.

For the non-studentized statistic $T_{\ns}^{\rII}$, an argument similar to that leading to \eqref{cv.ubd'} implies $ {\rm cv}_{\ns, \alpha}^{\rII} \leq [  1+ \{ \log( p)\}^{-1} ]  \s{n} \, \la(p,\alpha) \cdot  \max_{1\leq k\leq p} \hat{\sigma}_{ k}$. The rest of the proof is almost immediate. \qed





%===================================================================================================%
%=================================== Need revise ===================================================%
\section{Proof of Theorem~\ref{two-step.one-sample.theory}}

Let $\varrho=\sqrt{2}+\frac{\sqrt{2}}{2\log p}+\sqrt{\frac{2\log(1/\alpha)}{\log p}}$. First we prove the limiting null property for $\Psi^{f,\rI}_{\rs ,\alpha}$. To begin with, define the event $\mathcal{E}_{1n}= \{ \widehat{{\cal S}}_{1}   =  \{1,\ldots,p\} \}$ and observe that under $H_0^{\rI}$, $\mathbb{P}_{H_0^{\rI}}\big( \mathcal{E}_{1n}^{{\rm c}}  \big)  \leq \mathbb{ P}_{H_0^{\rI}} \big\{   T^{\rI}_{\rs}  >  \varrho\sqrt{\log(p)} \big\}$. This, together with \eqref{cv.ubd} and Theorem~\ref{one-sample.size}, implies $\limsup_{n\to \infty}\mathbb{P}_{H_0^{\rI}}\big( \mathcal{E}_{1n}^{{\rm c}} \big)  \leq \alpha$. On the other hand, it is easy to see that $\Psi^{f,\rI}_{\nu ,\alpha}=0$ on the event $\mathcal{E}_{1n}$ for $\nu\in\{\rs,\ns\}$. Together with the inequality $ \mathbb{P}_{H_0^{\rI}} \big\{ \Psi^{f,\rI}_{\nu ,\alpha} =1  \big\} \leq \mathbb{P}_{H_0^{\rI}}  \big\{ \Psi^{f,\rI}_{\nu ,\alpha} =1, \mathcal{E}_{1n} \big\} + \mathbb{P}_{H_0^{\rI}}\big(
\mathcal{E}_{1n}^{{\rm c}} \big)$, this completes the proof of (i).

Next we study the power of $\Psi^{f,\rI}_{\rs ,\alpha}$ under $H^{\rI}_1$. Let $\widehat{T}_{{\rm s}}^{\rI}=\max_{k\notin\widehat{\mathcal{S}}_1}\sqrt{n}|\bar{X}_k|/\hat{\sigma}_{1k}$ and define the event $\mathcal{E}_{2n}= \big\{ T_{\rs}^{\rI} = \widehat{T}_{\rs}^{\rI} \big\}$. By \eqref{cv.ubd} and \eqref{power.lbd}, we have $\lim_{n\rightarrow \infty}\mathbb{P}_{H^{\rI}_1} \big\{ T_{\rs}^{\rI} >  \varrho \sqrt{\log (p)} \big\} \rightarrow 1$, which further implies \be \lim_{n\rightarrow \infty}\mathbb{P}_{H^{\rI}_1} \big( \mathcal{E}_{2n}^{{\rm c}} \big) = 0 .  \label{event.E2n} \ee Further, since $\widehat{{\cal S}}_{1} \subseteq \{1,\ldots,p\}$, $\mathbb{P}\big\{ \max_{ k \notin \widehat{{\cal S}}_{1 }  } |W^{\rI}_{{\rs},k}|  >{\rm cv}_{\rs ,\alpha}^{\rI}  \big\} \leq  \mathbb{P}\big\{   | \bW^{\rI}_{\rs} |_\infty  >{\rm cv}_{\rs ,\alpha}^{\rI}  \big\} \leq \alpha$, and hence $ {\rm cv}_{\rs ,\alpha}^{\rI}(\widehat{{\cal S}}_{1 }) \leq {\rm cv}_{\rs ,\alpha}^{\rI}$. Together
with \eqref{event.E2n}, this leads to
\begin{align*}
    \mathbb{P}_{H^{\rI}_1} \big\{\Psi^{f,\rI}_{\rs ,\alpha} =1 \big\} &  \geq \mathbb{P}_{H^{\rI}_1} \big\{\Psi^{f,\rI}_{\rs ,\alpha} =1 , \, \mathcal{E}_{2n}\big\}    \\
    & \geq \mathbb{P}_{H^{\rI}_1} \big\{ T_{\rs}^{\rI} > {\rm cv}^{\rI}_{\rs,\alpha} , \, \mathcal{E}_{2n}\big\}\\
    & \geq \mathbb{P}_{H^{\rI}_1} \big\{ T_{\rs}^{\rI} > {\rm cv}^{\rI}_{\rs,\alpha} \big\}   - \mathbb{P}_{H^{\rI}_1}\big( \mathcal{E}_{2n}^{{\rm c}} \big) \\
    &  \to 1
\end{align*}
as $n\rightarrow \infty$. Similarly, we can construct the result for its non-studentized analogue $\Psi^{f,\rI}_{\ns ,\alpha}$. Hence, we completes the proof of Theorem~\ref{two-step.one-sample.theory}. \qed

