Multi-sample hypothesis testing of high-dimensional mean vectors under covariance heterogeneity

Abstract

In this paper, we focus on the hypothesis testing problem of the mean vectors of high-dimensional data in the multi-sample case. We propose two maximum-type statistics and apply a parametric bootstrap technique to compute the critical values. Unlike previous hypothesis testing methods that heavily depend on the structural assumptions of the unknown covariance matrix, the proposed methods accommodate a general covariance structure. Additionally, we introduce screening-based testing procedures to enhance the power of our tests. These test procedures do not require the use of approximate limiting distributions for the test statistics. Finally, we obtain and verify the theoretical properties through simulation studies.

A. Appendix

In this section, we present the technical proofs of the results stated in Sect. 3. Before that, we give some auxiliary results for convenient use later.

Lemma 1

(Lemma 3.1 in Chernozhukov et al.(2013) ${\varvec{X}}$ and ${\varvec{Y}}$ are p-dimensional random vectors, we assume that ${\varvec{X}} \sim N\left( {\varvec{0}}, {\varvec{\varSigma }}_{1}\right) , {\varvec{Y}} \sim N\left( {\varvec{0}}, {\varvec{\varSigma }}_{2}\right)$ and there exist constants $C_{1}>c_1>0$ such that for every $1 \le l \le p$, $c_{1} \le \sigma _{1, ll},\ \sigma _{2, ll} \le C_1$, where $\sigma _{1, ll}=diag\left( {\varvec{\varSigma }}_1\right)$ and $\sigma _{2, ll}=diag\left( {\varvec{\varSigma _2}}\right)$. Then, there exists a constant $C'$ depending only on $C_1$ and $c_{1}$ such that

$$\begin{aligned} \sup _{x \in R} \left|P\left( \mathop {\max }\limits _{1 \le l \le p} X_{l} \le x\right) - P\left( \mathop {\max }\limits _{1 \le l \le p} Y_{l} \le x\right) \right|\le C' \varLambda ^{\frac{1}{3}} \left\{ 1\vee \log (p/\varLambda )\right\} ^{\frac{2}{3}}, \end{aligned}$$

where $\varLambda =\left|{\varvec{\varSigma }}_{1}-{\varvec{\varSigma }}_{2}\right|_{\propto }.$

Lemma 2

(Lemma 3 in Chang et al. 2017) Suppose that $n_{i}$, $p>2$ and $\log (p) \le n_{i},$ $\theta _{n_{i}, p}=pn^{1-\frac{r}{2}}_{i}, \nu _{ir}({\varvec{X}})=\mathop {\max }\limits _{1 \le l \le p}(E|D^{-1/2}_{i}X_{i1l}|^{r})^{1/r}, R_{i}=(r_{i,kl}), {\hat{R}}_{i}=({\hat{r}}_{i,kl}), {\hat{r}}_{i,kl}=\sum _{j=1}^{n_i}(X_{ijk}-{\bar{X}}_{ik})(X_{ijl}-{\bar{X}}_{il})/ \sqrt{\sum _{j=1}^{n_i}(X_{ijk}-{\bar{X}}_{ik})^{2}\sum _{j=1}^{n_i}(X_{ijl}-{\bar{X}}_{il})^{2}},$ for all $i=1,\dots , K$ established.

(i) Assume that condition (A.1) holds. Then, there exist constants $C_2, C_3>0$ independent of $n_{i}$ and p such that, with a probability of at least $1-C_2\left\{ n_{i}^{-1}+\theta ^{\frac{2}{(2+r)}}_{n_{i}, p}\right\}$,

$$\begin{aligned}&\max \left( \left|{\varvec{D}}^{-\frac{1}{2}}_i\hat{{\varvec{\varSigma }}}_i {\varvec{D}}_{i}^{-\frac{1}{2}} - {\varvec{R}}_{i} \right|_{\propto }, \left|\hat{{\varvec{R}}}_i-{\varvec{R}}_i\right|_{\propto } \right) \\&\le C_3 \left[ \nu ^{2}_{i4} n_{i}^{-\frac{1}{2}} \left\{ \log (pn_{i})\right\} ^{\frac{1}{2}}+\nu ^{2}_{ir} \theta ^{\frac{2}{(r+2)}}_{n_{i}, p}+\nu ^{2}_{ir} \theta ^{\frac{2}{r}}_{n_{i}, p} \log (p)\right] . \end{aligned}$$

(ii) Assume that condition (A.2) holds. Then, there exist constants $C_4, C_5>0$ independent of $n_{i}$ and p such that, with a probability of at least $1-C_4 n^{-1}_{i},$

$$\begin{aligned} \max \left( \left| {\varvec{D}}^{-\frac{1}{2}}_i\hat{{\varvec{\varSigma }}}_{i} {\varvec{D}}_{i}^{-\frac{1}{2}} - {\varvec{R}}_{i} \right| _{\propto }, \left| \hat{{\varvec{R}}}_i-{\varvec{R}}_i \right| _{\propto }\right) \le C_5 \left[ n_{i}^{-\frac{1}{2}} \left\{ \log (pn_{i})\right\} ^{\frac{1}{2}} +n_{i}^{-1} {\log (pn_{i})}^{\frac{2}{\gamma }}\right] . \end{aligned}$$

(iii) Assume that $\nu _{i4}, i=1, \dots , K$ are uniformly bounded, then for $0 <t \le {\sqrt{n}_{i}},$

$$\begin{aligned} P\left\{ \sqrt{n}_{i} \left| \hat{{\varvec{D}}}^{-\frac{1}{2}}_i \left( \hat{{\varvec{\mu }}}_{i}-{\varvec{\mu }}_{i}\right) \right| _{\propto }\ge t\right\} \le Cp\exp (-ct^2) +n_{i}^{-1}, \end{aligned}$$

where $\hat{{\varvec{D}}}_i=diag\left( {\hat{\sigma }}^2 _{i1},\dots , {\hat{\sigma }}^2 _{ip}\right) ,\ \hat{{\varvec{\mu }}}_{i}=\left( {\bar{X}}_{i1},\dots , {\bar{X}}_{ip}\right) '$ and $C, c>0$ are constants independent of p.

The following states the auxiliary theory of the procedures for testing the equality of means, which is based on the idea of Gaussian approximation. ${\varvec{H}}_1, {\varvec{H}}_2, \dots , {\varvec{H}}_{N}$ are independent random vectors in ${\mathbb {R}}^{Kp}$, suppose each ${\varvec{H}}_i$ is centered and has a finite covariance matrix $E\left[ {\varvec{H}}_i{\varvec{H}}_i'\right]$. Which ${\varvec{H}}_i=\left( H_{i1},\dots , H_{iKp}\right) ',\ {\varvec{B}}_{i}=diag\left( {E\left[ {\varvec{H}}_i{\varvec{H}}_i'\right] }\right) ,$ $\sigma ^2_{il}=var(H_{il})=\sigma _{i, ll}$ and define

$$\begin{aligned} T_0:=\mathop {\max }\limits _{1\le l \le Kp} N^{-\frac{1}{2}}\sum _{i=1}^{N}\frac{H_{il}}{\sigma _{il}}. \end{aligned}$$

Let ${\varvec{G}}_1, \dots , {\varvec{G}}_N$ are a sequence of independent centered Gaussian random vectors in ${\mathbb {R}}^{Kp}$ such that each ${\varvec{G}}_i$ has the same covariance matrix as ${\varvec{H}}_i$ and ${\varvec{G}}_i \sim N\left( {\varvec{0}}, E\left[ {\varvec{H}}_i{\varvec{H}}_i'\right] \right)$. The following $Z_0$ is the Gaussian analogue of $T_0$ can be defined as

$$\begin{aligned} Z_0 :=\mathop {\max }\limits _{1 \le l \le Kp} N^{-\frac{1}{2}} \sum ^{N}_{i=1} \frac{G_{il}}{\sigma _{il}}. \end{aligned}$$

For $l=1,\dots , Kp$ and $r\ge 1,$ define the moments

$$\begin{aligned} \ M_r\left( {\varvec{H}}\right) :=\mathop {\max }\limits _{1 \le l \le Kp}\left( N^{-1}\sum _{i=1}^NE|V_{il}|^r\right) ^{1/r}, \end{aligned}$$

where ${\varvec{V}}_i={\varvec{V}}_{i}({\varvec{H}})=\left( V_{i1},\dots , V_{iKp}\right) '={\varvec{B}}_{i}^{-\frac{1}{2}}{\varvec{H}}_{i}$, $i=1, \dots , N$. For ${0 \le t \le 1}$, set $u\left( t\right) =\max \left( u_X\left( t\right) , u_G\left( t\right) \right)$, which is the maximum  $\left( 1-t\right)$-quantiles of $\mathop {\max }\limits _{1 \le i \le N} |{\varvec{V}}_i |_{\propto }$ and $\mathop {\max }\limits _{1 \le i \le N} \left|{\varvec{B}}_{i}^{-\frac{1}{2}}{\varvec{G}}_i \right|_{\propto }$, respectively.

Lemma 3

(Theorem 2.2 in Chernozhukov et al. 2013) Assume that $\sigma _{i, ll}$ is bounded away from 0 and $\propto$. Then, for any $0 \le t \le 1$,

$$\begin{aligned}&\sup _{x \in R} \left|P\left( T_0 \le x\right) -P\left( Z_0 \le x\right) \right|\nonumber \\&\le C\left[ \left( M_3^{\frac{3}{4}} \vee M_4 ^{\frac{1}{2}}\right) N^{-\frac{1}{8}} \left\{ \log (KpN/t) \right\} ^{\frac{7}{8}} + u\left( t\right) N^{-\frac{1}{2}} \left\{ \log (KpN/t) \right\} ^{\frac{3}{2}}+t\right] , \end{aligned}$$

(5)

where $C>0$ is a constant independent of N, p and t.

