TEST FOR HIGH DIMENSIONAL CORRELATION MATRICES

Shurong Zheng; Guanghui Cheng; Jianhua Guo; Hongtu Zhu

doi:10.1214/18-AOS1768

. Author manuscript; available in PMC: 2020 Oct 1.

Published in final edited form as: Ann Stat. 2019 Aug 3;47(5):2887–2921. doi: 10.1214/18-AOS1768

TEST FOR HIGH DIMENSIONAL CORRELATION MATRICES

Shurong Zheng ^1,^*, Guanghui Cheng ², Jianhua Guo ^3,^†, Hongtu Zhu ^4,^‡

PMCID: PMC6709985 NIHMSID: NIHMS1000738 PMID: 31452551

Abstract

Testing correlation structures has attracted extensive attention in the literature due to both its importance in real applications and several major theoretical challenges. The aim of this paper is to develop a general framework of testing correlation structures for the one-, two-, and multiple sample testing problems under a high-dimensional setting when both the sample size and data dimension go to infinity. Our test statistics are designed to deal with both the dense and sparse alternatives. We systematically investigate the asymptotic null distribution, power function, and unbiasedness of each test statistic. Theoretically, we make great efforts to deal with the non-independency of all random matrices of the sample correlation matrices. We use simulation studies and real data analysis to illustrate the versatility and practicability of our test statistics.

Keywords: MSC 2010 subject classifications, Primary 62H15, secondary 62H10, Dense alternatives, Global testing, Sample correlation matrices, Sparse alternatives

1. Introduction.

Consider random samples obtained from K independent populations. Let z^(ℓ) be a p−dimensional random vector for ℓ = 1, …, K. We denote $z_{1}^{(ℓ)}, \dots, z_{n_{ℓ}}^{(ℓ)}$ to be the n_ℓ independent samples of z(^ℓ) for the ℓ−th population and ${\bar{z}}^{(ℓ)} = {({\bar{z}}_{1}^{(ℓ)}, \dots, {\bar{z}}_{p}^{(ℓ)})}^{T} = n_{ℓ}^{- 1} \sum_{i = 1}^{n_{ℓ}} z_{i}^{(ℓ)}$ as its sample mean. Then, the sample covariance matrix and sample correlation matrix of $\{z_{i}^{(ℓ)} = {(z_{1 i}^{(ℓ)}, \dots, z_{p i}^{(ℓ)})}^{T} : i = 1, \dots, n_{ℓ}\}$ are, respectively, given by

S_{ℓ} = {(n_{ℓ} - 1)}^{- 1} \sum_{i = 1}^{n_{ℓ}} (z_{i}^{(ℓ)} - {\bar{z}}^{(ℓ)}) {(z_{i}^{(ℓ)} - {\bar{z}}^{(ℓ)})}^{T} and {\hat{R}}_{ℓ} = {[diag (S_{ℓ})]}^{- 1 / 2} S_{ℓ} {[diag (S_{ℓ})]}^{- 1 / 2},

where diag(S_n) is a diagonal matrix constructed from the diagonal elements of S_ℓ. There has been growing interest in the development of methods and theory for hypothesis testing on correlation structures ${\{R_{ℓ}\}}_{ℓ = 1}^{K}$ in different settings (Kullback, 1967; Aitkin, 1969; Jennrich, 1970; Schott, 1996; Browne, 1978; Cole, 1968; Schott, 2005; Gao et al., 2017; Zhou et al., 2015; Debashis and Alexander, 2014). See, for example Anderson (2003) and Cai (2017) for overviews of statistical challenges associated with such developments.

1.1. Existing Literature.

Under the classical setting with fixed p as min_ℓ{n_ℓ} → ∞, there are three major testing problems corresponding to K = 1, K = 2, and K > 2, respectively. As K = 1, it is one sample testing problem that focuses on testing H₀₁ : R₁ = R_∗ against H_A1 : R₁ ≠ R_∗, where R₁ is the population correlation matrix and R_∗ is a specific correlation matrix. An interesting asymptotic result is that the test statistic

(n_{1} - 1) \log (| R_{*} | / {| \hat{R}}_{1} |) - p + tr (R_{*}^{- 1} {\hat{R}}_{1})

is asymptotically distributed as a linear form in 0.5p(p−1) independent $χ_{1}^{2}$ random variables, and not in general $χ_{0.5 p (p - 1)}^{2}$ unless R_∗ = I_p (Kullback, 1967; Aitkin, 1969; Bartlett and Rajalakshman, 1953), where I_p is the p×p identity matrix. This fact shows that testing the correlation matrix is a more difficult task than testing the covariance matrix. As K = 2, it is two sample testing problem that tests H₀₂ : R₁ = R₂ against H_A2 : R₁ ≠ R₂, where R₁ and R₂ are two population correlation matrices. Several test statistics as distances between ${\hat{R}}_{1}$ and ${\hat{R}}_{2}$ and their asymptotic distributions have been studied in the literature (Aitkin, 1969; Jennrich, 1970; Larntz and Perlman, 1985). As K > 2, it is multiple sample testing problem that tests H_0K : R₁ = … = R_K against H_AK : not H_0K. Many test statistics and their asymptotic distributions have been extended from the case K = 2 to K > 2 (Kullback, 1967; Schott, 1996; Browne, 1978; Cole, 1968; Gupta et al., 2013).

Recently, ultra-high dimensional data arise from a variety of applications, including neuroimaging and genetics; that is, both p and min_ℓ{n_ℓ} converge to infinity. Testing correlation structures ${\{R_{ℓ}\}}_{ℓ = 1}^{K}$ in this high-dimensional setting has attracted extensive attention in the past decade due to both its importance in real applications and two major theoretical challenges, including high dimensionality and dependency (Cai, 2017; Debashis and Alexander, 2014). In this case, the test statistics developed for the classical setting either do not perform well or are no longer applicable. Therefore, under the high-dimensional setting, a collection of new testing statistics have been developed in the last few years for both the one- and two-population testing problems (Cai, 2017; Zhou et al., 2015; Cai and Zhang, 2016; Bodwin et al., 2016; Schott, 2005; Gao et al., 2017). For the one-sample case, the existing results focus on the test of short-range dependence, which includes independency as a special case, since the standard random matrix theory results are not directly applicable for a composite null. Moreover, the existing testing statistics are particularly powerful under either a “sparse” alternative or a dense alternative. For instance, Zhou et al. (2015) proposed several extreme value statistics to test the equality of two large U-statistic based correlation matrices, which include the rank-based correlation matrices as special cases.

1.2. Our Contributions.

The aim of this paper is to provide a general framework of testing correlation structures ${\{R_{ℓ}\}}_{ℓ = 1}^{K}$ for the one-, two-, and multiple sample testing problems as p → ∞. Compared with the existing literature discussed above, we make four major contributions as follows.

(I)
For the first time, we develop a set of test statistics to test correlation structures ${\{R_{ℓ}\}}_{ℓ = 1}^{K}$ for the one-, two-, and multiple sample testing problems under the high-dimensional setting. Our test statistics are designed to deal with both the dense and sparse alternatives. Specifically, they are the sum or the maximum of two terms, including a term for the dense alternative and the other for the sparse alternative.
(II)
We propose the test statistics for testing H₀₁: R₁ = R_∗ as K = 1 and then derive its asymptotic distribution and power function, even when R_∗ is an arbitrary correlation matrix. We make great efforts to deal with the non-independent elements of population random vectors during the derivation. In contrast, the existing results based on the standard random matrix theory (Bai and Silverstein, 2004) are limited to the covariance matrix or independent correlation (Gao et al., 2017; Li and Xue, 2015; Shao and Zhou, 2014; Qiu and Chen, 2012).
(III)
Similar to testing H₀₁ : R₁ = R_∗, we derive the asymptotic distribution of the test statistics and stride to deal with the non-independency of the two random matrices of the sample correlation matrices for testing H₀₂ : R₁ = R₂.
(IV)
To the best of our knowledge, we propose the first test statistic for testing H_0K : R₁ = ··· = R_K under the high-dimensional setting and then establish its asymptotic distribution under both H_0K and H_AK without assuming the normality. We also stride to deal with the non-independency of all random matrices of the sample correlation matrices.