%=============================================================================================%
%=============================================================================================%


%=============================================================================================%
%=============================================================================================%
\section{More simulation results}


\subsection{One-sample case}
In this subsection, we report the empirical powers (Figures \ref{s1} to \ref{s5}) of the proposed tests against different alternatives in Model 1$^{\rm{(I)}}$ to Model 5$^{\rm{(I)}}$ for $p=120,360$, along with those of the test by \cite{ZhongChenXu_2013} (ZCX).




\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p120f1.png}
     \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p360f1.png}\\
         \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p120f2.png}
        \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p360f2.png}\\
         \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p120f3.png}
 \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p360f3.png}\\
  \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p120f4.png}
\includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized with screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample}, along with the power of the test by \cite{ZhongChenXu_2013} (ZCX) at 5\% nominal significance for the Gaussian data and bandable covariance matrices in Model 1$^{\rm{(I)}}$ when $n=80$ and $p=120, 360$.}
  \label{s1}
\end{figure}


%=============================================================================================%


\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S1/M2S1/p120f1.png}
        \includegraphics[height=5cm,width=5cm]{figure/S1/M2S1/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S1/M2S1/p120f2.png}
       \includegraphics[height=5cm,width=5cm]{figure/S1/M2S1/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S1/M2S1/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S1/M2S1/p360f3.png}\\
  \includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p120f4.png}
\includegraphics[height=5cm,width=5cm]{figure/S1/M1S1/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized with screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample}, along with the power of the test by \cite{ZhongChenXu_2013} (ZCX) at 5\% nominal significance for the Gaussian data and bandable covariance matrices in Model 2$^{\rm{(I)}}$ when $n=80$ and $p=120, 360$.}
  \label{s2}
\end{figure}


%=============================================================================================%


%=============================================================================================%

\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p120f1.png}
        \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p120f2.png}
    \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p360f3.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p120f4.png}
 \includegraphics[height=5cm,width=5cm]{figure/S1/M3S1/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample} at 5\% nominal significance for the Gaussian data and long range dependence covariance matrices in Model 3$^{\rm{(I)}}$. For simulations when $n=80$ and $p=120, 360$.}
  \label{s3}
\end{figure}

%=============================================================================================%
\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p120f1.png}
         \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p120f2.png}
      \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p360f3.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p120f4.png} \includegraphics[height=5cm,width=5cm]{figure/S1/M4S1/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample} at 5\% nominal significance for the autoregressive process model, Model 4$^{\rm{(I)}}$, with $t$-distributed innovations when $n=80$ and $p=120, 360$.}
    \label{s4}
\end{figure}
%=============================================================================================%


\begin{figure}[h!]
    \centering
     \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p120f1.png}
       \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p120f2.png}
       \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p120f3.png}
 \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p360f3.png}\\
  \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p120f4.png}
 \includegraphics[height=5cm,width=5cm]{figure/S1/M5S1/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the one-sample problem \eqref{eq:onesample} at 5\% nominal significance for the moving average process with Beta distributed innovations in Model 5$^{\rm{(I)}}$ when $n=80$ and $p=120, 360$.}
  \label{s5}

\end{figure}

%=============================================================================================%
%=============================================================================================%

\subsection{Two-sample case}
In this subsection, we report the empirical powers of the proposed tests (Figures \ref{s11} to \ref{s15}) against different alternatives in Model 1$^{\rm{(II)}}$ to Model 5$^{\rm{(II)}}$ for $p=120,360$, along with those of the tests by \cite{ChenQin_2010} (CQ), \cite{CaiLiuXia_2014} (CLX), and \cite{DelaigleHallJin_2011} (HC2) at 5\% nominal significance.