Remark 2

The bound of (5) can be easily reduced under some conditions, respectively. According to Markov’s inequality, for $u>0$,

$$\begin{aligned} P\left( \mathop {\max }\limits _{1 \le i \le N}\left|{\varvec{B}}_{i}^{-\frac{1}{2}}{\varvec{G}}_{i}\right|_{\propto }>u\right) \le 2KNp\left( 1-\varPhi \left( u\right) \right) \le u^{-1}\exp \left\{ \log \left( KpN\right) -u^2/2\right\} . \end{aligned}$$

Since

$$\begin{aligned} u_{G}\left( t\right) =\inf \left\{ u\ge 0:P\left( \mathop {\max }\limits _{1 \le i \le N}\left|{\varvec{B}}_{i}^{-\frac{1}{2}}{\varvec{G}}_{i}\right|_{\propto }>u\right) \le t\right\} , \end{aligned}$$

for $0 \le t \le 1,\ u_{G}\left( t\right) \le \sqrt{2\log (KpN/t)}$. If $\mathop {\max }\limits _{1 \le i \le N}\mathop {\max }\limits _{1 \le l \le Kp}\left\{ E(\left|V_{il}\right|^{r})\right\} ^{\frac{1}{r}} \le C^{'}_4$ for $r \ge 4$ and $C^{'}_{4}>0$ holds, according to Markov’s inequality, for all $u>0$, let

$$\begin{aligned} \mu _{N, r}:=\left\{ E(\mathop {\max }\limits _{1 \le i \le N}|{\varvec{V}}_{i}|^{r}_{\propto })\right\} ^{\frac{1}{r}}, \end{aligned}$$

thus $P\left( \mathop {\max }\limits _{1 \le i \le N}|{\varvec{V}}_{i}|_{\propto }>u\right) \le u^{-r}E\left( \mathop {\max }\limits _{1 \le i \le N}|{\varvec{V}}_{i}|^{r}_{\propto }\right) =u^{-r}\mu ^{r}_{N, r}.$ And since

$$\begin{aligned} u_{X}\left( t\right) :=\inf \left\{ u \ge 0:P\left( \mathop {\max }\limits _{1 \le i \le N}|{\varvec{V}}_{i}|_{\propto }\right) >u) \le t\right\} , \end{aligned}$$

hence $u_{X}\left( t\right) \le t^{-\frac{1}{r}} \mu _{N, r}.$ By the inequality that

$$\begin{aligned} \mu _{N, r}=\left\{ E\left( \mathop {\max }\limits _{1 \le i \le N}\left|{\varvec{V}}_{i}\right|^{r}_{\propto }\right) \right\} ^{\frac{1}{r}} \le \left( KpN\right) ^{\frac{1}{r}} \mathop {\max }\limits _{1\le l \le Kp,1 \le i \le N}\left( E\left( \left|{\varvec{V}}_{il}\right|^{r}\right) \right) ^{\frac{1}{r}}. \end{aligned}$$

And taking $t=\min \left\{ 1, \left[ \mu _{N, r}N^{-\frac{1}{2}}\left\{ \log \left( KpN\right) \right\} ^{\frac{3}{2}}\right] ^{\frac{r}{(r+1)}} \right\} ,$ the bound of (5) is

$$\begin{aligned} N^{-\frac{1}{8}}\left\{ \log \left( KpN\right) \right\} ^{\frac{7}{8}}+\theta ^{\frac{1}{(r+1)}}_{N, p}\left\{ \log \left( KpN\right) \right\} ^{\frac{3}{2}}. \end{aligned}$$

(6)

Moreover, if $\mathop {\max }\limits _{1 \le i \le N}\mathop {\max }\limits _{1 \le l \le Kp}E(\exp (C^{'}_{5}\left|V_{il}\right|^{\gamma })) \le C^{'}_{6}$ for some $C^{'}_{5}>0,\ C^{'}_{6}>1$ and $0 < \gamma \le 2$ holds, then $u_{X}(t) \le \left\{ \log \left\{ KpN/t\right\} \right\} ^{\frac{1}{\gamma }}.$ Let $t=N^{-\frac{1}{2}},$ thus the bound of (5) is

$$\begin{aligned} N^{-\frac{1}{8}}\left\{ \log \left( KpN\right) \right\} ^{\frac{7}{8}}+N^{-\frac{1}{2}}\left\{ \log \left( KpN\right) \right\} ^{\frac{3}{2}+\frac{1}{\gamma }}, \end{aligned}$$

(7)

where $\theta _{N, p}=KpN^{1-\frac{r}{2}}.$

In the sequel, we give the proofs of Proposition 1 and Theorems 1–4. For simplicity, we only consider the three-sample case, i.e., $K=3$, and the proofs are analogous when $K >3$.

1.1 A. 1 Proof of Proposition 1

Because the non-studentized statistic $T_{nsmax}$ can be handled similarly, we only provide the proof of the studentized statistic $T_{smax}$ for the sake of clarity. To begin with, observe that for every $x \in {\mathbb {R}}$, $|x|= \max (-x, x)$. For $t>0$ and the definition of $T_{smax}$, the distribution of test statistic can be written as follows:

$$\begin{aligned}&P\left( T_{smax}>t\right) \\ =&P\Bigg (\mathop {\max }\limits _{1 \le l \le p} \Bigg (\frac{|{\bar{X}}_{1l} -{\bar{X}}_{2l}|}{\sqrt{{\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{2l}/n_2}}, \frac{|{\bar{X}}_{1l} -{\bar{X}}_{3l}|}{\sqrt{{\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{3l}/n_3}}, \frac{|{\bar{X}}_{2l} -{\bar{X}}_{3l}|}{\sqrt{{\hat{\sigma }}^2_{2l}/n_2+{\hat{\sigma }}^2_{3l}/n_3}}\Bigg ) \ge t\Bigg ) \\ =&P\Bigg (\mathop {\max }\limits _{1 \le l \le p} \Bigg (\max \Bigg (\frac{{\bar{X}}_{1l}-{\bar{X}}_{2l}}{\sqrt{{\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{2l}/n_2}}, -\frac{{\bar{X}}_{1l}-{\bar{X}}_{2l}}{\sqrt{{\hat{\sigma }}^2_{1k}/n_1+{\hat{\sigma }}^2_{2l}/n_2}}\Bigg ),\\ \;&~~~~~~\max \Bigg (\frac{{\bar{X}}_{1l}-{\bar{X}}_{3l}}{\sqrt{{\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{3l}/n_3}}, -\frac{{\bar{X}}_{1l}-{\bar{X}}_{3l}}{\sqrt{{\hat{\sigma }}^2_{2k}/n_2+{\hat{\sigma }}^2_{3l}/n_3}}\Bigg ), \\ \;&~~~~~~\max \Bigg (\frac{{\bar{X}}_{2l}-{\bar{X}}_{3l}}{\sqrt{{\hat{\sigma }}^2_{2l}/n_2+{\hat{\sigma }}^2_{3l}/n_3}}, -\frac{{\bar{X}}_{2l}-{\bar{X}}_{3l}}{\sqrt{{\hat{\sigma }}^2_{2l}/n_2+{\hat{\sigma }}^2_{3l}/n_3}}\Bigg )\Bigg )\ge t\Bigg ) . \end{aligned}$$

Establish a new set of dilated random vectors ${\varvec{X}}^{e}_{11}, \dots , {\varvec{X}}^{e}_{1n_1}, \ {\varvec{X}}^{e}_{21}, \dots , {\varvec{X}}^{e}_{2n_2},\ {\varvec{X}}^{e}_{31}, \dots , {{\varvec{X}}}^{e}_{3n_3}$ taking values in ${\mathbb {R}}^{2p}$, given by

$$\begin{aligned}&{\varvec{X}}^{e}_{1i}=\left( X^{e}_{1i1}, \dots , X^{e}_{1i2p}\right) '=\left( {\varvec{X}}'_{1i}, -{\varvec{X}}'_{1i}\right) ',\\&{\varvec{X}}^{e}_{2i}=\left( X^{e}_{2i1}, \dots , X^{e}_{2i2p}\right) '=\left( {\varvec{X}}'_{2i}, -{\varvec{X}}'_{2i}\right) ',\\&{\varvec{X}}^{e}_{3i}=\left( X^{e}_{3i1}, \dots , X^{e}_{3i2p}\right) '=\left( {\varvec{X}}'_{3i}, -{\varvec{X}}'_{3i}\right) '. \end{aligned}$$

From this point of view, we have $P\left( T_{smax}>t\right) =P\left( T^{e}_{smax}>t\right) ,$ where

$$\begin{aligned}&{\bar{X}}_{1l}^{e} =n^{-1}_{1} \sum _{i=1}^{n_1}X^{e}_{1il},\quad \left( {\hat{\sigma }}^{e}_{1l}\right) ^2=n^{-1}_1 \sum _{i=1}^{n_1}(X^{e}_{1il}-{\bar{X}}^{e}_{1l})^2,\\ {}&T^{e}_{s1}=\mathop {\max }\limits _{1 \le l \le p}\frac{{\bar{X}}^{e}_{1l}-{\bar{X}}^{e}_{2l}}{\sqrt{({\hat{\sigma }}^{e}_{1l})^2/n_1+({\hat{\sigma }}^{e}_{2l})^2/n_2}};\\&{\bar{X}}_{2l}^{e} =n^{-1}_{2} \sum _{i=1}^{n_2}X^{e}_{2il},\quad \left( {\hat{\sigma }}^{e}_{2l}\right) ^2=n^{-1}_2 \sum _{i=1}^{n_2}\left( X^{e}_{2il}-{\bar{X}}^{e}_{2l}\right) ^2,\\ {}&T^{e}_{s2}=\mathop {\max }\limits _{1 \le l \le p}\frac{{\bar{X}}^{e}_{1l}-{\bar{X}}^{e}_{3l}}{\sqrt{\left( {{\hat{\sigma }}^{e}_{1l}}\right) ^2/n_{1}+\left( {\hat{\sigma }}^{e}_{3l}\right) ^2/n_3}};\\&{\bar{X}}_{3l}^{e} =n^{-1}_{3} \sum _{i=1}^{n_3}X^{e}_{3il},\quad \left( {\hat{\sigma }}^{e}_{3l}\right) ^2=n^{-1}_3 \sum _{i=1}^{n_3}(X^{e}_{3il}-{\bar{X}}^{e}_{3l})^2,\\ {}&T^{e}_{s3}=\mathop {\max }\limits _{1 \le l \le p}\frac{{\bar{X}}^{e}_{2l}-{\bar{X}}^{e}_{3l}}{\sqrt{\left( {\hat{\sigma }}^{e}_{2l}\right) ^2/n_2+({\hat{\sigma }}^{e}_{3l})^2/n_3}};\\&T^{e}_{smax}=\max (T^{e}_{s1},T^{e}_{s2},T^{e}_{s3}). \end{aligned}$$