The rest of this paper is organized as follows. Section 2 focuses on the one-sample problem, whereas Section 3 focuses on two- and multiple- sample testing problems. In each section, we propose the test statistics and establish its asymptotic distribution, power function, and unbiasedness. Section 5 will present simulation studies. We apply the test statistics to the ADHD data sets in Section 6. All proofs are collected in the Appendices.

2. Test Statistics for One Sample Testing Problem.

In this section, we focus on the one-sample problem of testing H₀₁ : R₁ = R_∗ against H_A1 : R₁ ≠ R_∗. This section consists of three parts. In Section 2.1, we describe two proposed test statistics. We characterize its asymptotic null distributions in Section 2.2 and its power properties in Section 2.3.

2.1. Test statistics.

We first introduce two terms as follows:

L_{n, 1} = tr [{({\hat{R}}_{1} - R_{*})}^{2}] and T_{n, 1} = \max_{1 \leq h < j \leq p} n_{1} {({\hat{r}}_{1 h j} - r_{* 1 h j})}^{2} ({\hat{θ}}_{1 h j}^{- 1} δ_{\{R_{*} \neq I_{p}\}} + δ_{\{R_{*} = I_{p}\}}),

where ${\hat{θ}}_{1 h j} = n_{1}^{- 1} \sum_{i = 1}^{n_{1}} {\{(z_{h i}^{(1)} - {\bar{z}}_{h}^{(1)}) (z_{j i}^{(1)} - {\bar{z}}_{j}^{(1)}) / {(s_{1 h h} s_{1 j j})}^{1 / 2} - 0.5 {\hat{r}}_{1 h j} [{(z_{h i}^{(1)} - {\bar{z}}_{h}^{(1)})}^{2} / s_{1 h h} + {(z_{j i}^{(1)} - {\bar{z}}_{j}^{(1)})}^{2} / s_{1 j j}]\}}^{2}$ with ${\hat{R}}_{1} = {({\hat{r}}_{1 h j})}_{h, j = 1}^{p}$ , $R_{*} = {(r_{* 1 h j})}_{h, j = 1}^{p}$ , $S_{1} = {(s_{1 h j})}_{h, j = 1}^{p}$ and δ_{∙} is an indicator function. The first term L_n,1 is designed for the dense alternative, whereas T_n,1 is for the sparse alternative.

Based on L_n,1 and T_n,1, we propose a weighted test statistic M_n,1 as follows:

M_{n, 1} = L_{n, 1} + C_{0} δ_{\{T_{n, 1} > s^{*} (n_{1}, p)\}},

(2.1)

where the second term of M_n,1 is a hard thresholding, C₀ is a large positive number and s^∗(n₁, p) is a scalar threshold depending on (n₁, p). The choices of C₀ and s^∗(n₁, p) will be given in the following Remark 2.1. For a given significance level α, we construct the acceptance region of M_n,1 to be

\{(z_{1}^{(1)}, \dots, z_{n_{1}}^{(1)}) : (M_{n, 1} - μ_{z 0}) / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})] \leq q_{1 - α}\},

(2.2)

where q_1−α is the (1 − α)100% quantile of N (0, 1) and µ_z0 will be specified below.

We also propose a maximum test statistic $M_{n, 1}^{'}$ as follow:

M_{n, 1}^{'} = \max \{(L_{n, 1} - μ_{z 0}) / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})], C_{0}^{'} (T_{n, 1} - 4 \log p + \log \log p)\}

(2.3)

where $C_{0}^{'}$ is a positive constant and different $C_{0}^{'}$ represents different contributions from L_{n, 1} and T_n,1 to $M_{n, 1}^{'}$ . For a given significance level α, we construct the acceptance region of $M_{n, 1}^{'}$ to be

\{(z_{1}^{(1)}, \dots, z_{n_{1}}^{(1)}) : M_{n, 1}^{'} \leq c_{α}\},

where c_α is a critical value and the choices of c_α and $C_{0}^{'}$ will be given in Remark 2.1.

2.2. Null distribution.

Our first theoretical result is to characterize the limiting null distribution of L_n,1. We introduce two assumptions that will be used later. Assumption (a) specifies the moment assumption of $z_{i}^{(ℓ)}$ . Assumption (b) specifies the ratio of the dimension of $z_{i}^{(ℓ)}$ to the sample size n_ℓ. We then introduce Assumption (a) as follows.

Assumption (a). $z_{i}^{(ℓ)}$ has the independent component structure $z_{i}^{(ℓ)} = μ_{z}^{(ℓ)} + Σ_{ℓ}^{1 / 2} w_{z i}^{(ℓ)}$ , where $w_{z i}^{(ℓ)} = {(w_{z 1 i}^{(ℓ)}, \dots, w_{z p i}^{(ℓ)})}^{T}$ with independently and identically distributed (iid) elements $w_{z j i}^{(ℓ)} s^{'}$ , $E (w_{z j i}^{(ℓ)}) = 0$ , $E [{(w_{z j i}^{(ℓ)})}^{2}] = 1$ and the kurtosis of $w_{z j i}^{(ℓ)}$ is equal to $β_{ℓ} = E [{(w_{z j i}^{(ℓ)})}^{4}] - 3$ . That is, $\{w_{z j i}^{(ℓ)}\}$ are standardized iid random variables only requiring that the fourth moment exists. The spectral norm of R_ℓ is bounded.

Assumption (a) imposes the independent component structure on $z_{i}^{(ℓ)}$ , which has been commonly used in random matrix theory (Bai and Silverstein, 2004; Chen et al., 2010). It only requires the existence of moments until the fourth order. The identically distributed assumption is not critical for most theoretical developments below.

We state Assumption (b) as follows.

Assumption (b). The ratio of the dimension p to the sample size n_ℓ tends to a constant, that is, p/n_ℓ → y_ℓ ϵ (0,∞).

Assumption (b) gives the convergence regime of the data dimension and the sample sizes. It assumes that the data dimension increases proportionally with the sample size, even when the limit y_ℓ can be extremely small (or large). Therefore, the data dimension may be much smaller (or greater) than the sample size.

Our first theoretical result quantifies the limiting distribution of the statistic $tr [{({\hat{R}}_{1} - R_{*})}^{2}]$ . Let $\overset{L}{\to}$ denote the convergence in distribution.

Theorem 2.1. If Assumptions (a)-(b) hold for ℓ = 1 and under H₀₁, we conclude that

(I.1). $(L_{n, 1} - μ_{z 0}) / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})] \overset{L}{\to} N (0, 1);$

(I.2). $(M_{n, 1} - μ_{z 0}) / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})] \overset{L}{\to} N (0, 1)$ holds under some additional conditions, including s*(n_1,p) – 4 log p→ + ∞ and (C1), (C2), and (C3) in Cai, Liu and Xia (2013) for R_* and $z_{i}^{(1)}$ , where µ_z0 is defined as

\frac{(n_{1}^{2} - n_{1} - 1) p^{2}}{n_{1} {(n_{1} - 1)}^{2}} - \frac{2 n_{1}^{2} + n_{1} + 1}{{(n_{1} - 1)}^{3}} tr (R_{*}^{2}) + \frac{(n_{1}^{2} - 3 n_{1})}{{(n_{1} - 1)}^{3}} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} {(r_{* 1 j j^{'}})}^{4} + \frac{n_{1} p β_{1}}{{(n_{1} - 1)}^{2}} - \frac{2 n_{1} β_{1}}{{(n_{1} - 1)}^{2}} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} r_{* \frac{3}{2} j j^{'}} {(r_{* \frac{1}{2} j j^{'}})}^{3} + \frac{(n_{1}^{2} - 5 n_{1}) β_{1}}{2 {(n_{1} - 1)}^{3}} \sum_{j = 1}^{p} [r_{* 2 j j} \sum_{j = 1}^{p} {(r_{* \frac{1}{2} j j^{'}})}^{4}] + \frac{(n_{1}^{2} - 3 n_{1}) β_{1}}{2 {(n_{1} - 1)}^{3}} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} [{(r_{* 1 j j^{'}})}^{2} \sum_{h = 1}^{p} {(r_{* \frac{1}{2} h j})}^{2} {(r_{* \frac{1}{2} h j^{'}})}^{2}]

with r_*khj being the (h, j) entry of $R_{*}^{k}$ for k=1/2, 1, 3/2,and 2.