\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p120f1.png}
       \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p120f2.png}
        \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p360f3.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p120f4.png}
 \includegraphics[height=5cm,width=5cm]{figure/S2/M1S2/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against the alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ) and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance for the Gaussian data and block diagonal covariances in Model 1$^{(\rm{II})}$ when $n=m=80$ and $p=120, 360$.}
  \label{s11}
\end{figure}
%=============================================================================================%


\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p120f1.png}
        \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p120f2.png}
    \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p360f3.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p120f4.png}
 \includegraphics[height=5cm,width=5cm]{figure/S2/M2S2/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized with screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ),  \cite{CaiLiuXia_2014} (CLX) and \cite{DelaigleHallJin_2011} (HC2) at 5\% nominal significance for the Gaussian data and non-sparse covariances in Model 2$^{(\rm{II})}$ when $n=m=80$ and $p=120, 360$.}
  \label{s12}
\end{figure}

%=============================================================================================%



\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p120f1.png}
      \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p120f2.png}
     \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p360f3.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p120f4.png}
 \includegraphics[height=5cm,width=5cm]{figure/S2/M3S2/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ) and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance for the Gaussian data and long range dependence covariances in Model 3$^{(\rm{II})}$ when $n=m=80$ and $p=120, 360$.}
  \label{s13}
\end{figure}

%=============================================================================================%



\begin{figure}[h!]
    \centering
    \includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p120f1.png}
\includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p120f2.png}
     \includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p360f3.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p120f4.png}
 \includegraphics[height=5cm,width=5cm]{figure/S2/M4S2/p360f4.png}
  \caption{
Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the test by \cite{ChenQin_2010} (CQ) at 5\% nominal significance for the autoregressive process model, Model 4$^{\rm{(II)}}$, with $t$-distributed innovations when $n=m=80$ and $p=120, 360$.}
\label{s14}
\end{figure}


%=============================================================================================%


\begin{figure}[h!]
    \centering
     \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p120f1.png}
          \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p360f1.png}\\
    \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p120f2.png}
        \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p360f2.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p120f3.png}
  \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p360f3.png}\\
 \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p120f4.png} \includegraphics[height=5cm,width=5cm]{figure/S2/M5S2/p360f4.png}
  \caption{Empirical powers of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) against alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$) for the two-sample problem \eqref{eq:twosample}, along of those of the tests by \cite{ChenQin_2010} (CQ) and \cite{CaiLiuXia_2014} (CLX) at 5\% nominal significance for the moving average process with Gamma distributed innovations in Model 5$^{\rm{(II)}}$ when $n=m=80$ and $p=120, 360$.}
  \label{s15}
\end{figure}


%=============================================================================================%
%=============================================================================================%

\subsection{Numerical experiments on perfect correlated variables}

We conducted extra numerical experiments to demonstrate that the proposed tests can be applied to perfectly correlated variables. We have the results summarized in Tables \ref{t0} and \ref{t1}. The model under considerations are as following, where the settings are similar to Section \ref{section:numerical}.

\begin{enumerate}
\item[(i)] (One-sample problem): We considered independent random variables $\bX_i=\bGamma\bZ_i$ with $\bGamma\bGamma'=\bSigma_1$ where $\bZ_i$ consists of $p$ independent centred Gamma$(4,1)$ random variables. For the covariance matrix $\bSigma_1=(\sigma_{1,k\ell})_{1\leq k,\ell\leq p}$, let $\sigma_{1,k k}=1$; randomly drew an index set $\mathcal{J}=\{s_1,\cdots, s_{\lfloor p/20\rfloor}\}\subset \{1,\ldots, \lfloor p/10\rfloor\}$,
let $\sigma_{1,k \ell}=1$ for $10(t-1)+1\leq k\neq \ell\leq 10 t$ given $t\in \mathcal{J}$ and
let $\sigma_{1,k \ell}=0.7$ for $10(t-1)+1\leq k\neq \ell\leq 10 t$ when $t\in \{1,\ldots, \lfloor p/10\rfloor\}\setminus\mathcal{J}$; and $\sigma_{1,k \ell}=0$ otherwise.