Without loss of generality, we only need to focus on

$$\begin{aligned}&T^+_{smax}=\mathop {\max }\limits _{1\le l \le p}\left\{ \frac{{\bar{X}}_{1l}-{\bar{X}}_{2l}}{\left( {\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{2l}/n_2\right) ^{1/2}},\ \frac{{\bar{X}}_{1l}-{\bar{X}}_{3l}}{\left( {\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{3l}/n_3\right) ^{1/2}},\ \frac{{\bar{X}}_{2l}-{\bar{X}}_{3l}}{\left( {\hat{\sigma }}^2_{2l}/n_2+{\hat{\sigma }}^2_{3l}/n_3\right) ^{1/2}}\right\} . \end{aligned}$$

Let $N=n_1+n_2+n_3$, denote by

$$\begin{aligned}&s^2_{1l}=\frac{n_1}{N}\left( \sigma ^{2}_{1l}+\frac{n_1}{n_2} \sigma ^{2}_{2l}\right) ,\ {\hat{s}}^2_{1l}=\frac{n_1}{N}\left( {\hat{\sigma }}^{2}_{1l}+\frac{n_1}{n_2} {\hat{\sigma }}^{2}_{2l}\right) ,\ l =1, \dots , p, \\&{\varvec{D}}_1=\begin{bmatrix} s^2_{11} &{} &{} \\ {} &{} \ddots &{} \\ {} &{} &{} s^2_{1p} \end{bmatrix}, \ \hat{{\varvec{D}}}_1=\begin{bmatrix} {\hat{s}}^2_{11} &{} &{} \\ {} &{} \ddots &{} \\ {} &{} &{} {\hat{s}}^2_{1p} \end{bmatrix}. \\&s^2_{2l}=\frac{n_1}{N}\left( \sigma ^{2}_{1l}+\frac{n_1}{n_3} \sigma ^{2}_{3l}\right) ,\ {\hat{s}}^2_{2l}=\frac{n_1}{N}\left( {\hat{\sigma }}^{2}_{1l}+\frac{n_1}{n_3} {\hat{\sigma }}^{2}_{3l}\right) ,\ l=1, \dots , p, \\&{\varvec{D}}_2=\begin{bmatrix} s^2_{21} &{} &{} \\ {} &{} \ddots &{} \\ {} &{} &{} s^2_{2p} \end{bmatrix},\ \hat{{\varvec{D}}}_2=\begin{bmatrix} {\hat{s}}^2_{21} &{} &{} \\ {} &{} \ddots &{} \\ {} &{} &{} {\hat{s}}^2_{2p} \end{bmatrix}. \\&s^2_{3l}=\frac{n_2}{N}\left( \sigma ^{2}_{2l}+\frac{n_2}{n_3} \sigma ^{2}_{3l}\right) ,\ {\hat{s}}^2_{3l}=\frac{n_2}{N}\left( {\hat{\sigma }}^{2}_{2l}+\frac{n_2}{n_3} {\hat{\sigma }}^{2}_{3l}\right) ,\ l=1, \dots , p, \\&{\varvec{D}}_3=\begin{bmatrix} s^2_{31} &{} &{} \\ {} &{} \ddots &{} \\ {} &{} &{} s^2_{3p} \end{bmatrix},\ \hat{{\varvec{D}}}_3=\begin{bmatrix} {\hat{s}}^2_{31} &{} &{} \\ {} &{} \ddots &{} \\ {} &{} &{} {\hat{s}}^2_{3p} \end{bmatrix}. \end{aligned}$$

And

$$\begin{aligned} {\varvec{D}}=\begin{bmatrix}{\varvec{D}}_1 &{} &{} \\ {} &{} {\varvec{D}}_2 &{}\\ {} &{} &{} {\varvec{D}}_3 \end{bmatrix},\ \hat{{\varvec{D}}}=\begin{bmatrix}\hat{{\varvec{D}}}_1 &{} &{} \\ {} &{} {\hat{{\varvec{D}}}_2} &{}\\ {} &{} &{} {\hat{{\varvec{D}}}_3} \end{bmatrix}. \end{aligned}$$

First, we define a pooled sequence of random vectors $\left\{ {\varvec{\alpha }}_i=\left( \alpha _{i1}, \dots , \alpha _{i3p}\right) '\right\} _{i=1}^{N}$ as follows:

$$\begin{aligned}&\begin{aligned}&{\varvec{\alpha }}_{i}=\\&\left\{ \begin{array}{ll} \begin{bmatrix} {\varvec{I}}_{p \times p} \\ {\varvec{I}}_{p \times p} \\ {\varvec{0}}_{p \times p} \end{bmatrix} \begin{bmatrix} {\varvec{X}}_{1i}-{\varvec{\mu }}_1 \end{bmatrix}=\begin{bmatrix} {\varvec{X}}_{1i}-{\varvec{\mu }}_1 \\ {\varvec{X}}_{1i}-{\varvec{\mu }}_1\\ {{\varvec{0}}_{p}} \end{bmatrix}, &{} 1 \le i \le n_1; \\ \begin{bmatrix} -\frac{n_1}{n_2} {\varvec{I}}_{p\times p} \\ {\varvec{0}}_{p \times p} \\ {\varvec{I}}_{p \times p} \end{bmatrix} \begin{bmatrix} {\varvec{X}}_{2(i-n_1)}-{\varvec{\mu }}_2 \end{bmatrix}=\begin{bmatrix} - \frac{n_1}{n_2}\left( {\varvec{X}}_{2(i-n_1)}-{\varvec{\mu }}_2\right) \\ {{\varvec{0}}_{p}} \\ {\varvec{X}}_{2(i-n_1)}-{\varvec{\mu }}_2 \end{bmatrix}, &{} n_1+1 \le i \le n_1+n_2; \\ \begin{bmatrix} {\varvec{0}}_{p \times p}\\ -\frac{n_1}{n_2}{\varvec{I}}_{p \times p} \\ -\frac{n_2}{n_3}{\varvec{I}}_{p \times p}\end{bmatrix} \begin{bmatrix} {\varvec{X}}_{3(i-n_1-n_2)}-{\varvec{\mu }}_3 \end{bmatrix}=\begin{bmatrix} {{\varvec{0}}_{p}} \\ -\frac{n_1}{n_3} \left( {\varvec{X}}_{3(i-n_1-n_2)}-{\varvec{\mu }}_3 \right) \\ -\frac{n_2}{n_3}\left( {\varvec{X}}_{3(i-n_1-n_2)}-{\varvec{\mu }}_3\right) \end{bmatrix},&n_1+n_2+1 \le i \le n_1+n_2+n_3; \end{array} \right. \end{aligned} \end{aligned}$$

$$\begin{aligned}&\quad \tilde{{\varvec{\alpha }}}_{i}={\varvec{D}}^{-1/2} {\varvec{\alpha }}_{i},\ \hat{{\varvec{\alpha }}}_{i}=\hat{{\varvec{D}}}^{-1/2} {\varvec{\alpha }}_{i}. \end{aligned}$$

By the above notation and under the null hypothesis ${\varvec{H}}_0$, we have that

$$\begin{aligned}&T^{*}_{smax}=N^{-1/2}\mathop {\max }\limits _{1 \le l \le 3p}\sum \nolimits_{i=1}^{N}{\tilde{\alpha }}_{il}\\&=\mathop {\max }\limits _{1\le l \le p}\left\{ \frac{{\bar{X}}_{1l}-{\bar{X}}_{2l}}{\left( \sigma ^2_{1l}/n_1+\sigma ^2_{2l}/n_2\right) ^{1/2}},\ \frac{{\bar{X}}_{1l}-{\bar{X}}_{3l}}{\left( \sigma ^2_{1l}/n_1+\sigma ^2_{3l}/n_3\right) ^{1/2}},\ \frac{{\bar{X}}_{2l}-{\bar{X}}_{3l}}{\left( \sigma ^2_{2l}/n_2+\sigma ^2_{3l}/n_3\right) ^{1/2}}\right\} \\ \end{aligned}$$

and

$$\begin{aligned}&T^{+}_{smax}=N^{-1/2}\mathop {\max }\limits _{1 \le l \le 3p}\sum \nolimits_{i=1}^{N}{\hat{\alpha }}_{il}\\&=\mathop {\max }\limits _{1\le l \le p}\left\{ \frac{{\bar{X}}_{1l}-{\bar{X}}_{2l}}{\left( {\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{2l}/n_2\right) ^{1/2}},\ \frac{{\bar{X}}_{1l}-{\bar{X}}_{3l}}{\left( {\hat{\sigma }}^2_{1l}/n_1+{\hat{\sigma }}^2_{3l}/n_3\right) ^{1/2}},\ \frac{{\bar{X}}_{2l}-{\bar{X}}_{3l}}{\left( {\hat{\sigma }}^2_{2l}/n_2+{\hat{\sigma }}^2_{3l}/n_3\right) ^{1/2}}\right\} . \end{aligned}$$

Conditional on $\{{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\},$ the 3p-dimensional random vector ${\varvec{W}}_s=\left( W_{s1}, \dots , W_{s3p}\right) '$ is a centered Gaussian random vector with covariance matrix $\tilde{{\varvec{R}}}_s$. Let

$$\begin{aligned} Z^{+}_{smax}=\mathop {\max }\limits _{1 \le l \le 3p} W_{sl}. \end{aligned}$$