Remark 2.1. When R_∗ = I_p, Cai and Jiang (2011) proved that $\max_{1 \leq h < j \leq p} n_{1} {({\hat{r}}_{1 h j} - r_{* 1 h j})}^{2} - 4 \log p + \log \log p$ converged to a type I extreme value distribution function F(t) = exp[−(8π)^−1/2 exp(−t/2)] under H₀₁. When R_* ≠ I_p, similar to (22) of Cai and Zhang (2016), we conclude that $\max_{1 \leq h < j \leq p} n_{1} {({\hat{r}}_{1 h j} - r_{* 1 h j})}^{2} {\hat{θ}}_{1 h j}^{- 1} - 4 \log p + \log \log p$ converges to the type I extreme value distribution function under H₀₁ and (C1), (C2) and (C3) in Cai, Liu and Xia (2013) for R_* and $z_{i}^{(1)}$ . The choices of s*(n1,p), C₀, $C_{0}^{'}$ and c_r are given as follows:

Choice of the threshold s^∗(n₁,p): The test statistic M_n,1 mainly targets at L_n,1. For simplicity, the threshold is taken to be
$s^{*} (n_{1}, p) = [4 + {(\log \log n_{1} - 1)}^{2}] (\log p - 0.25 \log \log p) + u_{0},$
where u₀ satisfies exp[−(8π)^−1/2 exp(−u0/2)] = 0.99. The threshold ensures that even if n₁ and p are small, the probability of the event {T_n,1 > s^∗(n₁,p)} is bounded by 0.01 under H₀₁. The probability of the event {T_n,1 > s∗(n₁, p)} becomes negligible under H₀₁ when either n₁ or p is relatively large.

Choice of the constant C₀: The role of C₀ is to ensure that the second term of M_n,1 acts as the main term in M_n,1 when T_n,1 > s^∗(n₁, p). It is enough to require that $C_{0} / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})]$ is far away from q_1−α. For simplicity, let C₀ be p² throughout this paper.
Choice of the constant $C_{0}^{'}$ and the critical value c_α: Theorem 2.1 shows that $(L_{n, 1} - μ_{z 0}) / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})]$ is asymptotically distributed as N (0, 1) under H₀₁. To balance the contribution of L_n,1 and that of T_n,1, $C_{0}^{'}$ should be relatively small for extremely dense R₁ − R_∗, whereas $C_{0}^{'}$ should be large for extremely sparse R₁ − R_∗. However, it is unknown whether R1 − R_∗ is dense or sparse, so we choose $C_{0}^{'}$ such that $(L_{n, 1} - μ_{z 0}) / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})]$ and $C_{0}^{'} (T_{n, 1} - 4 \log p + \log \log p)$ have the same (1 − α/2)100% quantile, where α is the significance level. That is, we have $C_{0}^{'} = q_{1 - α / 2} / u_{0}^{'}$ and $c_{α} = q_{1 - α / 2}$ , where $u_{0}^{'}$ satisfies $\exp [- {(8 π)}^{- 1 / 2} \exp (- u_{0}^{'} / 2)] = 1 - α / 2$ . Then, we have $P (M_{n, 1}^{'} > q_{1 - α / 2}) \leq α$ under H₀₁

Proof. We will give the skeletons of the proof of Theorem 2.1. The details of the proof are placed in Appendices. The proof proceeds in three steps.

Skeleton of Step 1. To obtain the expansion of $tr [{({\hat{R}}_{1} - R_{*})}^{2}]$ as follows

\begin{array}{l} tr [{({\hat{R}}_{1} - R_{*})}^{2}] = tr (S_{1}^{2}) + tr (R_{*}^{2}) - 2 tr (S_{1} R_{*}) \\ - 2 tr \{S_{1}^{2} [diag (S_{1}) - I_{p}]\} + 2 tr \{[diag (S_{1}) - I_{p}] S_{1} R_{*}\} \\ + 2 tr \{S_{1}^{2} {[diag (S_{1}) - I_{p}]}^{2}\} - 1.5 tr \{{[diag (S_{1}) - I_{p}]}^{2} S_{1} R_{*}\} \\ + tr \{S_{1} [diag (S_{1}) - I_{p}] S_{1} [diag (S_{1}) - I_{p}]\} \\ - 0.5 tr \{[diag (S_{1}) - I_{p}] S_{1} [diag (S_{1}) - I_{p}] R_{*}\} + o_{p} (1), \end{array}

we assume $Cov (z_{1}^{(1)}) = R_{1}$ without loss of generality. This step is mainly to use the Taylor expansions of $s_{1 j j}^{- 1} = 1 - (s_{1 i j} - 1) + {(s_{1 i j} - 1)}^{2} + o (p^{- 1})$ and $s_{1 j j}^{- 1 / 2} = 1 - \frac{1}{2} (s_{1 i j} - 1) + \frac{3}{8} {(s_{1 i j} - 1)}^{2} + o (p^{- 1})$ with $S_{1} = {(s_{1 i j})}_{i, j = 1}^{p}$ .

Skeleton of Step 2. We want to derive the limits of the following four terms $tr \{S_{1}^{2} {[diag (S_{1}) - I_{p}]}^{2}\}$ , $tr \{{[diag (S_{1}) - I_{p}]}^{2} S_{1} R_{*}\}$ , $tr \{S_{1} [diag (S_{1}) - I_{p}] S_{1} [diag (S_{1}) - I_{p}]\}$ and $tr \{[diag (S_{1}) - I_{p}] S_{1} [diag (S_{1}) - I_{p}] R_{*}\}$ in probability.

Skeleton of Step 3. We want to derive the limiting null distribution of $(tr (S_{1}^{2}) - Etr (S_{1}^{2})$ , $tr (S_{1} R_{*}) - Etr (S_{1} R_{*})$ , $tr \{S_{1}^{2} [diag (S_{1}) - I_{p}]\} - Etr \{S_{1}^{2} [diag (S_{1}) - I_{p}]\}$ , $tr \{[diag (S_{1}) - I_{p}] S_{1} R_{*}\} - Etr \{[diag (S_{1}) - I_{p}] S_{1} R_{*}\}$ . Thus by the delta method, we obtain the central limit theorem (CLT) of $tr [{({\hat{R}}_{1} - R_{*})}^{2}]$ . Because these terms involve diag(S₁) − I_p, we cannot directly use the random matrix theory on linear spectral statistics of S₁ to obtain the limiting distribution of these terms. To solve the problem, we construct four martingale difference sequences to establish the CLT of these terms. Especially, the derivation of the CLT for the case R_∗ ≠ Ip is much more difficult than the derivation for the case R_∗ = I_p.

Corollary 2.1. Under the assumptions of Theorem 2.1, we have the following results:

If R_∗ is the identity matrix I_p, then µ_z0 reduces to
$μ_{z 0} = \frac{(n_{1}^{2} - n_{1} - 1) p^{2}}{n_{1} {(n_{1} - 1)}^{2}} - \frac{(n_{1}^{2} + 4 n_{1} + 1) p}{{(n_{1} - 1)}^{3}} - \frac{3 p n_{1} β_{1}}{{(n_{1} - 1)}^{3}} .$

If the population is Gaussian, then µ_z0 reduces to
$\frac{(n_{1}^{2} - n_{1} - 1) p^{2}}{n_{1} {(n_{1} - 1)}^{2}} - \frac{2 n_{1}^{2} + n_{1} + 1}{{(n_{1} - 1)}^{3}} tr (R_{*}^{2}) + \frac{(n_{1}^{2} - 3 n_{1})}{{(n_{1} - 1)}^{3}} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} {(r_{* 1 j j^{'}})}^{4}$

where r_∗1jj' is the (j, j') entry of R_∗.

Theorem 2.1 provides a unified framework of testing H₀₁ : R₁ = R_∗ for an arbitrary R_∗. Our test statistics account for both dense and sparse alternatives, and they work for $z_{i}^{(1)}$ satisfying the independent component structure specified in Assumption (a) and a ratio of p/n₁ = y₁ in Assumption (b). Technically, to prove Theorem 2.1, we develop a set of novel tools to deal with the dependence between S₁ and diag(S₁), which is technically nontrivial and is of independent interest for handling the sample correlation in more general settings. In contrast, Gao et al. (2017) only established the CLT of the sample correlation matrices of a high dimensional vector whose elements have an identity correlated structure R_∗ = I_p. Moreover, their theoretical result involves some two-dimensional contour integrals, which can be difficult to compute.