\item[(ii)] (Two-sample problem):  We considered independent and identically distributed $p$-variate random vectors $\bX_i\sim t_{\omega_1}(\bmu_1,\bSigma_1), \bY_j\sim t_{\omega_2}(\bmu_2,\bSigma_2)$ for $i=1,\cdots, n$ and $j=1,\cdots, m$, where $t_{\omega_1}(\bmu_1,\bSigma_1)$ and $t_{\omega_2}(\bmu_2,\bSigma_2)$ are the non-central multivariate $t$-distributions with non-central parameters $\bmu_1,\bmu_2$, degree freedoms $\omega_1=5, \omega_2=7$. For the covariance matrices $\bSigma_1=(\sigma_{1,k\ell})_{1\leq k,\ell\leq p}$ and $\bSigma_2=(\sigma_{2,k\ell})_{1\leq k,\ell\leq p}$, we randomly drew an index set $\mathcal{J}=\{s_1,\cdots, s_{\lfloor p/20\rfloor}\}\subset \{1,\ldots, \lfloor p/10\rfloor\}$, and let $\sigma_{1,k k}=\sigma_{2,k k}=1$, let $\sigma_{1,k \ell}=\sigma_{2,k \ell}=1$ for $10(t-1)+1\leq k\neq \ell\leq 10 t$ given $t\in \mathcal{J}$ and let $\sigma_{1,k \ell}=0.7$ and $\sigma_{2,k \ell}=0.85$ for $10(t-1)+1\leq k\neq \ell\leq 10 t$ when $t\in \{1,\ldots, \lfloor p/10\rfloor\}\setminus\mathcal{J}$, and $\sigma_{1,k \ell}=\sigma_{2,k \ell}=0$ otherwise.
\end{enumerate}

The alternatives for examining the empirical powers were similar to those in Sections \ref{subsection:numerical one sample} and \ref{subsection:numerical two sample}. That is, under the alternative, there were $\lfloor kp^r \rfloor$ coordinates presenting signals whose strengths are $\{2\beta\sigma_{1,\ell\ell}\log(p)/n\}^{1/2}$ for the one-sample problem and $\{2\beta\sigma_{\ell\ell}\log(p)(1/n+1/m)\}^{1/2}$ with $\sigma_{\ell\ell}$ the $\ell$th diagonal entry of the pooled covariance $\bSigma_{1,2}$. Particularly, we considered four alternatives: $r=0$ (with $k=8$) with $\beta=0.2,0.6$ and $\beta=0.01$ with $r=0.5,0.85$. From these simulation studies, we observe that the proposed tests are applicable to variables with perfect correlations and the empirical performance are consistent with results reported in Section  \ref{section:numerical}.



\begin{table}[h!]
\centering
\def\arraystretch{1.1}
  \begin{tabular}{cccccccc}\toprule
 && $p$ &$\Psi_{\ns,\alpha}$ & $\Psi_{\rs,\alpha}$ & $\Psi_{\ns,\alpha}^f$& $\Psi_{\rs,\alpha}^f$ \\ \midrule
\multirow{3}{*}{Size}&\multirow{3}{*}{$\beta=0$} & 120 &   0.048& 0.054&0.058 & 0.100 \\
& & 360 & 0.038&0.044&0.086 & 0.134 \\
& & 1080 & 0.036&0.048&0.098&0.152  \\\midrule

\multirow{12}{*}{Power}& \multirow{3}{*}{$\beta=0.01, r=0.85$} & 120 & 0.056&0.102&0.152&0.214 \\
& & 360 & 0.038&0.088&0.116&0.248 \\
& & 1080 &  0.060&0.150&0.146&0.320\\[.5ex]  \cline{2-7}

&\multirow{3}{*}{$\beta=0.01, r=0.5$}  & 120 & 0.044&0.088&0.148&0.241   \\
& & 360 &  0.050&0.108&0.136&0.234\\
& & 1080 & 0.024&0.148&0.122&0.334  \\[.5ex]  \cline{2-7}