We will demonstrate that the distribution of $T^{+}_{smax}$ is sufficiently close to the conditional distribution of $Z^+_{smax}$ given $\{{\mathbf{\mathcal{X}}}_{1}, {\mathbf{\mathcal{X}}}_2, {\mathbf{\mathcal{X}}}_3\}$ with high probability. We define the following random vectors. When $1 \le i \le n_1,$ let ${\varvec{g}}_i \sim N\left( {\varvec{0}}, {\varvec{\varSigma }}_1\right)$,

$$\begin{aligned} {\varvec{G}}_i=\begin{bmatrix} {\varvec{I}}_{p \times p}\\ {\varvec{I}}_{p \times p}\\ {\varvec{0}}_{p \times p} \end{bmatrix} {\left( {\varvec{g}}_i\right) } \text{ and } \tilde{{\varvec{G}}}_i={\varvec{D}}^{-1/2}{\varvec{G}}_i. \end{aligned}$$

When $n_1+1 \le i \le n_1+n_2,$ let ${\varvec{g}}_i \sim N\left( {\varvec{0}}, {\varvec{\varSigma }}_2\right)$,

$$\begin{aligned} {\varvec{G}}_i=\begin{bmatrix} -\frac{n_1}{n_2}{\varvec{I}}_{p \times p}\\ {\varvec{0}}_{p \times p}\\ {\varvec{I}}_{p \times p} \end{bmatrix}{\varvec{g}}_i \text{ and } \tilde{{\varvec{G}}}_i={\varvec{D}}^{-1/2}{\varvec{G}}_i. \end{aligned}$$

When $n_1+n_2+1 \le i \le n_1+n_2+n_3,$ let ${\varvec{g}}_{i} \sim N\left( {\varvec{0}}, {\varvec{\varSigma }}_{3}\right)$,

$$\begin{aligned} {\varvec{G}}_i=\begin{bmatrix} {\varvec{0}}_{p \times p}\\ \frac{n_1}{n_3} {\varvec{I}}_{p \times p}\\ - \frac{n_2}{n_3}{\varvec{I}}_{p \times p} \end{bmatrix} {\varvec{g}}_i \text{ and } \tilde{{\varvec{G}}}_i={\varvec{D}}^{-1/2}{\varvec{G}}_i. \end{aligned}$$

By the notation of ${\varvec{G}}_i$ and $\tilde{{\varvec{R}}}_1$, we define

$$\begin{aligned} Z^*_{smax}=\mathop {\max }\limits _{1 \le l \le 3p}N^{-\frac{1}{2}} \sum \nolimits_{i=1}^{N}{\tilde{G}}_{il}. \end{aligned}$$

Note that $Z^*_{smax}$ and $T^*_{smax}$ are random variables. By Lemma 3, we have the following Berry–Esseen type bound

$$\begin{aligned} d=\sup _{x \in R} \left|P\left( T^*_{smax}\le x\right) -P\left( Z^*_{smax} \le x\right) \right|, \end{aligned}$$

(8)

where the order of d depends on the moment conditions imposed on ${\varvec{\alpha }}_i$ as described in (6) and (7). For the Gaussian maximum $Z^*_{smax}$ and $Z^+_{smax}$ given above, obtained from Lemma 1 to bound:

$$\begin{aligned} {\hat{d}}=\sup _{x \in R} |P\left( Z^*_{smax}\le x\right) -{P\left( Z^+_{smax} \le x|{\mathbf{\mathcal{X}}}_1, {\mathbf{\mathcal{X}}}_2, {\mathbf{\mathcal{X}}}_3\right) }|\le C\varTheta ^{\frac{1}{3}}_s \left\{ 1+{\log }({3p}/{\varTheta _s})\right\} ^{\frac{2}{3}}, \end{aligned}$$

(9)

where $\varTheta _{s}=|{\tilde{R}}_s-R_{s}|_{\propto }.$ For $0 < t_{12} \le \frac{1}{2}$, define the event

$$\begin{aligned} {\varvec{\varepsilon }}_{n_1, n_2}\left( t_{12}\right) =\left\{ \mathop {\max }\limits _{1 \le l \le p} \left|\frac{{\hat{\sigma }}^2_{\nu l}}{\sigma ^2_{\nu l}}-1\right|\le t_{12}, \nu =1, 2\right\} . \end{aligned}$$

On ${\varvec{\varepsilon }}_{n_1, n_2}\left( t_{12}\right) ,$ we have $1-t_{12} \le \left( {\hat{s}}_{1l}/s_{1l}\right) ^2 \le 1+t_{12}$ for all $l=1, \dots , p$. The inequality $1+\frac{1}{2} \mu -\frac{1}{2} \mu ^2 \le \left( 1+\mu \right) ^{1/2} \le 1+\frac{1}{2} \mu$ holds for $\mu \ge {-1},$ and thus, we can apply the inequality for $0 < t_{12} \le 1/2$ to obtain that

$$\begin{aligned} {\varvec{\varepsilon }}_{n_1, n_2}(t_{12}) \subseteq {\varvec{A}}_{n_1, n_2}(t_{12})=\left\{ \mathop {\max }\limits _{1 \le l \le p} \left|\frac{{\hat{s}}_{1l}}{s_{1l}}-1\right|\le \frac{t_{12}\left( 1+t_{12}\right) }{2}\right\} . \end{aligned}$$

Similarly, we define the event

$$\begin{aligned} {\varvec{\varepsilon }}_{n_1, n_3}(t_{13})=\left\{ \mathop {\max }\limits _{1 \le l \le p} \left|\frac{{\hat{\sigma }}^2_{\nu l}}{\sigma ^2_{\nu l}}-1\right|\le t_{13},\ \nu =1, 3\right\} , \end{aligned}$$

and

$$\begin{aligned} {\varvec{\varepsilon }}_{n_2, n_3}\left( t_{23}\right) =\left\{ \mathop {\max }\limits _{1 \le l \le p} \left|\frac{{\hat{\sigma }}^2_{\nu l}}{\sigma ^2_{\nu l}}-1\right|\le t_{23}, \ \nu =2, 3\right\} . \end{aligned}$$

We have that

$$\begin{aligned} {\varvec{\varepsilon }}_{n_1, n_3}\left( t_{13}\right) \subseteq {\varvec{A}}_{n_{1}, n_{3}}\left( t_{13}\right) =\left\{ \mathop {\max }\limits _{1 \le l \le p} \left|{\hat{s}}_{2l}/s_{2l}-1\right|\le \frac{t_{13}(1+t_{13})}{2}\right\} \end{aligned}$$

and

$$\begin{aligned} {\varvec{\varepsilon }}_{n_2, n_3}({t_{23}}) \subseteq {\varvec{A}}_{n_{2}, n_{3}}\left( t_{23}\right) =\left\{ \mathop {\max }\limits _{1 \le l \le p} \left|{\hat{s}}_{3l}/s_{3l}-1\right|\le \frac{t_{23}(1+t_{23})}{2}\right\} . \end{aligned}$$

Apply Theorem 3 in Chernozhukov et al. (2015),

$$\begin{aligned} P\left( x-\varepsilon <Z^*_{smax} \le x\right) \le 4\varepsilon \left( 1+\left( 2\log \left( 3p\right) \right) ^{1/2}\right) :=\varDelta _{1}. \end{aligned}$$

(10)

Consequently, combination of inequalities (8) (9) (10) gives, for arbitrary $x \in {\mathbb {R}}$ and $\varepsilon >0$,

$$\begin{aligned} \begin{aligned} P\left( T^+_{smax}>x\right)&\le P\left( T^*_{smax}>x-\varepsilon \right) +P\left( \left( T^+_{smax}-T^*_{smax}\right)> \varepsilon \right) \\&\le \sup _{x \in R} \left| P\left( T^*_{smax}>x-\varepsilon \right) -P\left( Z^*_{smax}>x-\varepsilon \right) \right| \\&~~~~+P\left( Z^*_{smax}>x-\varepsilon \right) +P\left( T^+_{smax}-T^*_{smax}>\varepsilon \right) \\&\le d+\varDelta _1+P(Z^*_{smax}>x){+}P((T^+_{smax}-T^*_{smax})>\varepsilon )\\&\le d+\varDelta _1 + {\hat{d}}+P(Z^+_{smax}>x|{\mathbf{\mathcal{X}}}_{1}, {\mathbf{\mathcal{X}}}_{2}, {\mathbf{\mathcal{X}}}_{3})+P((T^+_{smax}-T^*_{smax})>\varepsilon ). \end{aligned} \end{aligned}$$

(11)

From the above inequalities, we can get the following conclusion

$$\begin{aligned}&P\left( T^+_{smax}>x\right) -P\left( Z^+_{smax}>x|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) \\ {}&\le d+{\hat{d}}+\varDelta _1+P\left( \left( T^+_{smax}-T^*_{smax}\right) >\varepsilon \right) . \end{aligned}$$

Similarly, we can get the inverse inequality as follows

$$\begin{aligned}&P\left( Z^+_{smax}>x\right) -P\left( T^+_{smax}>x|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) \le d+{\hat{d}}+\varDelta _1+P\left( \left( T^*_{smax}-T^+_{smax}\right) >\varepsilon \right) . \end{aligned}$$

Thus, we can obtain the following result. For every $x \in {\mathbb {R}}$, we have that

$$\begin{aligned}&\sup _{x \in R}\left| P\left( T^+_{smax}>x\right) -P\left( Z^+_{smax}>x| {\mathbf{\mathcal{X}}}_{1}, \ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) \right| \nonumber \\&\le \varDelta _1+d +{\hat{d}}+{\mathop {\max }\limits \left\{ P\left( \left( T^+_{smax}-T^*_{smax}\right)>\varepsilon \right) , P\left( \left( T^*_{smax}-T^+_{smax}\right) >\varepsilon \right) \right\} .} \end{aligned}$$

(12)

The next major task is to calculate the probability of

$$\begin{aligned} P\left( (T^+_{smax}-T^*_{smax})>\varepsilon \right) \hbox { ~ and~} P\left( (T^*_{smax}-T^+_{smax})>\varepsilon \right) . \end{aligned}$$