2.3. Power properties and optimality.

We examine the power properties of M_n,1 and $M_{n, 1}^{'}$ . We first establish the asymptotic distribution of the statistic $tr [{({\hat{R}}_{1} - R_{*})}^{2}]$ under the alternative hypothesis H_A1.

Theorem 2.2. Assuming that Assumptions (a) and (b) hold for ℓ = 1, we have

\frac{tr [{({\hat{R}}_{1} - R_{*})}^{2}] - μ_{z A}}{σ_{z A}} \overset{L}{\to} N (0, 1),

where µ_zA and σ_zA depend on the alternative population correlation matrix R₁ and will be given in Appendix.

Given the result in Theorem 2.2, we can characterize the properties of the power functions, which is given by

g_{1} (R_{1}, α) = P ((M_{n, 1} - μ_{z 0}) / [2 {(n_{1} - 1)}^{- 1} tr (R_{*}^{2})] > q_{1 - α}), g_{1}^{'} (R_{1}, α) = P (M_{n, 1}^{'} > c_{α}) .

In the following, we will study the properties of the power functions g₁(R₁, α) and $g_{1}^{'} (R_{1}, α)$ .

Corollary 2.2. Assuming that Assumptions (a) and (b) hold for ℓ = 1, we have the following results:

If tr[(R₁ − R_∗)²] > c₀ > 0, then g₁(R₁, α) > α when the sample size n₁ is large enough and c₀ is any given small constant;
If tr[(R₁ − R_∗)²] tends to infinity, then g₁(R₁, α) and g'₁(R₁, α) are close to one as n₁ → ∞;
If the absolute value of at least one entry of R₁−R_∗ is greater than $n_{1}^{- 1 / 2} \sqrt{\log (p) \log (n_{1})}$ and the conditions (C1), (C2), and (C3) in Cai et al. (2013) hold for R₁ and $z_{i}^{(1)}$ then g₁(R₁, α) and $g_{1}^{'} (R_{1}, α)$ are close to one as n₁ → ∞.

Corollary 2.2 shows that the proposed test M_n,1 is asymptotically unbiased. In Appendix, we will prove that (i) For the dense alternative tr[(R₁ – R_*)²] → ∞, the power functions tend to one; (ii). For the sparse alternative, if the absolute value of at least one entry of R₁ − R_∗ is greater than $n_{1}^{- 1 / 2} \sqrt{\log (p) \log (n_{1})}$ , then the power functions will be close to one.

Similar to Cai and Ma (2013), we define

Θ_{1}^{*} (b_{1}, b_{10}) = \{R_{1} : {‖R_{1} - R_{*}‖}_{F} > b_{1} \sqrt{p / n_{1}} o r {‖R_{1} - R_{*}‖}_{\infty} > b_{10} \sqrt{\log p / n_{1}}\},

where b₁, b₁₀ are positive constants, ${‖R_{1} - R_{*}‖}_{F} = {\{tr [{(R_{1} - R_{*})}^{2}]\}}^{1 / 2}$ and ${‖R_{1} - R_{*}‖}_{\infty} = \max_{1 \leq i \leq j \leq p} |e_{i}^{T} (R_{1} - R_{*}) e_{j}|$ with e_i and e_j being the ith column and jth column of the p × p identity matrix, respectively.

Theorem 2.3. Let 0 < α < β < 1. Suppose that as p/n₁ → y₁ > 0 as n₁ → + ∞. Then there exist two constants 0 < b₁, b₁₀ < 1 such that for any test ϕ with the significance level α for testing H₀₁ : R₁ = R_∗, we have

\underset{n_{1} \to \infty}{\lim \sup} \inf_{R_{1} \in Θ_{1}^{*} (b_{1}, b_{10})} E_{R_{1}} ϕ < β,

where $E_{R_{1}}$ is the expectation under the population correlation matrix being R₁.

Theorem 2.3 shows that no level α test can distinguish the null hypothesis from the alternative hypothesis with the power tending to one as p/n₁ → y₁ > 0, ${‖R_{1} - R_{*}‖}_{F} = O (\sqrt{p / n_{1}})$ or ${‖R_{1} - R_{*}‖}_{\infty} > b_{10} \sqrt{\log p / n_{1}}$ . Then, Theorem 2.3 gives the lower bound for the optimality of our proposed procedure.

3. Extensions to Two and Multiple Sample Testing Problems.

This section consists of two parts. In Section 3.1, we focus on the two-sample problem of testing H₀₂ : R₁ = R₂ against H_A2 : R₁ ≠ R₂. In Section 3.2, we consider the multiple sample testing problem.

3.1. Extension to Two Sample Testing Problem.

3.1.1. Test statistics and their null distributions for two-sample testing problem.

Let ${\hat{R}}_{ℓ} = {({\hat{r}}_{ℓ h j})}_{h, j = 1}^{p}$ and $S_{ℓ} = {(s_{ℓ h j})}_{h, j = 1}^{p}$ for ℓ = 1, 2. We introduce two terms as follows:

L_{n, 2} = tr [{({\hat{R}}_{1} - {\hat{R}}_{2})}^{2}] and T_{n, 2} = \max_{1 \leq h < j \leq p} {(n_{1}^{- 1} {\hat{θ}}_{1 h j} + n_{2}^{- 1} {\hat{θ}}_{2 h j})}^{- 1} {({\hat{r}}_{1 h j} - {\hat{r}}_{2 h j})}^{2},

where ${\hat{θ}}_{ℓ h j}$ is defined as

{\hat{θ}}_{ℓ h j} = n_{ℓ}^{- 1} {\sum_{i = 1}^{n_{ℓ}} \{\frac{(z_{h i}^{(ℓ)} - {\bar{z}}_{h}^{(ℓ)}) (z_{j i}^{(ℓ)} - {\bar{z}}_{j}^{(ℓ)})}{{(s_{ℓ h h} s_{ℓ j j})}^{1 / 2}} - 0.5 {\hat{r}}_{ℓ h j} [\frac{{(z_{h i}^{(ℓ)} - {\bar{z}}_{h}^{(ℓ)})}^{2}}{s_{ℓ h h}} + \frac{{(z_{j i}^{(ℓ)} - {\bar{z}}_{j}^{(ℓ)})}^{2}}{s_{ℓ j j}}]\}}^{2} .

The first term L_n,2 is introduced to deal with the dense alternative, whereas the second term T_n,2 is for the sparse alternative.

We propose a weighted test statistic M_n,2 as follows:

M_{n, 2} = L_{n, 2} + C_{0, 2} δ_{\{T_{n, 2} > s (n_{1}, n_{2}, p)\}},

(3.1)

where C_0,2 and the threshold s(n₁, n₂, p) will be given in Remark 3.1. For a given significance level α, we construct an acceptance region of M_n,2 to be

\{{\{z_{i}^{(ℓ)} : i = 1, \dots, n_{ℓ}\}}_{ℓ = 1}^{2} : (M_{n, 2} - {\hat{μ}}_{z 12}) / \{2 [{(n_{1} - 1)}^{- 1} + {(n_{2} - 1)}^{- 1}] \hat{a}\} \leq q_{1 - α}\},

where ${\hat{μ}}_{z 12}$ and $\hat{a}$ will be defined below.

We also propose a maximum test statistic $M_{n, 2}^{'}$ as follows:

M_{n, 2}^{'} = \max \{(L_{n, 2} - {\hat{μ}}_{z 12}) / \{2 [{(n_{1} - 1)}^{- 1} + {(n_{2} - 1)}^{- 1}] \hat{a}\}, C_{0, 2}^{'} (T_{n, 2} - 4 \log p + \log \log p)\}

(3.2)

where $C_{0, 2}^{'}$ is a positive constant. For a given significance level α, we construct an acceptance region of $M_{n, 2}^{'}$ to be

\{{\{z_{i}^{(ℓ)} : i = 1, \dots, n_{ℓ}\}}_{ℓ = 1}^{2} : M_{n, 2}^{'} \leq c_{α, 2}\},

where the positive constant $C_{0, 2}^{'}$ and the critical value c_α,2 will be given in Remark 3.1.

We establish the asymptotic null distribution of L_n,2 as follows.

