11

A test for high‐dimensional covariance matrices
via the extended cross‐data‐matrix methodology
Kohei Endo

Graduate School of Pure and Applied Sciences
University of Tsukuba
Kazuyoshi Yata
Institute of Mathematics

University of Tsukuba
Makoto Aoshima
Institute of Mathematics

University of Tsukuba

Abstract

In this paper, we consider two‐sample tests for covariance matrices in high‐dimensionalsettings. We introduce the extended cross‐data‐matrix (ECDM) methodology. We con‐struct test statistics by using the ECDM methodology. We show that the ECDM test
statistics have the consistency property and the asymptotic normality in high‐dimensionalsettings. We propose a new test procedure based on the ECDM test statistics and evaluate
its asymptotic size and power from theoretical and numerical aspects.

Keywords and phrases: Asymptotic normality; ECDM; HDLSS; Large p , small n

1

Introduction

High‐dimension, low‐sample‐size (HDLSS) data situations occur in many areas of modern
science such as genetic microarrays, medical imaging, text recognition, finance, chemometrics,
and so on. In recent years, substantial work has been done on HDLSS asymptotic theory inwhich the sample size n is fixed or narrow\infty while n/parrow 0 as the data dimension parrow\infty.

Hall et al. [6] and Yata and Aoshima [11] explored several types of geometric representa‐
tions of HDLSS data. Yata and Aoshima [11] developed the noise‐reduction methodology andgave consistent estimators of both the eigenvalues and eigenvectors together with principal

component (PC) scores in the HDLSS context. The HDLSS asymptotic theory was created2
under the assumption that either the population distribution is Gaussian or random vari‐ables in a certain sphered data matrix have the \rho‐mixing dependency. However, Yata andAoshima [10] developed the HDLSS asymptotic theory without such the assumptions. More‐over, they created a new principal component analysis (PCA) called the cross‐data‐matrix(CDM) methodology that is applicable to constructing an unbiased estimator in HDLSS non‐parametric settings. Aoshima and Yata [2] developed a variety of inference for HDLSS datasuch as given‐bandwidth confidence regions, two‐sample tests, tests of the equality of twocovariance matrices, classification, variable selection, regression, tests of the correlation co‐efficients and so on, and also discussed the sample size determination to ensure prespecified

accuracy for each inference. Yata and Aoshima [12] improved the test of the correlation co‐efficients by using the extended cross‐data‐matrix (ECDM) methodology that is an extensionof the CDM method. One of the advantages of the ECDM methodoıogy is to produce anunbiased estimator having small asymptotic variance at a low computational cost.
In this paper, we consider two‐sample tests for high‐dimensional covariance matrices.

Let x_{h1},x_{hn_{h}} be independent and identically distributed (i.i. d. )samples of a p‐‐variate

random variable from populations \pi h(h=1,2) . We assume n_{1}/n_{2}arrow\theta\in(0, \infty) and nh<pfor h=1,2 . We assume that x_{hj} has an unknown mean vector \mu_{h} and unknown covariancematrix \Sigma_{h}(\geq O) for h=1,2 . We consider a test problem as follows:
H_{0}:\Sigma_{1}=\Sigma_{2}vs.

H_{1} : \Sigma_{1}\neq\Sigma_{2} .

(1)

The test problem has been studied in the conventional low‐dimensional settings. In particular,

the likelihood ratio test (LRT) is commonly used and enjoys certain optimality under regular‐
ity conditions, see Anderson [1]. However, in the high‐dimensional settings, the conventional
test procedures such as the LRT perform poorly or are not even defined since the sample
covariance matrix S , the simplest estimator of the population covariance matrix \Sigma , performs

poorly. Li and Chen [8] gave a test statistic based on U‐statistics for high‐dimensional data.
However, it requires high computational cost. Srivastava et al. [9] improved the statistic interms of computational cost. In this paper, we shall produce statistics by the ECDM method
and propose a new test procedure for (1).
The rest of the paper is organized as follows. In Section 2, we state assumptions requiredin the construction of a test procedure for (1). In Section 3, we introduce the ECDM method‐
ology. In Section 4, we produce test statistics for (1) by using the ECDM methodology. We
show that the ECDM test statistics have the consistency property and the asymptotic nor‐mality in high‐dimensional settings. We propose a new test procedure based on the ECDM
test statistics and evaluate its asymptotic size and power theoretically. Finally, in Section 5,
we give simulation studies to check the performance of the proposed test procedure.