& \multirow{3}{*}{$\beta=0.2, r=0~ (k=8)$}  & 120 &  0.090&0.118&0.214&0.272  \\
& & 360 & 0.084&0.136&0.206&0.288  \\
& & 1080 &   0.046&0.164&0.178&0.400 \\[.5ex] \cline{2-7}


& \multirow{3}{*}{$\beta=0.6, r=0~ (k=8)$}  & 120 &  0.292&0.330&0.506&0.530  \\
& & 360 &  0.242&0.308&0.470&0.530 \\
& & 1080 &  0.186&0.278&0.416&0.518         \\ \bottomrule
  \end{tabular}
   \caption{Empirical performance of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) for perfectly correlated variables in one-sample problems $H^{\rI}_{0} : \bmu_{1}=\bzero ~~ {versus} ~~ H^{\rI}_{ 1}:\bmu_1\neq\bzero$. The empirical powers are against
the alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$)
at 5\% nominal significance. In the simulation, $n=80$.}\label{t0}
     \end{table}





     \begin{table}[h!]
\centering
\def\arraystretch{1.1}
  \begin{tabular}{ccccccccc}\toprule
 && $p$ &$\Psi_{\ns,\alpha}$ & $\Psi_{\rs,\alpha}$ & $\Psi_{\ns,\alpha}^f$& $\Psi_{\rs,\alpha}^f$ %& CQ & CLX
 \\ \midrule
\multirow{3}{*}{Size}&\multirow{3}{*}{$\beta=0$} & 120 &   0.040&0.070&0.052&0.078%&0.032&0.064
\\
 && 360 & 0.040&0.050&0.048&0.086%&0.050&0.048
 \\
 && 1080 & 0.042&0.090&0.060&0.094%&0.046&0.054
  \\ \midrule

\multirow{8}{*}{Power}&\multirow{2}{*}{$\beta=0.01, r=0.85$}
 & 120 & 0.056 & 0.094 & 0.074 & 0.088 %& 0.046& 0.088
  \\
& & 360 &  0.042&0.084&0.062&0.070%&0.036&0.068
 %\\ & 1080 &  0.056&1&0.076&1&0.054&1
 \\ [.5ex] \cline{2-7}

&\multirow{2}{*}{$\beta=0.01, r=0.5$}
 & 120 & 0.060 & 0.082&0.052&0.060%& 0.034&0.072
   \\
& & 360 &  0.052&0.070&0.046&0.060%&0.048&0.098
% \\ & 1080 &  0.050&0.996&0.034&1&0.044&1
  \\ [.5ex] \cline{2-7}


& \multirow{2}{*}{$\beta=0.2, r=0~ (k=8)$}
 & 120 & 0.092&0.128&0.140&0.198%&0.056&0.128
 \\
 && 360 & 0.084&0.114&0.132&0.182%&0.048&0.098
%  \\ & 1080 &   0.044&0.958&0.052&0.982&0.046&0.934
 \\ [.5ex] \cline{2-7}


 &\multirow{2}{*}{$\beta=0.6, r=0~ (k=8)$}
 & 120 &  0.472&0.564&0.536&0.588%&0.394&0.492
  \\
& & 360 &  0.416&0.520&0.514&0.548%&0.360&0.420
% \\ & 1080 &   0.100&0.988&0.130&0.998&0.050&0.984
         \\ \bottomrule

 \end{tabular}
   \caption{Empirical performance of the proposed tests (non-studentized without screening $\Psi_{\ns,\alpha}$, studentized without screening $\Psi_{\rs,\alpha}$, non-studentized with screening $\Psi_{\ns,\alpha}^f$, and studenzied with screening $\Psi_{\rs,\alpha}^f$) for perfectly correlated variables in two-sample problems $H^{\rII}_{0} : \bmu_{1}=\bmu_{2} ~~ {versus} ~~ H^{\rII}_{ 1}:\bmu_{1}\neq \bmu_{2}$. The empirical powers are against
the alternatives with different levels of the signal strength ($\beta$) and sparsity ($1-r$)
at 5\% nominal significance. In the simulation, $n=m=80$.}\label{t1}
     \end{table}


%=============================================================================================%
%=============================================================================================%

\section{More results on real data analysis}
In the following tables, for the ALL data, we display the top 15 gene-sets in the CC and MF categories that were identified to te the diseases associated by our propose two-step test without studentization, $\Psi_{\ns,\alpha}^f$, but not by the test in \cite{ChenQin_2010}.