Theorem 3.1. Let R be the common correlation matrix R = R₁ = R₂. Assuming that Assumptions (a) and (b) hold for ℓ = 1 and 2 and under H₀₂, we conclude that

(II.1). $(L_{n, 2} - μ_{z 12}) / \{2 [{(n_{1} - 1)}^{- 1} + {(n_{2} - 1)}^{- 1}] tr (R^{2})\} \overset{L}{\to} N (0, 1)$ ;

(II.2). $(M_{n, 2} - μ_{z 12}) / \{2 [{(n_{1} - 1)}^{- 1} + {(n_{2} - 1)}^{- 1}] tr (R^{2})\} \overset{L}{\to} N (0, 1)$ holds under some additional conditions, including s(n₁, n₂, p) − 4 log p → + ∞ and (C1), (C2), and (C3) of Cai, Liu and Xia (2013) for R and $z_{i}^{(ℓ)}$ , where µ_z12 is given by

\begin{array}{l} μ_{z 12} = \sum_{ℓ = 1}^{2} \frac{(n_{ℓ}^{2} - n_{ℓ} - 1) p^{2}}{n_{ℓ} {(n_{ℓ} - 1)}^{2}} - \sum_{ℓ = 1}^{2} \frac{2 n_{ℓ}^{2} + n_{ℓ} + 1}{{(n_{ℓ} - 1)}^{3}} tr (R^{2}) + \sum_{ℓ = 1}^{2} \frac{β_{ℓ} n_{ℓ} p}{{(n_{ℓ} - 1)}^{2}} \\ - \sum_{ℓ = 1}^{2} \frac{β_{ℓ} 2 p n_{ℓ}}{{(n_{ℓ} - 1)}^{2}} b_{0} + \sum_{ℓ = 1}^{2} \frac{β_{ℓ} p (n_{ℓ}^{2} - 5 n_{ℓ})}{2 {(n_{ℓ} - 1)}^{3}} c_{0} + \sum_{ℓ = 1}^{2} \frac{p (n_{ℓ} - 3) n_{ℓ}}{2 {(n_{ℓ} - 1)}^{3}} d_{ℓ}, \end{array}

with r_0khj being the (h, j) entry of R^k for k = 1/2, 1, 2/3, and 2 and

b_{0} = p^{- 1} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} r_{0 \frac{3}{2} j j^{'}} {(r_{0 \frac{1}{2} j j^{'}})}^{3},

c_{0} = p^{- 1} \sum_{j = 1}^{p} r_{02 j j} \sum_{j^{'} = 1}^{p} {(r_{0 \frac{1}{2} j j^{'}})}^{4},

d_{ℓ} = p^{- 1} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} [2 {(r_{01 j j^{'}})}^{4} + β_{ℓ} {(r_{01 j j^{'}})}^{2} \sum_{h = 1}^{p} {(r_{0 \frac{1}{2} h j})}^{2} {(r_{0 \frac{1}{2} h j^{'}})}^{2}] .

Remark 3.1. Similar to (22) of Cai and Zhang (2016), we conclude that

\max_{1 \leq h < j \leq p} {(n_{1}^{- 1} {\hat{θ}}_{1 h j} + n_{2}^{- 1} {\hat{θ}}_{2 h j})}^{- 1} {({\hat{r}}_{1 h j} - {\hat{r}}_{2 h j})}^{2} - 4 \log p + \log \log (p)

converges to the type I extreme value distribution function under H₀₂ and (C1), (C2), and (C3) in Cai, Liu and Xia (2013) for R and $z_{i}^{(ℓ)}$ . The choices of s(n₁, n₂, p), C_0,2, $C_{0, 2}^{'}$ and c_α,2 are given as follows:

Choice of the threshold s(n₁, n₂, p): The test statistic M_n,2 mainly targets at L_n,2. For simplicity, we set s(n₁, n₂, p) as
$s (n_{1}, n_{2}, p) = [4 + {(\log \log (n_{1} + n_{2}) - 1)}^{2}] (\log p - 0.25 \log \log p) + u_{0}^{'},$
Where $u_{0}^{'}$ satisfies $\exp [- {(8 π)}^{- 1 / 2} \exp (- u_{0}^{'} / 2)] = 0.99$ . The threshold ensures that even for small n₁, n₂ and p, the probability of the event {T_n,2 > s(n₁, n₂, p)} is bounded by 0.01 under H₀₂. The probability of the event {T_n,2 > s(n₁, n₂, p)} becomes negligible under H₀₂ when either n₁, n₂ or p is moderately large.

Choice of C_0,2, $C_{0, 2}^{'}$ and c_α,2: The constants C_0,2, $C_{0, 2}^{'}$ and c_α,2 are the same as C₀, $C_{0}^{'}$ and c_α in Remark 2.1. Moreover, $P (M_{n, 2}^{'} > q_{1 - α / 2}) \leq α$ under H₀₂.

Proof. We will give the skeletons of the proof of Theorem 3.1. The details of the proof are placed in Appendices. The proof proceeds in three steps.

Skeleton of Step 1. It is assumed that $Cov (z_{i}^{(ℓ)}) = R_{ℓ}$ holds for ℓ = 1, 2 without loss of generality. We obtain the expansion of $tr [{({\hat{R}}_{1} - {\hat{R}}_{2})}^{2}]$ as follows

\begin{array}{l} tr [{({\hat{R}}_{1} - {\hat{R}}_{2})}^{2}] \\ = tr (S_{1}^{2}) + tr (S_{2}^{2}) - 2 tr (S_{1} S_{2}) - 2 tr [S_{1}^{2} (diag (S_{1}) - I_{p})] - 2 tr [S_{2}^{2} (diag (S_{2}) - I_{p})] \\ + 2 tr \{[diag (S_{1}) - I_{p}] S_{1} S_{2}\} + 2 tr \{[diag (S_{2}) - I_{p}] S_{1} S_{2}\} \\ + 2 \sum_{ℓ = 1}^{2} tr [S_{ℓ}^{2} {(diag (S_{ℓ}) - I_{p})}^{2}] + \sum_{ℓ = 1}^{2} tr [S_{ℓ} (diag (S_{ℓ}) - I_{p}) S_{ℓ} (diag (S_{ℓ}) - I_{p})] \\ - 1.5 tr \{{[diag (S_{1}) - I_{p}]}^{2} S_{1} S_{2}\} - 1.5 tr \{{[diag (S_{2}) - I_{p}]}^{2} S_{1} S_{2}\} \\ - 0.5 tr \{[diag (S_{1}) - I_{p}] S_{1} [diag (S_{1}) - I_{p}] S_{2}\} \\ - 0.5 tr \{S_{1} [diag (S_{2}) - I_{p}] S_{2} [diag (S_{1}) - I_{p}]\} \\ - tr \{[diag (S_{2}) - I_{p}] [diag (S_{1}) - I_{p}] S_{1} S_{2}\} \\ - tr \{S_{1} [diag (S_{1}) - I_{p}] S_{2} [diag (S_{2}) - I_{p}]\} + o_{p} (1) . \end{array}

This step is mainly to use the Taylor expansions of $s_{ℓ j j}^{- 1} = 1 - (s_{ℓ j j} - 1) + {(s_{ℓ j j} - 1)}^{2} + o (p^{- 1})$ and $s_{ℓ j j}^{- 1 / 2} = 1 - \frac{1}{2} (s_{ℓ j j} - 1) + \frac{3}{8} {(s_{ℓ j j} - 1)}^{2} + o (p^{- 1})$ with $S_{ℓ} = {(s_{ℓ j j})}_{i, j = 1}^{p}$ for ℓ=1,2.

Skeleton of Step 2. We want to derive the limits of the following ten terms in probability: $tr [S_{1}^{2} {(diag (S_{1}) - I_{p})}^{2}]$ , $tr [S_{2}^{2} {(diag (S_{2}) - I_{p})}^{2}]$ , $tr \{{[diag (S_{1}) - I_{p}]}^{2} S_{1} S_{2}\}$ , $tr \{{[diag (S_{2}) - I_{p}]}^{2} S_{1} S_{2}\}$ , $tr [S_{1} (diag (S_{1}) - I_{p}) S_{1} (diag (S_{1}) - I_{p})]$ , $tr [S_{2} (diag (S_{2}) - I_{p}) S_{2} (diag (S_{2}) - I_{p})]$ , $tr \{[diag (S_{1}) - I_{p}] S_{1} [diag (S_{1}) - I_{p}] S_{2}\}$ , $tr \{S_{1} [diag S_{2} - I_{p}] S_{2} [diag S_{2} - I_{p}]\}$ , $tr \{[diag (S_{2}) - I_{p}] [diag (S_{1}) - I_{p}] S_{1} S_{2}\}$ , $tr \{S_{1} [diag (S_{1}) - I_{p}] S_{2} [diag (S_{2}) - I_{p}]\}$ .