2

Assumptions
In this section, we introduce the basic assumptions for the test of hypotheses (1). The

eigen‐decomposition of \Sigma_{h} is given by \Sigma_{h}=H_{h}\Lambda_{h}H_{h}^{T} , where \Lambda_{h}=diag(\lambda_{h1}, \ldots, \lambda_{hp}) is a3diagonal matrix of eigenvalues, \lambda_{h1}\geq
corresponding eigenvectors.
Now, we assume the following model:\geq\lambda_{hp}\geq 0 , and H_{h} is an orthogonal matrix of thex_{hj}=\Gamma_{h}w_{hj}+\mu_{h} ,for h=1,2,

where \Gamma_{h}=(\gamma_{h1}, \ldots, \gamma_{hq_{h} ) is a p\cross q_{h} matrix for some q_{h}>0 such that \Gamma_{h}\Gamma_{h}^{T}=\Sigma_{h} , andw_{hj}=(w_{h1j}, \ldots, w_{hq_{h}j})^{T}, j=1, n_{h} , are i.i. d . random vectors having E(w_{hj})=0 andVar(w_{hj})=I_{q_{h}} . Here, I_{q_{h}} denotes the identity matrix of dimension q_{h} . Let Var(w_{hrj}^{2})=M_{hr},r=1,q_{h};h=1,2 . We assume that \lim\sup_{parrow\infty}M_{hr}<\infty for all h, r . Similar to Aoshima
and Yata [3] and Bai and Saranadasa [4], we assume that

E(w_{hrj}^{2}w_{hsj}^{2})=E(w_{hrj}^{2})E(w_{hsj}^{2})=1 and E(w_{hrj}w_{hsj}w_{htj}w_{huj})=0 for all r\neq s, t,

(A‐i)

u.

We assume the following assumption instead of (A‐i) as necessary:(A‐ii)

E(w_{hr_{1}j}^{\alpha_{1} w_{hr_{2}j}^{\alpha_{2} \cdots w_{hr_{v}j} ^{\alpha_{v} )=E(w_{hr_{1}j}^{\alpha_{1} )E(w_{hr_{2}j}^{\alpha_{2} )\cdots E(w_ {hr_{v}j}^{\alpha_{v} ) for all r_{1}\neq r_{2}\neq[1, q_{h}] and \alpha_{i}\in[1,4], i=1,v

\neq r_{v}\in, where v\leq 8 and \sum_{\iota={\imath} ^{v}\alpha_{i}\leq 8.

See Chen and Qin [5] about (A‐ii). Note that (A‐ii) implies (A‐i). Note that (A‐ii) is naturally

satisfied when x_{hj} is Gaussian. We assume the following assumption for \Sigma_{h} as necessary:( A‐iii)\frac{tr(\Sigma_{h}^{4}){tr(\Sigma_{h}^{2})^{2}ar ow0

Note thatas parrow\infty for h=1,2.

tr(\Sigma_{h}^{4})/tr(\Sigma_{h}^{2})^{2}arrow 0 as parrow\infty for h=1,2 ” is equivalent to \lambda_{h1}/tr(\Sigma_{h}^{2})^{ \imath}/2}arrow 0
h=1,2 ”. Let m= \min\{p, n_{1}, n_{2}\} and \triangle=||\Sigma_{1}-\Sigma_{2}||_{F}^{2}=tr\{(\Sigma_{1}-\Sigma_{2})^{2}\}.

a\mathfrak{Z}parrow\infty for

We assume one of the following two assumptions as necessary:for\frac{tr(\Sigma_{h}^{2}) {n_{h}\triangle}ar ow0 as(A‐v)
1 i_{M}\sup_{ar ow\infty}\{\frac{n_{h}\triangle}{tr(\Sigma_{h}^{2}) \}<\infty for(A‐iv)

3h=1,2 ;

marrow\infty

h=1,2.