Under the null hypothesis, we know that

$$\begin{aligned}{} & {} \begin{aligned}&\left|\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 \sigma ^2_{1l}+n_1 \sigma ^2_{2l}}} \right|\\&\le \mathop {\max }\limits _{1 \le l \le p} \left|{\hat{s}}_{1l}/s_{1l} -1 \right|\sqrt{n_1} \left|\hat{{\varvec{D}}}_{1, 2} \left( \hat{{\varvec{\mu }}}_1-\hat{{{\varvec{\mu }}}}_2\right) \right|_{\propto }, \end{aligned} \end{aligned}$$

(13)

$$\begin{aligned}{} & {} \begin{aligned}&\left|\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}({\bar{X}}_{1l}-{\bar{X}}_{3l})}{\sqrt{n_3 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}({\bar{X}}_{1l}-{\bar{X}}_{3l})}{\sqrt{n_3 \sigma ^2_{1l}+n_1 \sigma ^2_{3l}}} \right|\\&\le \mathop {\max }\limits _{1 \le l \le p} \left|{\hat{s}}_{2l}/s_{2l} -1 \right|\sqrt{n_1} \left|\hat{{\varvec{D}}}_{1, 3} (\hat{{\varvec{\mu }}}_1-\hat{{\varvec{\mu }}}_3)\right|_{\propto }, \end{aligned} \end{aligned}$$

(14)

$$\begin{aligned}{} & {} \begin{aligned}&\left|\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{2l}+n_2 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 \sigma ^2_{2l}+n_2 \sigma ^2_{3l}}} \right|\\&\le \mathop {\max }\limits _{1 \le l \le p} \left|{\hat{s}}_{3l}/s_{3l} -1 \right|\sqrt{n_2} \left|\hat{{\varvec{D}}}_{2, 3} \left( \hat{{\varvec{\mu }}}_2-\hat{{\varvec{\mu }}}_3\right) \right|_{\propto }, \end{aligned} \end{aligned}$$

(15)

where

$$\begin{aligned}&\hat{{\varvec{D}}}_{1, 2}=diag\left( \hat{{\varvec{\varSigma }}}_1+\frac{n_{1}}{n_{2}} \hat{{\varvec{\varSigma }}}_2\right) ,~~ \hat{{\varvec{D}}}_{2, 3}=diag\left( \hat{{\varvec{\varSigma }}}_2+\frac{n_{2}}{n_{3}} \hat{{\varvec{\varSigma }}}_3\right) ,~~ \hat{{\varvec{D}}}_{1, 3}=diag\left( \hat{{\varvec{\varSigma }}}_1+\frac{n_{1}}{n_{3}} \hat{{\varvec{\varSigma }}}_3\right) . \end{aligned}$$

By the above definitions, we obtain that

$$\begin{aligned} \begin{aligned}&P((T^+_{smax}-T^*_{smax})>\varepsilon ) \\&\le P\left( \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-T^*_{smax}>\varepsilon , \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{3l}}}-T^*_{smax}>\varepsilon , \right. \\&~~~~~~~~~\left. \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{3l}+n_3 {\hat{\sigma }}^2_{2l}}}-T^*_{smax}>\varepsilon \right) \\&\le P\left( \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-T^*_{smax}>\varepsilon \right) +P\left( \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{3l}}}-T^*_{smax}>\varepsilon \right) \\&~~~+P\left( \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{2l}+n_2 {\hat{\sigma }}^2_{3l}}}-T^*_{smax}>\varepsilon \right) \\&\le P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 \sigma ^2_{1l}+n_1 \sigma ^2_{2l}}}\right|>\varepsilon \right) \\&~+P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 \sigma ^2_{1l}+n_1 \sigma ^2_{3l}}}\right|>\varepsilon \right) \\&~+P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{2l}+n_2 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 \sigma ^2_{2l}+n_2 \sigma ^2_{3l}}}\right| >\varepsilon \right) . \end{aligned} \end{aligned}$$

Similarly, the following inequalities

$$\begin{aligned} \begin{aligned}&P((T^*_{smax}-T^+_{smax})>\varepsilon )\le P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 \sigma ^2_{1l}+n_1 \sigma ^2_{2l}}}\right|>\varepsilon \right) \\&\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~+P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 \sigma ^2_{1l}+n_1 \sigma ^2_{3l}}}\right|>\varepsilon \right) \\&\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~+P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{2l}+n_2 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 \sigma ^2_{2l}+n_2 \sigma ^2_{3l}}}\right| >\varepsilon \right) \end{aligned} \end{aligned}$$

can be obtained. Thus, $P((T^+_{smax}-T^*_{smax})>\varepsilon )$ and $P((T^*_{smax}-T^+_{smax})>\varepsilon )$ can be controlled by the same boundary. First, we consider the upper bounds of

$$\begin{aligned} P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 \sigma ^2_{1l}+n_1 \sigma ^2_{2l}}}\right| >\varepsilon \right) . \end{aligned}$$

It follows that

$$\begin{aligned}&{P\left( \left|\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2\sigma ^2_{1l}+n_1\sigma ^2_{2l}}} \right|\ge \varepsilon \right) } \nonumber \\&\le P\left( \left\{ \mathop {\max }\limits _{1 \le l \le p} \left|\frac{{\hat{s}}_{1l}}{s_{1l}}-1 \right|\sqrt{n_1} \left|\hat{{\varvec{D}}}^{-1/2}_{1, 2} \left( \hat{{\varvec{\mu }}}_1-\hat{{\varvec{\mu }}}_2\right) \right|_{\propto } \ge \varepsilon \right\} \bigcap \{ {\varvec{A}}_{n_1, n_2}(t_{12})\}\right) +P\left( \{ {\varvec{A}}^{c}_{n_1, n_2}(t_{12})\}\right) \nonumber \\&\le P\left( \mathop {\max }\limits _{1 \le l \le p} \sqrt{n_1} \left|\hat{{\varvec{D}}}^{-1/2}_{1, 2} \left( \hat{{\varvec{\mu }}}_{1}-\hat{{\varvec{\mu }}}_2\right) \right|_{\propto } \frac{t_{12}(1+t_{12})}{2} \ge \varepsilon \right) +P\left( {\varvec{A}}^{c}_{n_1, n_2}\left( t_{12}\right) \right) \nonumber \\&\le P\left( \mathop {\max }\limits _{1 \le l \le p} \sqrt{n_1} \left|\hat{{\varvec{D}}}^{-1/2}_{1, 2} \left( \hat{{\varvec{\mu }}}_{1}-\hat{{\varvec{\mu }}}_2\right) \right|_{\propto } \ge \frac{2 \varepsilon }{t_{12}(1+t_{12})}\right) +P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\left( t_{12}\right) \right) . \end{aligned}$$

(16)

By Lemma 2

$$\begin{aligned} P\left( \mathop {\max }\limits _{1 \le l \le p} \sqrt{n_1} \left|\hat{{\varvec{D}}}_{1, 2} \left( \hat{{\varvec{\mu }}}_{1}-\hat{{\varvec{\mu }}}_2\right) \right|_{\propto } \ge \frac{2 \varepsilon }{t_{12}(1+t_{12})}\right) \le cp\exp \left( -c \frac{4 \varepsilon ^2}{t_{12}^2(1+t_{12})^2}\right) +n_1^{-1}; \end{aligned}$$

(17)

and $P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\left( t_{12}\right) \right) \le P\left( \mathop {\max }\limits _{1 \le l \le p} \left|\frac{{\hat{\sigma }}^2_{1\,l}}{\sigma ^2_{1\,l}}-1\right|> t_{12}\right) +P\left( \mathop {\max }\limits _{1 \le l \le p} \left|\frac{{\hat{\sigma }}^2_{2\,l}}{\sigma ^2_{2\,l}}-1\right|> t_{12}\right) .$ Let

$$\begin{aligned} t_{12}\asymp \max \Big \{&\nu ^2_{14} n_1^{-\frac{1}{2}} \left\{ \log (pn_1)\right\} ^{\frac{1}{2}} +\nu ^2_{1r} \theta ^{\frac{2}{r+2}}_{n_1, p}+\nu ^2_{1r} \theta ^{\frac{2}{r}} _{n_1, p} \log (p),\\&~\nu ^2_{24} n_2^{-\frac{1}{2}} \left\{ \log (pn_2)\right\} ^{\frac{1}{2}} +\nu ^2_{2r} \theta ^{\frac{2}{r+2}}_{n_2, p}+\nu ^2_{2r} \theta ^{\frac{2}{r}} _{n_2, p} \log (p)\Big \}, \end{aligned}$$

we can obtain that

$$\begin{aligned} P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\left( t_{12}\right) \right) \le c\{n_1^{-1}+\theta ^{\frac{2}{r+2}}_{n_1, p}\}+c\{n_2^{-1}+\theta ^{\frac{2}{r+2}}_{n_2, p}\}, \end{aligned}$$

(18)

where $\theta _{n_1, p}=pn^{1-\frac{r}{2}}_1$ and $\theta _{n_2, p}=pn^{1-\frac{r}{2}}_2.$ From (13), (16), (17) and (18), we have that

$$\begin{aligned} \begin{aligned}&{ P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{2l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_2}\left( {\bar{X}}_{1l}-{\bar{X}}_{2l}\right) }{\sqrt{n_2\sigma ^2_{1l}+n_1\sigma ^2_{2l}}} \right| \ge \varepsilon \right) }\\&\le cp \exp \left( -c \frac{4 \varepsilon ^2}{t_{12}^2(1+t_{12})^2}\right) +n_{1}^{-1} +c\left\{ n_{1}^{-1}+\theta _{n_{1}, p}^{\frac{2}{r+2}} \right\} +c\left\{ n_{2}^{-1}+\theta _{n_{2}, p}^{\frac{2}{r+2}} \right\} . \end{aligned} \end{aligned}$$

(19)