Skeleton of Step 3. We want to derive the limiting null distribution of $(tr (S_{1}^{2}) - Etr (S_{1}^{2})$ , $tr (S_{2}^{2}) - Etr (S_{2}^{2})$ , $tr (S_{1} S_{2}) - Etr (S_{1} S_{2})$ , $tr [S_{1}^{2} (diag (S_{1}) - I_{p})] - Etr [S_{1}^{2} (diag (S_{1}) - I_{p})]$ , $tr [S_{2}^{2} (diag (S_{2}) - I_{p})] - Etr [S_{2}^{2} (diag (S_{2}) - I_{p})]$ , $tr \{[diag (S_{1}) - I_{p}] S_{1} S_{2}\} - Etr \{[diag (S_{1}) - I_{p}] S_{1} S_{2}\}$ , $tr \{[diag (S_{2}) - I_{p}] S_{1} S_{2}\} - Etr \{[diag (S_{2}) - I_{p}] S_{1} S_{2})\}$ . Thus by the delta method, we obtain the CLT of $tr [{({\hat{R}}_{1} - {\hat{R}}_{2})}^{2}]$ . Because these terms involve the product of any two or three terms among S₁, diag(S₁) − I_p, S₂ and diag(S₂) − I_p, the CLT for Theorem 2.1 is not directly applicable. Thus, in order to derive the CLT of these terms, eight new martingale difference sequences are constructed. Especially, the derivation of the CLT for the two population case R₁ = R₂ is very different from and more difficult than the derivation for the one population case R₁ = R_∗.

Remark 3.2. Under the null hypothesis H₀₂, we do not know the true R, so we have to estimate the terms related to R in the asymptotic mean and variance. Let

ω_{1} = n_{1} / (n_{1} + n_{2}), ω_{2} = 1 - ω_{1} a n d M_{ℓ i} = {(x_{i}^{(ℓ)})}^{T} f o r ℓ = 1, 2,

{\hat{a}}_{ℓ} = \frac{[tr ({\hat{R}}_{ℓ}^{2}) - (n_{ℓ}^{2} - n_{ℓ} - 1) p^{2} n_{ℓ}^{- 1} {(n_{ℓ} - 1)}^{- 2} - β_{ℓ} n_{ℓ} p {(n_{ℓ} - 1)}^{- 2}] {(n_{ℓ} - 1)}^{2}}{p (n_{ℓ}^{2} - n_{ℓ} + 2)}, ℓ = 1, 2,

where $x_{i}^{(ℓ)} = {(x_{1 i}^{(ℓ)}, \dots, x_{p i}^{(ℓ)})}^{T} = {[diag (S_{ℓ})]}^{- 1 / 2} (z_{i}^{(ℓ)} - {\bar{z}}^{(ℓ)})$ with i = 1,…,n_ℓ. Then, we estimate a₀ = p^–1tr(R²), b₀, c₀, and d_ℓ as follows:

{\hat{a}}_{0} = ω_{1} {\hat{a}}_{1} + ω_{2} {\hat{a}}_{2},

{\hat{b}}_{0} = \sum_{ℓ = 1}^{2} ω_{ℓ} β_{ℓ}^{- 1} p^{- 1} (n_{ℓ}^{- 1} \sum_{i = 1}^{n_{ℓ}} tr \{{\hat{R}}_{\{1, 2\} / \{ℓ\}} M_{ℓ i} [diag (M_{ℓ i}) - I_{p}]\} - 2 {\hat{a}}_{0}),

{\hat{c}}_{0} = \sum_{ℓ = 1}^{2} ω_{ℓ} β_{ℓ}^{- 1} p^{- 1} (n_{ℓ}^{- 1} \sum_{i = 1}^{n_{ℓ}} tr \{{\hat{R}}_{\{1, 2\} / \{ℓ\}} [{\hat{R}}_{ℓ} - {(n_{ℓ} - 1)}^{- 1} M_{ℓ i}] {[diag (M_{ℓ i}) - I_{p}]}^{2}\} - 2 {\hat{a}}_{0}),

{\hat{d}}_{ℓ} = p^{- 1} \{n_{ℓ}^{- 1} \sum_{i = 1}^{n_{ℓ}} tr \{[{\hat{R}}_{ℓ} - M_{ℓ i} {(n_{ℓ} - 1)}^{- 1}] [diag (M_{ℓ i}) - I_{p}] {\hat{R}}_{\{1, 2\} / \{ℓ\}} [diag (M_{ℓ i}) - I_{p}]\},

with letting $β_{ℓ}^{- 1} = 0$ if β_ℓ = 0, ${\hat{R}}_{\{1, 2\} / \{1\}} = {\hat{R}}_{2}$ and ${\hat{R}}_{\{1, 2\} / \{2\}} = {\hat{R}}_{1}$ . Finally, we can obtain an estimate of µ_z12 as follows:

\begin{array}{l} {\hat{μ}}_{z 12} = \sum_{ℓ = 1}^{2} \frac{(n_{ℓ}^{2} - n_{ℓ} - 1) p^{2}}{n_{ℓ} {(n_{ℓ} - 1)}^{2}} - \sum_{ℓ = 1}^{2} \frac{p (2 n_{ℓ}^{2} + n_{ℓ} + 1)}{{(n_{ℓ} - 1)}^{3}} {\hat{a}}_{0} + \sum_{ℓ = 1}^{2} \frac{β_{ℓ} n_{ℓ} p}{{(n_{ℓ} - 1)}^{2}} \\ - \sum_{ℓ = 1}^{2} \frac{β_{ℓ} 2 p n_{ℓ}}{{(n_{ℓ} - 1)}^{2}} {\hat{b}}_{0} + \sum_{ℓ = 1}^{2} \frac{β_{ℓ} p (n_{ℓ}^{2} - 5 n_{ℓ})}{2 {(n_{ℓ} - 1)}^{3}} {\hat{c}}_{0} + \sum_{ℓ = 1}^{2} \frac{p (n_{ℓ} - 3) n_{ℓ}}{2 {(n_{ℓ} - 1)}^{3}} {\hat{d}}_{ℓ} . \end{array}

Corollary 3.1. Under the same assumptions of Theorem 3.1, we concluded that

$(L_{n, 2} - {\hat{μ}}_{z 12}) / {2 [{(n_{1} - 1)}^{- 1} + {(n_{2} - 1)}^{- 1}] p {\hat{a}}_{0}} \overset{L}{\to} N (0, 1);$
$(M_{n, 2} - {\hat{μ}}_{z 12}) / {2 [{(n_{1} - 1)}^{- 1} + {(n_{2} - 1)}^{- 1}] p {\hat{a}}_{0}} \overset{L}{\to} N (0, 1)$ holds under some additional conditions, including s(n₁, n₂, p) − 4 log p → + ∞ and (C1), (C2), and (C3) in Cai, Liu and Xia (2013) for R and $z_{i}^{(ℓ)}$ .

3.1.2. Power properties and optimality.

In the following, we will study the power properties of the statistics M_n,2 and $M_{n, 2}^{'}$ .

Theorem 3.2. Under Assumptions (a) and (b), we have

σ_{A 12}^{- 1} \{tr [{({\hat{R}}_{1} - {\hat{R}}_{2})}^{2}] - μ_{A 12}\} \overset{L}{\to} N (0, 1),

where µ_A12 and σ_A12 depend on the alternative population correlation matrices R₁ and R₂ and will be given in Appendix.

Theorem 3.2 gives the asymptotic distribution of the statistic $tr [{({\hat{R}}_{1} - {\hat{R}}_{2})}^{2}]$ under the alternative hypothesis. The power function is given by

g_{2} (R_{1}, R_{2}, α) = P ((M_{n, 2} - {\hat{μ}}_{z 12}) / \{2 [{(n_{1} - 1)}^{- 1} + {(n_{2} - 1)}^{- 1}] p {\hat{a}}_{0}\} > q_{1 - α}),

and $g_{2}^{'} (R_{1}, R_{2}, α) = P (M_{n, 2}^{'} > c_{α, 2})$ . Then, g₂(R₁, R₂, α) and $g_{2}^{'} (R_{1}, R_{2}, α)$ have the following properties.