ECDM methodology

The ECDM methodology was developed by Yata and Aoshima [12, 13] as an extensionof the CDM method due to Yata and Aoshima [10]. Throughout this section, we omit thesubscript with regard to the population. Let n_{(1)}=\lceil n/2\rceil and n(2)=n-n_{(1)} , where \lceil x\rceil
denotes the smallest integer \geq x . Let

V_{n(1)(k)}=\{ begin{ar ay}{l} \{ lfo rk/2\rflo r-n_{(1)}+1,\lfo rk/2\rflo r\} if\lfo rk/2\rflo r \geqn({\imath}), \{1,\lfo rk/2\rflo r\} cup\{ lfo rk/2\rflo r+n_{(2)}+1,n\} otherwise; \end{ar ay}
V_{n(2)(k)}=\{ begin{ar ay}{l} \{ lfo rk/2\rflo r+1,\lfo rk/2\rflo r+n_{(2)}\ if\lfo rk/2\rflo r\leq n_{(1)}, \{1,\lfo rk/2\rflo r-n_{(1)}\ cup\{ lfo rk/2\rflo r+1,n\} otherwise \end{ar ay}

4
for k=3,2n-1 , whereof elements in a set S .

\lf o r x\rflo r denotes the largest integer \leq x . Let \# S denote the number

Note that\# V_{n(l)(k)}=n_{(l)}, l=1,2, V_{n(1)(k)}\cap V_{n(2)(k)}=\emptyset and
2n-1 . Also, note thati\in V_{n(1)(i+j)} andn} for k=3,
V_{n(1)(k)}\cup V_{n(2)(k)}= {ı,j\in V_{n(2)(i+j)} for i<j(\leq n) . Let\overline{x}_{n(1)(k)}=n_{(1)}^{-1}\sum_{j\in V_{n(1)(k)} x_{j}for k=3,2n-1 .

\overline{x}_{n(2)(k)}=n_{(2)}^{-1}\sum_{j\in V_{n(2)(k)} x_{j}andThen, Yata and Aoshima [12] gave an estimator of tr(\Sigma^{2}) by

W_{n}= \frac{2u_{n} {n(n-1)}\sum_{i<j}^{n}\{(x_{i}-\overline{x}_{n(1)(i+j)}) ^{T}(x_{j}-\overline{x}_{n(2)(i+j)})\}^{2}
where

,

(2)

u_{n}=n_{(1)}n_{(2)}/\{(n_{(1)}-1)(n_{(2)} -{\imath})\} . Note that E(W_{n})=tr(\Sigma^{2}) . Let m_{0}= \min\{p, n\}.

Aoshima and Yata [3] and Yata and Aoshima [13] gave the following result.
Theorem 3.1 ([3, 13]). Assume (A‐i). Then, it holds that as m_{0}arrow\infty

Var( \frac{W_{n} {tr(\Sigma^{2}) =(\frac{4}{n^{2} +\frac{8tr(\Sigma^{4})+ 4\sum_{r- 1}^{q}(M_{r}-2)(\gamma_{r}^{T}\Sigma\gamma_{r})^{2} {tr(\Sigma^{2}) ^{2}n})\{1+o(1)\}.
Remark 1. When x_{j} is Gaussian, it holds that asVar4m_{0}arrow\infty( \frac{W_{n} {tr(\Sigma^{2}) =(\frac{4}{n^{2} +\frac{8tr(\Sigma^{4}) {tr(\Sigma^{2})^{2}n})\{1+o(1)\}.