The bounds of terms

$$\begin{aligned} \begin{aligned} P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{1l}+n_1 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}\left( {\bar{X}}_{1l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3\sigma ^2_{1l}+n_1\sigma ^2_{3l}}} \right| \ge \varepsilon \right) \end{aligned} \end{aligned}$$

(20)

and

$$\begin{aligned} \begin{aligned}&{P\left( \left| \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3 {\hat{\sigma }}^2_{2l}+n_2 {\hat{\sigma }}^2_{3l}}}-\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_2n_3}\left( {\bar{X}}_{2l}-{\bar{X}}_{3l}\right) }{\sqrt{n_3\sigma ^2_{2l}+n_2\sigma ^2_{3l}}} \right| \ge \varepsilon \right) } \end{aligned} \end{aligned}$$

(21)

are analogous. Combining the results of the three parts, we obtain that

$$\begin{aligned} \begin{aligned} \max \{P((T^+_{smax}&-T^*_{smax})>\varepsilon ),P((T^*_{smax}-T^+_{smax})>\varepsilon )\}\\&\le Cp \exp \left( -c \frac{4 \varepsilon ^2}{t_{12}^2(1+t_{12})^2}\right) +n_{1}^{-1}+c\left\{ n_{1}^{-1}+\theta _{n_1, p}^{\frac{2}{r+2}}\right\} +c\left\{ n_{2}^{-1}+\theta _{n_2, p}^{\frac{2}{r+2}}\right\} \\&+Cp\exp \left( -c \frac{4 \varepsilon ^2}{t_{13}^2(1+t_{13})^2}\right) +n_{1}^{-1}+c\left\{ n_{1}^{-1}+\theta _{n_1, p}^{\frac{2}{r+2}}\right\} +c\left\{ n_{3}^{-1}+\theta _{n_3, p}^{\frac{2}{r+2} }\right\} \\&+Cp\exp \left( -c \frac{4 \varepsilon ^2}{t_{23}^2(1+t_{23})^2}\right) +n_{2}^{-1}+c\left\{ n_{2}^{-1}+\theta _{n_2, p}^{\frac{2}{r+2} }\right\} +c\left\{ n_{3}^{-1}+\theta _{n_3, p}^{\frac{2}{r+2} }\right\} . \end{aligned} \end{aligned}$$

Let $t=\max \left( t_{12}, t_{13}, t_{23}\right)$, we then have that

$$\begin{aligned} \begin{aligned} \max \{P((T^+_{smax}&-T^*_{smax})>\varepsilon ),P((T^*_{smax}-T^+_{smax})>\varepsilon )\}\\ \le&Cp \exp \left( -c \frac{4 \varepsilon ^2}{t^2\left( 1+t\right) ^2}\right) +n_{1}^{-1}+c\left\{ n_{1}^{-1}+\theta _{n_1, p}^{\frac{2}{r+2}}\right\} +n_{1}^{-1}\\&+c\left\{ n_{2}^{-1}+\theta _{n_2, p}^{\frac{2}{(r+2)}}\right\} +n_{2}^{-1}+c\left\{ n_{3}^{-1}+\theta _{n_3, p}^{\frac{2}{r+2} }\right\} . \end{aligned} \end{aligned}$$

(22)

Let $\varepsilon =ct\left( 1+t\right) \left\{ \log (pN)\right\} ^{\frac{1}{2}},$ thus

$$\begin{aligned} \varDelta _{1} \le 4c(1+t)\log \left( 3pN\right) . \end{aligned}$$

By Lemma 3, we can solve for the bound on d:

$$\begin{aligned} d \le C\left( N^{-\frac{1}{8}} \left\{ \log (3pN)\right\} ^{\frac{7}{8}}+\theta ^{\frac{1}{(r+1)}}_{N, p} \left\{ \log (3pN)\right\} ^{\frac{3}{2}}\right) , \end{aligned}$$

where $N=n_1+n_2+n_3,\ \theta _{N, p}=pN^{1-\frac{r}{2}}.$ Substituting the above result into (12) gives the result of Proposition 1 (i). Under condition (A.2), the proof is similar, and thus, it is omitted. $\square$

1.2 A.2 Proof of Theorem 1

The proof of the case of non-studentized test is similar to studentized, and thus, we only give proof of the test based on studentized statistic. Recall that

$$\begin{aligned} c_{s, \alpha } =\inf \left\{ t\in {\mathbb {R}} :P\left( \left|{\varvec{W}}_{s}\right|_{\propto }\ge t|{\mathbf{\mathcal{X}}}_1, \dots , {\mathbf{\mathcal{X}}}_K\right) \le \alpha \right\} , \end{aligned}$$

then it follows from Proposition 1 and Lemma 2 that, under (A.1) or (A.2)

$$\begin{aligned} |P_{{\varvec{H}}_0}\left( T_{smax}> c_{s,\alpha }\right) -P\left\{ |{\varvec{W}}_s|_{\propto } >c_{s, \alpha }|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right\} |{\mathop {\rightarrow }\limits ^{p}} 0, \end{aligned}$$

as $N \rightarrow \infty$ According to Theorem 3 in Chernozhukov et al. (2015), we know the following inequalities hold

$$\begin{aligned} \alpha&\ge P\left( |{\varvec{W}}_s|_{\propto }>c_{s, \alpha }|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) \end{aligned}$$

and

$$\begin{aligned}&P\left( |{\varvec{W}}_s|_{\propto }>c_{s, \alpha }|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) \\&\ge P\left( |{\varvec{W}}_s|_{\propto }>c_{s, \alpha }-N^{-1}|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) -P\left( c_{s, \alpha }-N^{-1} \le |{\varvec{W}}_s|\le c_{s, \alpha }|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right)&\\&\ge \alpha -CN^{-1}\left\{ \log (3p)\right\} ^{\frac{1}{2}}. \end{aligned}$$

Thus, we have that

$$\begin{aligned} \lim _{n_1,n_2, n_3,p\rightarrow \infty } P_{{\varvec{H}}_0}\left( T_{smax} >c_{s, \alpha }\right) = \alpha , \end{aligned}$$

which completes the proof of Theorem 1. $\square$

1.3 A.3 Proof of Theorem 2

For the statistic $T_{nsmax}$, the proof is similar, thus, we only give the proof of the test based on $T_{smax}$. Recall that for $1\le i<j\le 3,$

$$\begin{aligned}&T_{smax}=\begin{bmatrix} \hat{{\varvec{D}}}_{12}\sqrt{n_1n_2}\left( \bar{{\varvec{X}}}_1-\bar{{\varvec{X}}}_2\right) \\ \hat{{\varvec{D}}}_{13}\sqrt{n_1n_3}\left( \bar{{\varvec{X}}}_1-\bar{{\varvec{X}}}_3\right) \\ \hat{{\varvec{D}}}_{23}\sqrt{n_2n_3}\left( \bar{{\varvec{X}}}_2-\bar{{\varvec{X}}}_3\right) \end{bmatrix}_{\propto },~~{\varvec{\varepsilon }}_{n_i, n_j}\left( t_{ij}\right) =\left\{ \mathop {\max }\limits _{1 \le l \le p} |\frac{{\hat{\sigma }}^2_{\nu l}}{\sigma ^2_{\nu l}}-1|\le t_{ij}, \ \nu =i, j\right\} . \end{aligned}$$

Let ${\varvec{W}}_s|{\mathbf{\mathcal{X}}}_1, {\mathbf{\mathcal{X}}}_2, {\mathbf{\mathcal{X}}}_3 \sim N\left( {\varvec{0}}, \tilde{{\varvec{R}}}_s\right)$, Borell (1975) showed that for every $u>0,$

$$\begin{aligned} P\left\{ |{\varvec{W}}_s|_{\propto } \ge E\left( |{\varvec{W}}_s|_{\propto }|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) +u |{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right\} \le \exp \left( -\frac{u^2}{2}\right) . \end{aligned}$$

(23)

From the properties of the distribution of $|{\varvec{W}}_{s}|_{\propto }$, the following inequalities hold

$$\begin{aligned} E\left( |{\varvec{W}}_s|_{\propto }|{\mathbf{\mathcal{X}}}_{1},\ {\mathbf{\mathcal{X}}}_{2},\ {\mathbf{\mathcal{X}}}_{3}\right) \le (1+\left\{ 2\log (3p)\right\} ^{-1})\left\{ 2\log (3p)\right\} ^{\frac{1}{2}}, \end{aligned}$$

which implies

$$\begin{aligned} c_{s, \alpha } \le [1+\left\{ 2\log (3p)\right\} ^{-1}]\sqrt{2\log \left( 3p\right) }+\sqrt{2\log (1/\alpha )}. \end{aligned}$$

(24)