Corollary 3.2. Under the same assumptions of Theorem 3.2, we have the following results:

If tr[(R₂ − R₁)²] > c₀ > 0, where c₀ is a positive scalar, then g₂(R₁, R₂, α) > α when the sample size is large enough;
If tr[(R₁ − R₂)²] → ∞, then g₂(R₁, R₂, α) and $g_{2}^{'} (R_{1}, R_{2}, α)$ are close to one as n₁, n₂ → ∞;
If the absolute value of at least one entry of R₁ − R₂ is greater than ${[\log (p) \log (n_{1} + n_{2})]}^{1 / 2} / \sqrt{\min \{n_{1}, n_{2}\}}$ and the conditions (C1), (C2), and (C3) in Cai et al. (2013) hold for R_ℓ and $z_{i}^{(ℓ)}$ , ℓ = 1, 2, then g₂(R₁, R₂, α) and $g_{2}^{'} (R_{1}, R_{2}, α)$ are close to one as n₁, n₂ → ∞.

Corollary 3.2 shows that the proposed test M_n,2 is asymptotically unbiased. In Appendix, we will prove that for the dense alternative tr[(R₁ − R₂)²] → ∞, the power functions tend to one. For the sparse alternative, if the absolute value of at least one entry of R₁ − R₂ is greater than ${[\log (p) \log (n_{1} + n_{2})]}^{1 / 2} / \sqrt{\min \{n_{1}, n_{2}\}}$ , then the power functions are close to one.

Similar to Cai and Ma (2013), we define

\begin{array}{l} Θ_{2}^{*} (b_{2}, b_{20}) = \{R_{1}, R_{2} : {‖R_{1} - R_{2}‖}_{F} > b_{2} \min \{\sqrt{p / n_{1}}, \sqrt{p / n_{2}}\} \\ o r {‖R_{1} - R_{2}‖}_{\infty} > b_{20} \min \{\sqrt{\log p / n_{1}}, \sqrt{\log p / n_{2}}\}, \end{array}

where b₂, b₂₀ are positive constants, ${‖R_{1} - R_{2}‖}_{F} = {\{tr [{(R_{1} - R_{2})}^{2}]\}}^{1 / 2}$ and ${‖R_{1} - R_{2}‖}_{\infty} = \max_{1 \leq i \leq j \leq p} |e_{i}^{T} (R_{1} - R_{2}) e_{j}|$ with e_i and e_j being the ith column and jth column of the p × p identity matrix, respectively.

Theorem 3.3. Let 0 < α < β < 1. Suppose that p/n_i → y_i > 0 as n_i → ∞ for i = 1, 2. Then there exist two constants 0 < b₂, b₂₀ < 1 such that for any test ϕ with the significance level α for testing H₀₂ : R₁ = R₂, we have

\underset{n_{1}, n_{2} \to \infty}{\lim \sup} \inf_{R_{1}, R_{2} \in Θ_{2}^{*} (b_{2}, b_{20})} E_{R_{1}, R_{2}} ϕ < β,

where $E_{R_{1}, R_{2}}$ is the expectation under the two population correlation matrix being R₁ and R₂.

Theorem 3.3 shows that no level α test can distinguish between the null hypothesis and all alternative hypotheses with the power tending to one as p/n_i → y_i > 0 for i = 1, 2 and ${‖R_{1} - R_{2}‖}_{F} = O (\min \{\sqrt{p / n_{1}}, \sqrt{p / n_{2}}\})$ or ${‖R_{1} - R_{2}‖}_{\infty} > b_{20} \min \{\sqrt{\log p / n_{1}}, \sqrt{\log p / n_{2}}\}$ . Then, Theorem 3.3 gives the lower bound for the optimality of our proposed procedure.

3.2. Test statistic for multiple sample testing problem.

We extend the test statistic from two samples to K samples. The one weighted test statistic is constructed as

M_{n, K} = \sum_{1 \leq ℓ_{1} < ℓ_{2} \leq K} ω_{ℓ_{1}, ℓ_{2}} tr [{({\hat{R}}_{ℓ_{1}} - {\hat{R}}_{ℓ_{2}})}^{2}] + T_{n, K},

where

T_{n, K} = C_{0} \sum_{1 \leq ℓ_{1} < ℓ_{2} \leq K} ω_{ℓ_{1}, ℓ_{2}} δ_{\{\max_{1 \leq h < j \leq p} {(n_{ℓ_{1}}^{- 1} {\hat{θ}}_{ℓ_{1} h j} + n_{ℓ_{2}}^{- 1} {\hat{θ}}_{ℓ_{2} h j})}^{- 1} {({\hat{r}}_{ℓ_{1} h j} - {\hat{r}}_{ℓ_{2} h j})}^{2} > s (n_{ℓ_{1}}, n_{ℓ_{2}}, p)\}},

with $\{ω_{ℓ_{1}, ℓ_{2}}, 1 \leq ℓ_{1} < ℓ_{2} \leq K\}$ being a vector of weights.

For simplicity, we focus on the asymptotic distribution of M_n,K.

We first present a key lemma as follows.

Lemma 3.1. Assume that Assumptions (a) and (b) hold for ℓ = 1,…, K. Then, $\{tr [{({\hat{R}}_{ℓ_{1}} - {\hat{R}}_{ℓ_{2}})}^{2}] - μ_{A ℓ_{1} ℓ_{2}}, 1 \leq ℓ_{1} < ℓ_{2} \leq K\}$ are asymptotically distributed as a multivariate normal distribution. Moreover, we have

\{tr [{({\hat{R}}_{ℓ_{1}} - {\hat{R}}_{ℓ_{2}})}^{2}] - μ_{A ℓ_{1} ℓ_{2}}, tr [{({\hat{R}}_{ℓ_{3}} - {\hat{R}}_{ℓ_{4}})}^{2}] - μ_{A ℓ_{3} ℓ_{4}}\} \overset{L}{\to} N (0_{2}, Γ),

where $Γ = {\{γ_{A u u^{'}}\}}_{u, u^{'} = 1}^{2}$ with $γ_{A 11} = σ_{A ℓ_{1} ℓ_{2}}$ , $γ_{A 22} = σ_{A ℓ_{3} ℓ_{4}}$ and

γ_{A 12} = \{\begin{array}{l} σ_{A ℓ_{1} ℓ_{2} ℓ_{3} ℓ_{4}}, \\ 0, \end{array} \begin{array}{l} ℓ_{1} < ℓ_{2} = ℓ_{3} < ℓ_{4}; \\ ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4} a r e a l l u n e q u a l . \end{array}

Moreover, $μ_{A ℓ_{1} ℓ_{2}}$ , $μ_{A ℓ_{3} ℓ_{4}}$ and γ_Auu' have closed forms and will be given in Appendix.

Based on Lemma 3.1, we can establish the asymptotic null distribution of M_n,K as follows.