Two‐sample tests for covariance matrices

In this section, we consider two‐sample tests for covariance matrices using the ECDM
methodology. Let us consider the following hypotheses which are equivalent to (1):H_{0}:\triangle=0vs.

H_{1} : \triangle>0.

Note that \Delta=tr(\Sigma_{1}^{2})+tr(\Sigma_{2}^{2})-2tr(\Sigma_{1}\Sigma_{2}) . Li and Chen [8] proposed a test statistic asfollows:

U=A_{n_{1}}+A_{n_{2}}-2tr(S_{1n_{1}}S_{2n}2),5where S_{hn_{h}} is the sample covariance matrix having E(S_{hn_{h}})=\Sigma_{h} and

A_{n_{h} = \frac{1}{n_{h}(n_{h}-1)}\sum_{j\neq k}^{n_{h} (x_{hj}^{T}x_{hk})^{2} -\frac{2}{n_{h}(n_{h}-1)(n_{h}-2)}\sum_{j\neq k\neq l}^{n_{h} x_{hk}^{T}x_{hj}x_ {hj}^{T}x_{hl}+ \frac{1}{n_{h}(n_{h}-1)(n_{h}-2)(n_{h}-3)}\sum_{j\neq k\neq l\neq l'}^{n_{h} x_{hj}^{T}x_{hk}x_{hl}^{T}x_{hl'}.
Under (A‐ii), ( A‐iii) and some regularity conditions, they showed the following asymptotic
result:\frac{U-\triangle}{2tr(\Sigma_{1}^{2})/n_{1}+2tr(\Sigma_{2}^{2})/n_{2} \Rightar ow N(0,1) as

marrow\infty.

(3)Here, "\Rightarrow denotes the convergence in distribution and N(0,1) denotes a random variable
distributed as the standard normal distribution. However, the computational cost of A_{n_{h}} isof the order, O(pn_{h}^{4}) , which is inappropriate for practical use. On the other hand, Srivastava

et al. [9] modified A_{n_{h}} as\frac{(n_{h}-1)(n_{h}-2)tr(M_{h}^{2})-n_{h}(n_{h}-1)tr(D_{h}^{2})+tr(D_{h}) ^{2} {n_{h}(n_{h}-1)(n_{h}-2)(n_{h}-3)} ( =A_{n_{h}}^{*} , say),where

Y_{h}=(y_{h1}, \ldots, y_{hn_{h}}), y_{hj}=x_{hj}-\overline{x}_{h},j=1,

Y_{h}^{T}Y_{h}, D_{h}=diag(y_{h1}^{T}y_{h1}, \ldots, y_{hn_{h}}^{T}y_{hn_{h}}) , forof the order, O(pn_{h}^{2}) . Letn_{h},\overline{x}_{h}=n_{h}^{-1}\sum_{j=1}^{n_{h} x_{hj}, M_{h}=h=1,2 . The computational cost of A_{n_{h} ^{*} is\sigma U=2(\frac{1}{n_{1}-1}+\frac{1}{n_{2}-1})\frac{(n_{1}-1)tr(\Sigma_{1} ^{2})+(n_{2}-1)tr(\Sigma_{2}^{2})}{n_{1}+n_{2}-2}
and\hat{\sigma}_{U}=2( \frac{1}{n_{1}-} \frac{1}{n_{2}-1})\frac{(n_{1}-1)A_{n_{1} ^{*}+(n_{2}-1)A_{n2}^{*} {n_{1}+ n_{2}-2}.
ı+Then, they gave a test procedure for (1) by
rejecting

H_{0} \Leftrightar ow\frac{U}{\hat{\sigma}_{U} >z_{\alpha} ,(4)where z_{\alpha} is a constant such that P\{N(0,1)>z_{\alpha}\}=\alpha . Then, under (A‐ii) and ( A‐iii), it
holds that as

marrow\infty

Size

Also, we have the following result.