Thus the following inequalities hold:

$$\begin{aligned}&P(T_{smax} \ge c_{s, \alpha })\ge P(T^{+}_{smax}>c_{s, \alpha })\nonumber \\&\ge P\left( T^+_{smax}>\left( 1+\{2\log \left( 3p\right) \}^{-\frac{1}{4}}+\{2\log \left( 3p\right) \}^{-\frac{1}{2}}\right) \eta (p, \alpha ),\right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad ~~~~~~~~~~~~{\varvec{\varepsilon }}_{n_1, n_2}(t_{12}), {\varvec{\varepsilon }}_{n_1, n_3}\left( t_{13}\right) , {\varvec{\varepsilon }}_{n_2, n_3}\left( t_{23}\right) \Big )\nonumber \\&\ge 1-P\left( T^+_{smax}\le \left( 1+\{2\log \left( 3p\right) \}^{-\frac{1}{4}}+\{2\log \left( 3p\right) \}^{-\frac{1}{2}}\right) \eta (p, \alpha )\right) -P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\left( t_{12}\right) \right) \nonumber \\&\qquad -P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_3}\left( t_{13}\right) \right) -P\left( {\varvec{\varepsilon }}^{c}_{n_2, n_3}\left( t_{23}\right) \right) \nonumber \\&\ge 1-P\left( \mathop {\max }\limits _{1 \le l \le p} \frac{ \sqrt{n_1n_2}|{\bar{X}}_{1l}-{\bar{X}}_{2l}|}{n_2{\hat{\sigma }}^2_{1l}+n_1{\hat{\sigma }}^2_{2l}} \le \left( 1+\left\{ 2\log (3p)\right\} ^{-\frac{1}{4}}+\left\{ 2\log \left( 3p\right) \right\} ^{-\frac{1}{2}}\right) \eta (p, \alpha )\right) \nonumber \\&~~~~~-P\left( \mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_1n_3}|{\bar{X}}_{1l}-{\bar{X}}_{3l}|}{n_3{\hat{\sigma }}^2_{1l}+n_1{\hat{\sigma }}^2_{3l}} \le \left( 1+\left\{ 2\log \left( 3p\right) \right\} ^{-\frac{1}{4}}+\{2\log (3p)\}^{-\frac{1}{2}}\right) \eta (p, \alpha )\right) \nonumber \\&~~~~~-P\left( \mathop {\max }\limits _{1 \le l \le p} \frac{ \sqrt{n_2n_3}|{\bar{X}}_{2l}-{\bar{X}}_{3l}|}{n_3{\hat{\sigma }}^2_{2l}+n_2{\hat{\sigma }}^2_{3l}} \le (1+\{2\log (3p)\}^{-\frac{1}{4}}+\{2\log (3p)\}^{-\frac{1}{2}})\eta (p, \alpha )\right) \nonumber \\&~~~~~-P\left( {\varvec{\varepsilon }}^c_{n_1, n_2}\left( t_{12}\right) \right) -P\left( {\varvec{\varepsilon }}^c_{n_1, n_3}\left( t_{13}\right) \right) -P\left( {\varvec{\varepsilon }}^c_{n_2, n_3}\left( t_{23}\right) \right) . \end{aligned}$$

(25)

Define the following equations

$$\begin{aligned} \begin{aligned}&L_{ij}={\underset{1 \le l \le p}{{\arg \max }}}\ \sigma ^{-1}_{lij}|\mu _{il}-\mu _{jl}|,\ \sigma ^2_{lij}=\frac{\sigma ^2_{il}}{n_i}+\frac{\sigma ^2_{jl}}{n_j},\ {\hat{\sigma }}^2_{lij}=\frac{{\hat{\sigma }}^2_{il}}{n_i}+\frac{{\hat{\sigma }}^2_{jl}}{n_j}, 1\le i<j\le 3. \end{aligned} \end{aligned}$$

For $1\le i<j\le 3$, $0 < t_{ij} \le \frac{1}{2}$, assume without loss of generality that $\mu _{iL_{ij}}-\mu _{jL_{ij}}>0,$ then on the event ${\varvec{\varepsilon }}_{n_i, n_j}\left( t_{ij}\right) ,$

$$\begin{aligned} \begin{aligned}&\mathop {\max }\limits _{1 \le l \le p} \frac{\sqrt{n_in_j}|{\bar{X}}_{il}-{\bar{X}}_{jl}|}{\left( n_j{\hat{\sigma }}^2_{il}+n_i{\hat{\sigma }}^2_{jl}\right) ^{\frac{1}{2}}} \ge \frac{\left( {\bar{X}}_{iL_{ij}}-\mu _{iL_{ij}}\right) -\left( {\bar{X}}_{jL_{ij}}-\mu _{jL_{ij}}\right) }{{\hat{\sigma }}_{L_{ij}{ij}}} +\frac{\mu _{iL_{ij}}-\mu _{jL_{ij}}}{{\hat{\sigma }}_{L_{ij}{ij}}}&\\&~~~~~~~~\ge \frac{\left( {\bar{X}}_{iL_{ij}}-\mu _{iL_{ij}}\right) -\left( {\bar{X}}_{jL_{ij}}-\mu _{jL_{ij}}\right) }{{\hat{\sigma }}_{L_{ij}{ij}}} +\left( 1+\frac{1}{2}t_{ij}\right) ^{-1}{\frac{\mu _{iL_{ij}}-\mu _{jL_{ij}}}{\sigma _{L_{ij}{ij}}}}.&\end{aligned} \end{aligned}$$

Combining the above inequalities, let $(1+\{2\log (3p)\}^{-\frac{1}{4}}+\{2\log (3p)\}^{-\frac{1}{2}} )\eta (p, \alpha ))=C_{p, \eta }$, we have that

$$\begin{aligned}&(25)\ge 1-P(\frac{\left( {\bar{X}}_{1L_{12}}-\mu _{1L_{12}}\right) -\left( {\bar{X}}_{2L_{12}}-\mu _{2 L_{12}}\right) }{{\hat{\sigma }}_{L_{12} 12}} +\left( 1+ {t_{12}/2}\right) ^{-1}\frac{\mu _{1L_{12}}-\mu _{2L_{12}}}{{\sigma _{L_{12}{12}}}} \le C_{p, \eta })\nonumber \\&-P(\frac{\left( {\bar{X}}_{1L_{13}}-\mu _{1L_{13}}\right) -\left( {\bar{X}}_{3 L_{13}}-\mu _{3L_{13}}\right) }{{\hat{\sigma }}_{L_{13} {13}}} +\left( 1+{t_{13}/2}\right) ^{-1}\frac{\mu _{1L_{13}}-\mu _{3L_{13}}}{{\sigma _{L_{13}{13}}}} \le C_{p, \eta })\nonumber \\&-P(\frac{\left( {\bar{X}}_{2 L_{23}}-\mu _{2L_{23}}\right) -\left( {\bar{X}}_{3L_{23}}- \mu _{3L_{23}}\right) }{{\hat{\sigma }}_{L_{23} {23}}} +\left( 1+{t_{23}/2}\right) ^{-1}\frac{\mu _{2L_{23}}- \mu _{3L_{23}}}{{\sigma _{L_{23} {23}}}} \le C_{p, \eta })\nonumber \\&-P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\left( t_{12}\right) \right) -P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_3}\left( t_{13}\right) \right) -P\left( {\varvec{\varepsilon }}^{c}_{n_2, n_3}\left( t_{23}\right) \right) . \end{aligned}$$

(26)

Furthermore, we choose $u_{12}=u_{12, n, p},\ u_{13}=u_{13, n, p},\ u_{23}=u_{23, n, p}$ such that

$$\begin{aligned}&1+\xi _{n}=\left( 1+{t_{ij}/2}\right) \left( 1+\{2\log \left( 3p\right) \}^{-\frac{1}{4}}+\left\{ 2\log (3p)\right\} ^ {-\frac{1}{2}}+u_{ij}\right) ,1\le i<j \le 3. \end{aligned}$$

For $\xi _{n}>0$ such that $\xi _{n} \rightarrow 0$ and $\xi _{n}\sqrt{\log \left( 3p\right) } \rightarrow \infty$, $1\le i<j\le 3$, we have that

$$\begin{aligned} \frac{\mathop {\max }\limits _{1 \le l \le p} |\mu _{il}-\mu _{jl}|}{\left( \sigma ^2_{il}/n_i+\sigma ^2_{jl}/n_j\right) ^{\frac{1}{2}}}\ge \left( 1+{t_{ij}/2}\right) \left( 1+\{2\log \left( 3p\right) \}^{-\frac{1}{2}}+\{2\log \left( 3p\right) \}^{-\frac{1}{4}} +u_{ij}\right) \eta (p, \alpha ). \end{aligned}$$

By Lemma 2, the following inequalities are obtained to deduce (26)

$$\begin{aligned}&P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\right) \left( t_{12}\right) +P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_3}\right) \left( t_{13}\right) +P\left( {\varvec{\varepsilon }}^{c}_{n_2, n_3}\right) \left( t_{23}\right) \\&\le c\left\{ n_1^{-1}+\theta ^{\frac{2}{r+2}}_{n_1, p}\right\} +c\left\{ n_2^{-1}+\theta ^{\frac{2}{r+2}}_{n_2, p}\right\} +c\left\{ n_3^{-1}+\theta ^{\frac{2}{r+2}}_{n_3, p}\right\} , \end{aligned}$$

where

$$\begin{aligned}&t_{ij}\asymp \max \Big \{\nu ^2_{i4} n_i^{-\frac{1}{2}} \left\{ \log (pn_i)\right\} ^{\frac{1}{2}} +\nu ^2_{ir} \theta ^{\frac{2}{r+2}}_{n_i, p}+\nu ^2_{ir} \theta ^{\frac{2}{r}} _{n_i, p} \log (p),\\&\nu ^2_{j4} n_j^{-\frac{1}{2}} \left\{ \log (pn_j)\right\} ^{\frac{1}{2}} +\nu ^2_{jr}, \theta ^{\frac{2}{r+2}}_{n_j, p}+\nu ^2_{jr} \theta ^{\frac{2}{r}} _{n_j, p} \log (p)\Big \}, 1\le i<j\le 3; \end{aligned}$$

and under condition (A.1) with $p=O\left( n^{\frac{r}{2}-1}N^{-\delta }\right) ,$ thus $p=O\left( n_{i}^{\frac{r}{2}-1-\delta }\right) , i=1, 2, 3,$ we can get $\theta _{n_1, p}=O\left( n_1^{-\delta }\right) ,\ \theta _{n_2, p}=O\left( n_2^{-\delta }\right) ,\ \theta _{n_3, p}=O\left( n_3^{-\delta }\right)$. Under these conditions, for $1\le i<j \le 3$, let $t_{ij}\asymp \left\{ n_i^{-\frac{2\delta }{\left( r+2\right) }}, n_j^{-\frac{2\delta }{(r+2)}}\right\} ,$ we have that

$$\begin{aligned}&P\left( {\varvec{\varepsilon }}_{n_i, n_j}^c\left( t_{ij}\right) \right) \le C\left\{ n_i^{-1}+n_i^{-\frac{2\delta }{(r+2)}}\right\} +C\left\{ n_j^{-1}+n_j^{-\frac{2\delta }{(r+2)}}\right\} . \end{aligned}$$