\begin{table*}[h!]
\centering

\noindent{ \label{st1}
\begin{tabular}{ll}
\toprule
    GO ID & GO term description \\ \midrule
GO:0000323 &     lytic vacuole \\[.5ex]
GO:0005758& mitochondrial intermembrane space\\[.5ex]
GO:0009295& nucleoid\\[.5ex]
GO:0030863  & cortical cytoskeleton\\[.5ex]
GO:0032991& macromolecular complex\\[.5ex]
GO:1990234&  transferase complex\\[.5ex]
GO:0005765& lysosomal membrane\\[.5ex]
GO:0009898& cytoplasmic side of plasma membrane\\[.5ex]
GO:0016529& sarcoplasmic reticulum\\[.5ex]
GO:0044439& peroxisomal part\\[.5ex]
GO:0048770  &pigment granule\\[.5ex]
GO:0045178&  basal part of cell\\[.5ex]
GO:0005768& endosome\\[.5ex]
GO:0009986& cell surface\\[.5ex]
GO:0031907& microbody lumen
   \\ \bottomrule
\end{tabular}}

 \caption{Top 15 gene-sets in the CC category that were identified to be BCR/ABL associated by the proposed two-step test, $\Psi_{\ns,\alpha}^f$, but not by the test by \cite{ChenQin_2010}. The significance level was at 0.05 and the FDR was controlled at 0.015.
      }
\end{table*}




\begin{table*}[h!]
\centering
 \label{st2}
\noindent{%
\begin{tabular}{ll}
\toprule
    GO ID & GO term description \\ \midrule
GO:0004252  &serine-type endopeptidase activity\\[.5ex]
GO:0015485& cholesterol binding\\[.5ex]
GO:0042625& ATPase activity, coupled to transmembrane movement of ions\\[.5ex]
GO:0042626& ATPase activity, coupled to transmembrane movement of substances\\[.5ex]
GO:0061135& endopeptidase regulator activity\\[.5ex]
GO:1901681& sulfur compound binding \\[.5ex]
GO:0004860  & protein kinase inhibitor activity\\[.5ex]
GO:0008013  &beta-catenin binding\\[.5ex]
GO:0008022& protein C-terminus binding\\[.5ex]
GO:0008528  &G-protein coupled peptide receptor activity\\[.5ex]
GO:0016684  & oxidoreductase activity, acting on peroxide as acceptor\\[.5ex]
GO:0001882  &nucleoside binding \\[.5ex]
GO:0016788  &hydrolase activity, acting on ester bonds\\[.5ex]
GO:0030374  &ligand-dependent nuclear receptor transcription coactivator activity\\[.5ex]
GO:0032182  &ubiquitin-like protein binding
   \\ \bottomrule
\end{tabular}}

 \caption{Top 15 gene-sets in the MF category that were identified to be BCR/ABL associated by the proposed two-step test, $\Psi_{\ns,\alpha}^f$, but not by the test by \cite{ChenQin_2010}. The significance level was at 0.05 and the FDR was controlled at 0.015.
      }
\end{table*}




%===============================================================================================%
\baselineskip=16.5pt
\begin{thebibliography}{1} \itemsep 0.25cm
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\newcommand{\enquote}[1]{``#1''}


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\bibitem[{Chernozhukov, Chetverikov and Kato(2013)}]{ChernozhukovChetverikovKato_2013}
Chernozhukov, V., Chetverikov, D. and Kato, K. (2013).
Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. \AS~{\bf 41}, 2786--2819.

\bibitem[{Chernozhukov, Chetverikov and Kato(2014)}]{ChernozhukovChetverikovKato_2014}
Chernozhukov, V., Chetverikov, D. and Kato, K. (2014).
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\bibitem[{de la Pe\~na, Lai and Shao(2009)}]{PenaLaiShao_2009}
de la Pe\~na, V.~H., Lai, T.~L. and Shao, Q.-M. (2009). {\it Self-Normalized Processes: Limit Theory and Statistical Applications.} Springer, Berlin.




\bibitem[{Einmahl and Li(2008)}]{EinmahlLi_2008}
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Jing, B.-Y., Shao, Q.-M. and Wang, Q. (2003). Self-normalized Cram\'{e}r-type large deviations for independent random variables. \AP~{\bf 31}, 2167--2215.
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\fi
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