=\alpha+o(1) .

6Theorem 4.1. Assume (A‐ii) and ( A ‐iii). The test by (4) has that as marrow\inftyPower= \Phi(\frac{\triangle}{\sigma_{U} -z_{\alpha})+o(1) ,(5)where \Phi(\cdot) denotes the c.d.f. of N(0,1) .
In this paper, we give a more powerful test statistic compared with U.

4.1

A test statistic based on the ECDM methodology
Now, by using the ECDM methodology, we estimate \triangle by

T=W_{n_{1}}+W_{n_{2}}-2tr(S_{1n_{1}}S_{2n_{2}}),where W_{n_{h}}s are given by (2). Note that E(T)=\triangle . Also, note that the computational costof W_{n_{h}} is of the order, O(pn_{h}^{2}) . Let\sigma^{2}=\sum_{h=1}^{2}(\frac{4}{n_{h}^{2} tr(\Sigma_{h}^{2})^{2}+\frac{8} {n_{h} tr\{(\Sigma_{h}^{2}-\Sigma_{1}\Sigma_{2})^{2}\}+\frac{4}{n_{h} \sum_{r=1} ^{q_{h} (M_{hr}-2)\{\gam a_{hr}^{T}(\Sigma_{1}-\Sigma_{2})\gam a_{hr}\}^{2})+ \frac{8}{n_{1}n_{2} tr(\Sigma_{1}\Sigma_{2})^{2}.

Then, we have the foılowing result.

Lemma 4.1. Assume (A‐i). It holds that as marrow\infty

Var(T)=\sigma^{2}\{1+o(1)\}.
From Lemma 4.1 we have the following result.

Theorem 4.2. Assume (A‐i) and (A‐iv). It holds that as marrow\infty\frac{T}{\Delta}=1+o_{P}(1).

Next, we consider the case when (A‐iv) is not met. Let\sigma_{T}^{2}=\frac{4}{n_{1}^{2} tr(\Sigma_{1}^{2})^{2}+\frac{4}{n_{2}^{2} tr (\Sigma_{2}^{2})^{2}+\frac{8}{n_{1}n_{2} tr(\Sigma_{1}\Sigma_{2})^{2}
and\hat{\sigma}_{T}^{2}=\frac{4}{n_{1}^{2} W_{n_{1} ^{2}+\frac{4}{n_{2}^{2} W_{n_ {2} ^{2}+\frac{8}{n_{1}n_{2} tr(S_{1n1}S_{2n2})^{2}
We have the following result.

7
Lemma 4.2. Assume (A‐i), ( A ‐iii) and (A‐v). It holds that as marrow\infty\sigma^{2}=\sigma_{T}^{2}\{1+o(1)\}.
Note that as

marrow\infty\frac{\hat{\sigma}_{T}^{2} {\sigma_{T}^{2} =1+O_{P(1)}under (A‐i). From Lemma 4.2 we have the following result.

Theorem 4.3. Assume (A‐ii), ( A ‐iii) and (A‐v). It holds that as marrow\infty\frac{T-\triangle}{\hat{\sigma}_{T} =\frac{T-\triangle}{\sigma_{T} +o_{P}(1) \Rightar ow N(0,1).

Now, we consider a more powerful test statistic. Let

c= \max\{\frac{tr(\Sigma_{1}) {tr(\Sigma_{2}) , \frac{tr(\Sigma_{2}) {tr(\Sigma_{1}) \}and\hat{c}=\max\{\frac{tr(S_{1n1}) {tr(S_{2n_{2} )}, \frac{tr(S_{2n2}) {tr(S_{1n1}) \}.

Then, under (A‐i), we have that as marrow\infty\hat{C}=C+O_{P(1)} .