Summing up the above equations, we have the following inequalities

$$\begin{aligned} (26) \ge 1&-P\left( \frac{\left( {\bar{X}}_{1L_{12}}-\mu _{1L_{12}}\right) -\left( {\bar{X}}_{2L_{12}}-\mu _{2L_{12}}\right) }{{\hat{\sigma }}_{L_{12}{12}}} \le -u_{12} \eta (p, \alpha )\right) \\&-P\left( \frac{\left( {\bar{X}}_{1L_{13}}-\mu _{1L_{13}}\right) -\left( {\bar{X}}_{3L_{13}}-\mu _{3L_{13}}\right) }{{\hat{\sigma }}_{L_{13}{13}}} \le -u_{13} \eta (p, \alpha )\right) \\&-P\left( \frac{\left( {\bar{X}}_{2L_{23}}-\mu _{2L_{23}}\right) -\left( {\bar{X}}_{3L_{23}}-\mu _{3L_{23}}\right) }{{\hat{\sigma }}_{L_{23}{23}}} \le -u_{23} \eta (p, \alpha )\right) \\&-P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\left( t_{12}\right) )-P({\varvec{\varepsilon }}^{c}_{n_1, n_3}\left( t_{13}\right) )-P({\varvec{\varepsilon }}^{c}_{n_2, n_3}\left( t_{23}\right) \right) \\ \ge 1&-C_{21}e^{-cu_{12}^2 log\left( p\right) }-C_{22}e^{-cu_{13}^2 log\left( p\right) }-C_{23}e^{-cu_{23}^2 log\left( p\right) }-n_1^{-1}n_2^{-1}-n_2^{-1}n_3^{-1}\\&-n^{-1}_1n^{-1}_3-P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\right) \left( t_{12}\right) -P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_3}\right) \left( t_{13}\right) -P\left( {\varvec{\varepsilon }}^{c}_{n_2, n_3}\right) \left( t_{23}\right) \rightarrow 1. \end{aligned}$$

Under (A.2), for $1\le i<j\le 3$, let

$$\begin{aligned}&t_{ij} \asymp \max \{ n_i^{-\frac{1}{2}}\left( \log \left( pn_i\right) \right) ^{\frac{1}{2}}+n_i^{-1}\left( \log \left( pn_i\right) \right) ^{\frac{2}{\gamma }}, n_j^{-\frac{1}{2}}\left( \log \left( pn_j\right) \right) ^{\frac{1}{2}}+n_j^{-1}\left( \log \left( pn_j\right) \right) ^{\frac{2}{\gamma }}\}, \end{aligned}$$

thus

$$\begin{aligned}&P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_2}\left( t_{12}\right) \right) +P\left( {\varvec{\varepsilon }}^{c}_{n_1, n_3}\left( t_{13}\right) \right) +P\left( {\varvec{\varepsilon }}^{c}_{n_2, n_3}\left( t_{23}\right) \right) \le cn_1^{-1}+ cn_2^{-1}+cn_3^{-1}. \end{aligned}$$

Following from (25) and (26), we can get the conclusion. $\square$

1.4 A.4 Proof of Theorem 3

Define the event ${\mathcal {B}}_{1n}=\left\{ \hat{{\varvec{S}}}=\left\{ 1,\dots ,p \right\} \right\}$, ${\mathcal {B}}_{2n}=\{T_{smax}>(1+\{2\log \left( 3p\right) \}^{-1})\sqrt{2\log \left( 3p\right) }+\sqrt{0.0002 \log \left( 1/\alpha \right) }\}$ and ${\varvec{A}}= \left\{ T_{smax}>\left( 1+\left\{ 2\log \left( 3p\right) \right\} ^{-1}\right) \sqrt{2\log \left( 3p\right) }+\sqrt{2 \log \left( 1/\alpha \right) }\right\}$. Under ${\varvec{H}}_0$, we have that $P_{{\varvec{H}}_0}\left( {\mathcal {B}}^{c}_{1n}\right) \le P_{{\varvec{H}}_0}\left( {\mathcal {B}}_{2n}\right) .$ By the fact that

$$\begin{aligned} P_{{\varvec{H}}_0}\left( {\mathcal {B}}_{2n}\right) {=} P_{{\varvec{H}}_0}({\mathcal {B}}_{2n}\cap {\varvec{A}})+P_{{\varvec{H}}_0}({\mathcal {B}}_{2n}\cap {\varvec{A}}^c), \end{aligned}$$

and

$$\begin{aligned} P_{{\varvec{H}}_0}\left\{ \left\{ T^{s}_{smax} >c^s_{s, \alpha }\right\} , {{\mathcal {B}}_{1 n}}\right\} =0, \end{aligned}$$

we can obtain

$$\begin{aligned}&P_{{\varvec{H}}_0}\left\{ T^{s}_{smax}>c^s_{s, \alpha }\right\} \le P_{{\varvec{H}}_0}\left\{ \left\{ T^{s}_{smax}>c^s_ {s, \alpha }\right\} , {{\mathcal {B}}_{1 n}}\right\} +P_{{\varvec{H}}_0}\left\{ {{\mathcal {B}}^{c}_{1 n}}\right\} ,\\&\le P_{{\varvec{H}}_0}\left\{ \left\{ T^{s}_{smax} >c^s_{s, \alpha }\right\} , {{\mathcal {B}}_{1 n}}\right\} +P_{{\varvec{H}}_0}\left\{ {{\mathcal {B}}_{2n}}\right\} \le P_{{\varvec{H}}_0}({\mathcal {B}}_{2n}\cap {\varvec{A}})+P_{{\varvec{H}}_0}({\mathcal {B}}_{2n}\cap {\varvec{A}}^c). \end{aligned}$$

Thus, we have that

$$\begin{aligned} \limsup _{n_1,n_2, n_3,p \rightarrow \infty } P_{{\varvec{H}}_0}\left( T^{s}_{smax} >c^s_{s, \alpha }\right) \le \alpha + o_{p}(1). \end{aligned}$$

The proof of the non-studentized analog ${\varvec{T}}^{s}_{nsmax}$ can be constructed similarly, and thus, we complete the proof of Theorem 3. $\square$

1.5 A.5 Proof of Theorem 4

Define the set $\mathbf {{\mathcal{B}}}_{3n}=\left\{ T_{smax}=T^{s}_{smax}\right\}$, and we know that

$$\begin{aligned} T^{s}_{smax}=\mathop {\max }\limits _{l \notin \hat{{\varvec{S}}}} \left\{ \frac{\sqrt{n_1n_2}|{\bar{X}}_{1l}-{\bar{X}}_{2l}|}{\left( n_2{\hat{\sigma }}^{2}_{1l}+n_1{\hat{\sigma }}^{2}_{2l}\right) }, \frac{\sqrt{n_2n_3}|{\bar{X}}_{2l}-{\bar{X}}_{3l}|}{\left( n_3{\hat{\sigma }}^{2}_{2l}+n_2{\hat{\sigma }}^{2}_{3l}\right) }, \frac{\sqrt{n_1n_3}|{\bar{X}}_{1l}-{\bar{X}}_{3l}|}{\left( n_3{\hat{\sigma }}^{2}_{1l}+n_1{\hat{\sigma }}^{2}_{3l}\right) }\right\} . \end{aligned}$$

Because

$$\begin{aligned} c_{s, \alpha } \le \left( 1+\{2\log (3p)\}^{-1}\right) \sqrt{2\log (3p)}+\sqrt{2\log (1/\alpha )} \end{aligned}$$

and $\lim _{n_1, n_2, n_3, p \rightarrow \infty }P_{{\varvec{H}}_1}\left( T_{smax}>c_{s, \alpha }\right) =1,$ we have that

$$\begin{aligned} \lim _{n_1, n_2, n_3, p \rightarrow \infty } P_{{\varvec{H}}_1}\left( T^{s}_{smax}>\left( 1+\{2\log (3p)\}^{-1}\right) \sqrt{2\log (3p)}+\sqrt{2\log (1/\alpha )}\right) =1, \end{aligned}$$

which implies $\lim _{n_1, n_2, n_3,p \rightarrow \infty } P_{{\varvec{H}}_1}\left( \mathbf {{\mathcal{B}}}^{c}_{3n}\right) =0.$ Because $\hat{{\varvec{S}}}_{all}\subseteq \left\{ 1, \dots , 3p\right\}$ and the following three equations

$$\begin{aligned} P(\mathop {\max }\limits _{l \notin \hat{{\varvec{S}}}_{all}}\left\{ |{W_{sl}}|>c_{s, \alpha }\right\} ) \le P(|{\varvec{W}}_{s}|_{\propto }>c_{s, \alpha }),~~c_{s,\alpha }=\inf \{t \in R: P({|{\varvec{W}}_{s}|}_{\propto } \ge t) \le \alpha \},\\ c_{s,\alpha }^{s}=\inf \left\{ t \in R: P(\mathop {\max }\limits _{l \notin \hat{{\varvec{S}}}_{all}}{|W_{sl}|} \ge t) \le \alpha \right\} , \end{aligned}$$

we have that $c^s_{s, \alpha } \le c_{s, \alpha }.$ Then, we know that

$$\begin{aligned}&P_{{\varvec{H}}_1}\left( T^{s}_{smax}>c^{s}_{s, \alpha }\right) \ge P_{{\varvec{H}}_1} \left\{ \left( T^{s}_{smax}>c^{s}_ {s, \alpha }\right) ,\ \mathbf {{\mathcal{B}}}_{3n}\right\} \\ \ge&P_{{\varvec{H}}_1} \left\{ T_{smax}>c_{s, \alpha },\ \mathbf {{\mathcal{B}}}_{3n}\right\} \ge P_{{\varvec{H}}_1}\left\{ T_{smax}>c_{s, \alpha }\right\} -P_{{\varvec{H}}_1}\left\{ \mathbf {{\mathcal{B}}}^{c}_{3n}\right\} \rightarrow 1, \end{aligned}$$

as $n_1, n_2, n_3, p \rightarrow \infty .$ Similarly, the proof for $T^{s}_{nsmax}$ is similar, and then, we complete the proof of this theorem. $\square$