(6)

We propose the following test statistic:
T_{*}=\hat{c}T.

Note that T_{*}\geq T w.p.ı. Also, note that as

marrow\infty

T_{*}=T\{1+o_{P}(1)\}under (A‐i) and H_{0} . Then, from Theorems 4.2 and 4.3, we have the following results.
Corollary 4.1. Assume (A‐i) and (A‐iv). It holds that as marrow\infty\frac{T}{c\triangle}*=1+o_{P}(1).

Corollary 4.2. Assume (A‐ii), ( A ‐iii) and (A‐v). It holds that as marrow\infty\frac{T_{*}/c-\Delta}{\hat{\sigma}_{T} \Rightar ow N(0,1)

.

8
4.2

A more powerful two‐sample test
We give a test procedure for (1) by
rejecting

H_{0} \Leftrightar ow\frac{T}{\hat{\sigma}_{T} *>z_{\alpha} .

(7)Then, we have the following results.

Theorem 4.4. Assume (A‐ii) and ( A ‐iii). The test by (7) has that as marrow\infty
Size=\alpha+o(1)

and

Power

= \Phi(\frac{\triangle}{\sigma_{T} -z_{\alpha}/c)+o(1).

Corollary 4.3. Assume (A‐iv) under H_{1} . Assume also (A‐i). Then, the test by (7) has thatasmarrow\inftyPower=1+o(1) .

Remark 2. We consider testing (1) by (7) with T instead of T_{*} . Then, it has (5) as marrow\infty.
From Theorems 4.1 and 4.4, when c>1 , the asymptotic power of (7) is greater than thatof (4). Thus, we recommend to use the test by (7).
5Simulation studies
In this section, we summarize simulation studies of the findings.

We used computer

simulations to study performances of the test procedures by (4) and (7). Independent pseudo‐
random normal observations were generated from \pi_{h} : N_{p}(0, \Sigma_{h}) for h=1,2 . We set \alpha=0.05
and \Sigma_{1}=B(0.3^{|i-j|^{1/3}})B , where

B=diag[\{0.5+1/(p+1)\}^{1/2}, \{0.5+p/(p+1)\}^{1/2}].
As for the alternative hypothesis, we set \Sigma_{2}=1.2B(0.4^{|i-j|^{1/3}})B . We considered four cases
for p and n_{l}s :

(I)

(II)
(III)
(IV)

p=2^{S}(s=4, \ldots, 10), n_{1}=n_{2}=4\lceil(p/2)^{1/2}\rceil ;p=2^{S}(s=4, \ldots, 10), n_{1}=2\lceil(p/2)^{1/2}\rceil, n_{2}=4\lceil(p/2)^{1/2}\rceil ;p=1000, n_{1}=n_{2}=5s(s=1, \ldots, 10) ;p=1000, n_{1}=4s, n_{2}=8s(s=1, \ldots, 10) .

9g_{2}p

2p(I) p=2^{S}(s=4, \ldots, 10), n_{1}=n_{2}=4\lceil(p/2)^{1/2}\rceil\ulcornerg_{2}p2p(II) p=2^{s}(s=4, \ldots, 10), n_{1}=2\lceil(p/2)^{1/2}\rceil, n_{2}=4\lceil(p/2)^{1/2}\rceil(III) p=1000,n_{1}=n2=5s ( s= ı10)(IV) p=1000, n_{1}=4s, n_{2}=8s(s=1, \ldots, 10)
Figure 1: The performances of the test procedures by (4) and (7). For each panel, the valueof (4) is denoted by the dashed ıine and the value of (7) is denoted by the solid line.

10
For each case, we checked the performance by 2000 replications. We defined P_{r}=1 (or 0 )when H_{0} was falsely rejected (or not) for r=1 ,2000, and defined \overline{\alpha}=\sum_{r=1}^{2000}P_{r}/2000
to estimate the size. We also defined P_{r}=1 (or 0 ) when H_{1} was falsely rejected (or not)for r=1 ,2000, and defined 1- \overline{\beta}=1-\sum_{r=1}^{2000}P_{r}/2000 to estimate the power. Notethat their standard deviations are less than 0.0ıl. In Figure 1, we plotted \overline{\alpha} (left panel)and 1-\overline{\beta} (right panel) in case of (I) to (IV). We observed that both the test proceduresgave preferable performances for the size in (I) to (IV). However, the test procedure by (7)gave better performance compared to (4) with respect to the power. See Section 4.2 for thetheoretical reason.

Appendix
Proof of Theorem 4.1. Note that tr(\Sigma_{1}^{2})=tr(\Sigma_{2}^{2})\{1+o(1)\} as parrow\infty under (A‐v). Also,

note that\sum_{r=1}^{q_{h} \{ gam a_{hr}^{T}(\Sigma_{1}-\Sigma_{2})\gam a_{hr}\ ^{2} \leq [\{\Sigma_{h}(\Sigma_{1}-\Sigma_{2})\}^{2}]\leq\lambda_{h1}^{2}\triangle= o(tr(\Sigma_{h}^{2})\triangle)
tr(8)

under ( A‐iii) for h=1,2 . Then, it holds that \sigma/\sigma U=1+o(1) under ( A‐iii) and (A‐v). Thus,
from Theorems 1 and 2 in Li and Chen [8], under (A‐ii), ( A‐iii) and (A‐v), we have that as
marrow\inftyP( \frac{U}{\hat{\sigma}_{U} >z_{\alpha})=P(\frac{U-\triangle}{\sigma U} >z_{\alpha}-\frac{\triangle}{\sigma_{U} +o_{P}(1)(\begin{ar y}{l \underline{\triangle}-z_{\alpha} \sigma_{U} \end{ar y}) +o(1)
=\Phi

.

On the other hand, from (8), under (A‐ii) and (A‐iv), it holds that Var(U)/\triangle^{2}= o(ı), sothat U/\triangle=1+o_{P}(1) . Then, we have thatP( \frac{U}{\hat{\sigma}_{U} >z_{\alpha})=P(\frac{\triangle}{\sigma_{U} \{1+ o_{P}(1)\}>z_{\alpha})ar ow 1
from the fact that \sigma_{U}/\triangle=o(1) under (A‐iv). Thus, by considering the convergent subse‐\squarequence of \triangle/\sigma_{U} , we can conclude the result.
Proof of Lemma 4.1. Assume (A‐i). Recall that

T=W_{n_{1}}+W_{n_{2}}-2tr(S_{1n1}S_{2n_{2}})and

U=A_{n1}+A_{n2}-2tr(S_{1n_{1}}S_{2n_{2}}) .

11
11
Note that E(T)=\Delta . By noting that Cov(W_{n_{1}}, W_{n_{2}})=0 , it holds that

Var (T)=Var(W_{n_{1}})+Var(W_{n_{2}})+4Var (tr (S_{1n_{1}}S_{2n_{2}}) )-4Cov(W_{n_{{\imath}}}, tr(S_{1n_{1}}S_{2n2}))-4Cov(W_{n_{2}}, tr(S_{1n_{1}} S_{2n_{2}})).

From Theorem 3.1 and (6.2) in [8], we can claim that Var(W_{n_{h}})=Var(A_{n}h)\{1+o(1)\} as marrow\infty for h=1,2 . Also, we can claim that
Cov(W_{n_{h}}, tr(S_{1n_{1}}S_{2n_{2}}))=Cov(A_{n_{h}}, tr(S_{1n_{1}}S_{2n_{2}} ))\{1+\squareo(1)\} for h=1,2 . Then, from (2.5) and (6.ı) in [8], we can conclude the result.

Proof of Theorem 4.2. From (8), under (A‐i) and (A‐iv), it holds that Var(T)/\triangle^{2}that T/ \triangle= l + op(ı). It concludes the result.
Proof of Lemma 4.2. From (8), we can conclude the result.

=o(ı), so\square\squareProof of Theorem 4.3. Similarly to Proof of Lemma 3.1 in Ishii et al. [7], under (A‐i), we canclaim that W_{n_{h}}=A_{n_{h}}+o_{P}(\sigma) as marrow\infty for h=1,2 . From (8), it holds that \sigma=\sigmaT{l + o(ı)}
under ( A ‐iii) and (A‐v). Then, in a way similar to Proof of Theorem 1 in [8], we can conclude
the result.

\squareProofs of Corollary 4.1 and Corollary 4.2. From Theorems 4.2 and 4.3, by using Slutsky’s
theorem, we can conclude the results.

\squareProof of Theorem 4.4. Similarly to Proof of Theorem 4.1, by using Corollary 4.2, we canconclude the result.

\squareProof of Corollary 4.3. By using Corollary 4.1, we can conclude the result.

\square12
Acknowledgements
The research of the second author was partially supported by Grant‐in‐Aid for Young

Scientists (B), Japan Society for the Promotion of Science (JSPS), under Contract Number
26800078. The research of the third author was partially supported by Grants‐in‐Aid for

Scientific Research (A) and Challenging Research (Exploratory), JSPS, under Contract
Numbers 15H01678 and 17K19956.

References[ı] Anderson T. W. (2003). An introduction to multivariate statistical analysis 3rd ed. Wiley.
[2] Aoshima M., Yata K. (2011). Two‐stage procedures for high‐dimensional data. Sequential
Anal. (Editor’s special invited paper) 30, 356‐399.

[3] Aoshima M., Yata K. (2015). Asymptotic normality for inference on multisample, high‐dimensional mean vectors under mild conditions. Methodol. Comput. Appl. Probab. 17,
419‐439.

[4] Bai Z., Saranadasa H. (1996). Effect of high dimension: By an example of a two sample
problem. Statist. Sinica 6, 311‐329.

[5] Chen S. X., Qin Y.‐L. (2010). A two‐sample test for high‐dimensional data with appli‐
cations to gene‐set testing. Ann. Statist. 38, 808‐835.

[6] Hall P., Marron J.S., Neeman A. (2005). Geometric representation of high dimension,
low sample size data. J. R. Statist. Soc. Ser. B 67, 427‐444.

[7] Ishii A., Yata K., Aoshima M. (201\backslash 7) . Equality tests of high‐dimensional covariancematrices under the strongly spiked eigenvalue model, submitted.

[8] Li J., Chen S. X. (2012). Two sample tests for high‐dimensional covariance matrices.
Ann. Statist. 40, 908‐940.

[9] Srivastava M. S., Yanagihara H., Kubokawa T. (2014). Tests for covariance matrices inhigh dimension with less sample size. J. Multivariate Anal. 130, 289‐309.

[ı0] Yata K., Aoshima M. (2010). Effective PCA for high‐dimension, low‐sample‐size datawith singular value decomposition of cross data matrix. J. Multivariate Anal. 101, 2060‐
2077.

[11] Yata K., Aoshima M. (2012). Effective PCA for high‐dimension, low‐sample‐size datawith noise reduction via geometric representations. J. Multivariate Anal. ı05, 193‐215.

[12] Yata K., Aoshima M. (2013). Correıation tests for high‐dimensional data using extended
cross‐data‐matrix methodology. J. Multivariate Anal. ı17, 313‐331.

13[13] Yata K., Aoshima M. (2016). High‐dimensional inference on covariance structures via
the extended cross‐data‐matrix methodology. J. Multivariate Anal. 151, 151‐166.
Institute of Mathematics

University of Tsukuba
Ibaraki 305‐8571

Japan

E‐mail address: yata@math.tsukuba.ac.jp