Theorem 3.4. Under the same assumptions of Lemma 3.1 and H_0K, we conclude that

(III.1). $v_{K}^{- 1 / 2} \sum_{1 \leq ℓ_{1} < ℓ_{2} \leq K} ω_{ℓ_{1}, ℓ_{2}} \{tr [{({\hat{R}}_{ℓ_{1}} - {\hat{R}}_{ℓ_{2}})}^{2}] - μ_{z ℓ_{1} ℓ_{2}}\} \overset{L}{\to} N (0, 1)$ ;

(III.2). $v_{K}^{- 1 / 2} [M_{n, K} - \sum_{1 \leq ℓ_{1} < ℓ_{2} \leq K} ω_{ℓ_{1}, ℓ_{2}} μ_{z ℓ_{1} ℓ_{2}}] \overset{L}{\to} N (0, 1)$ under some additional conditions, including $s (n_{ℓ_{1}}, n_{ℓ_{2}}, p) - 4 \log p \to + \infty$ and (C1), (C2), and (C3) of Cai, Liu and Xia (2013) for R and $z_{i}^{(ℓ)}$ , where ν_K = 4[tr(R²)]²uK is given by

\begin{array}{l} u_{K} = \sum_{1 \leq ℓ_{1} < ℓ_{2} \leq K} ω_{ℓ_{1} ℓ_{2}}^{2} {[{(n_{ℓ_{1}} - 1)}^{- 1} + {(n_{ℓ_{2}} - 1)}^{- 1}]}^{2} \\ + 2 \sum_{1 \leq ℓ_{1} < ℓ_{2} = ℓ_{3} < ℓ_{4} \leq K} ω_{ℓ_{1} ℓ_{2}} ω_{ℓ_{3} ℓ_{4}} {(n_{ℓ_{2}} - 1)}^{- 2} \\ + 2 \sum_{1 \leq ℓ_{1} < ℓ_{3} < ℓ_{2} = ℓ_{4} \leq K} ω_{ℓ_{1} ℓ_{2}} ω_{ℓ_{3} ℓ_{4}} {(n_{ℓ_{2}} - 1)}^{- 2} \\ + 2 \sum_{1 \leq ℓ_{1} = ℓ_{3} < ℓ_{2} < ℓ_{4} \leq K} ω_{ℓ_{1} ℓ_{2}} ω_{ℓ_{3} ℓ_{4}} {(n_{ℓ_{2}} - 1)}^{- 2}, \end{array}

and $μ_{z ℓ_{1} ℓ_{2}}$ can be similarly defined as µ_z12.

Remark 3.3. There are two important issues associated with M_n,K. The first one is to determine the weights $ω_{ℓ_{1} ℓ_{2}}$ . Since the asymptotic variance of $tr [{({\hat{R}}_{ℓ_{1}} - {\hat{R}}_{ℓ_{2}})}^{2}] - μ_{z ℓ_{1} ℓ_{2}}$ is equal to $4 {[{(n_{ℓ_{1}} - 1)}^{- 1} + {(n_{ℓ_{2}} - 1)}^{- 1}]}^{2} {[tr (R^{2})]}^{2}$ , a reasonable choice of $ω_{ℓ_{1} ℓ_{2}}$ , is $ω_{ℓ_{1} ℓ_{2}} = {[{(n_{ℓ_{1}} - 1)}^{- 1} + {(n_{ℓ_{2}} - 1)}^{- 1}]}^{- 1}$ for 1 ≤ ℓ₁ < ℓ₂ ≤ K. The second one is to estimate the asymptotic mean and variance under H_0K, since we do not know what the true R is. A good estimate of p⁻¹tr(R²) is $\sum_{ℓ = 1}^{K} (n_{ℓ} - 1) {(n_{1} + \dots + n_{K} - K)}^{- 1} {\hat{a}}_{ℓ}$ , where ${\hat{a}}_{ℓ}$ was defined in Remark 3.2. Then the estimate of ν_K

{\hat{v}}_{K} = \sum_{ℓ = 1}^{K} (n_{ℓ} - 1) {(n_{1} + \dots + n_{K} - K)}^{- 1} p {\hat{a}}_{ℓ} u_{K} .

Furthermore, the estimate ${\hat{μ}}_{z ℓ_{1} ℓ_{2}}$ can be obtained by replacing 1 and 2 by ℓ₁ and ℓ₂ in ${\hat{μ}}_{z 12}$ in Remark 3.2. The C₀ is the same as C_0,2 in Remark 3.1. The threshold $s (n_{ℓ_{1}}, n_{ℓ_{2}}, p)$ is obtained by replacing 1 and 2 by ℓ₁ and ℓ₂ in s(n₁, n₂, p) in Remark 3.2.

4. Estimation of the kurtosis β₁.

To estimate β₁ in Theorems 2.1 and 3.1, we consider two cases as follows.

Case 1: When R₁ is unknown, the covariance matrix Σ₁ is unknown. We may use an estimator of β₁ as follows:

{\tilde{β}}_{1} = \frac{n_{1} \hat{V} - 2 \{(n_{1} - 1) tr (S_{1}^{2}) - {[tr (S_{1})]}^{2}\}}{n_{1} \sum_{j = 1}^{p} s_{1 j j}^{2}},

where $S_{1} = {(s_{1 ℓ j})}_{ℓ, j = 1}^{p}$ and

\hat{V} = {(n_{1} - 1)}^{- 1} \sum_{i = 1}^{n_{1}} {\{{(z_{i}^{(1)} - {\bar{z}}^{(1)})}^{T} (z_{i}^{(1)} - {\bar{z}}^{(1)}) - n_{1}^{- 1} \sum_{i = 1}^{n_{1}} [{(z_{i}^{(1)} - {\bar{z}}^{(1)})}^{T} (z_{i}^{(1)} - {\bar{z}}^{(1)})]\}}^{2} .

Case 2: For R₁ = R_∗ for a pre-specified correlation matrix R_∗, we may estimate β₁ as follows:

{\hat{β}}_{1} = p^{- 1} \sum_{ℓ = 1}^{p} [\frac{{(n_{1} - 1)}^{- 1} \sum_{i = 1}^{n_{1}} {(ξ_{ℓ i} - {\bar{ξ}}_{ℓ})}^{2} - 2 {\bar{ξ}}_{ℓ}^{2}}{{\bar{ξ}}_{ℓ}^{2} \sum_{j = 1}^{p} {(r_{* \frac{1}{2} ℓ j})}^{4}}],

where $ξ_{ℓ i} = {(z_{ℓ i}^{(1)} - {\bar{z}}_{ℓ i}^{(1)})}^{2}$ , ${\bar{ξ}}_{ℓ} = n_{1}^{- 1} \sum_{i = 1}^{n_{1}} ξ_{ℓ i}$ and $R_{*}^{1 / 2} = {(r_{* \frac{1}{2} ℓ j})}_{ℓ, j = 1}^{p}$

The following lemma gives the consistency of the estimator ${\hat{β}}_{1}$ under the null hypothesis H₀₁ : R₁ = R_∗.

Lemma 4.1. Suppose that $E [{(z_{ℓ 1}^{(1)})}^{8}] \leq c$ holds for all ℓ = 1,…, p, where c is a positive constant. Under the null hypothesis H₀₁ and assumptions (a)–(b), we have

{\hat{β}}_{1} = β_{1} + o_{p} (1) .

Proof. Without loss of generality, assume $E z_{ℓ 1}^{(1)} = 0$ . Let e_ℓ be the ℓ-th column of the p × p identity matrix. We can show the following results:

n_{1}^{- 1} \sum_{i = 1}^{n_{1}} z_{ℓ i}^{(1)} = o_{p} (1),

(4.1)

n_{1}^{- 1} \sum_{i = 1}^{n_{1}} {(z_{ℓ i}^{(1)})}^{2} = E {(z_{ℓ 1}^{(1)})}^{2} + o_{p} (1),

(4.2)

n_{1}^{- 1} \sum_{i = 1}^{n_{1}} {(z_{ℓ i}^{(1)})}^{4} = E {(z_{ℓ 1}^{(1)})}^{4} + o_{p} (1),

(4.3)

where o_p(1) is uniform for ℓ = 1,…, p. For instance, to prove (4.1), we have

E [{(n_{1}^{- 1} \sum_{i = 1}^{n_{1}} z_{ℓ i}^{(1)} - E z_{ℓ 1}^{(1)})}^{2}] = n_{1}^{- 1} E {(z_{ℓ 1}^{(1)} - E z_{ℓ 1}^{(1)})}^{2} = n_{1}^{- 1} e_{ℓ}^{T} \sum_{*} e_{ℓ} \leq n_{1}^{- 1} c = o (1),

where o(1) is uniform for all ℓ = 1,…, p. It follows that

{\bar{ξ}}_{ℓ} = n_{1}^{- 1} \sum_{i = 1}^{n_{1}} {(z_{ℓ 1}^{(1)})}^{2} - {[n_{1}^{- 1} \sum_{i = 1}^{n_{1}} z_{ℓ i}^{(1)}]}^{2} = e_{ℓ}^{T} \sum_{*} e_{ℓ} + o_{p} (1),

n_{1}^{- 1} \sum_{i = 1}^{n_{1}} ξ_{ℓ i}^{2} = E {(z_{ℓ 1}^{(1)})}^{4} + o_{p} (1) .

Therefore, we have

\begin{matrix} {(n_{1} - 1)}^{- 1} \sum_{i = 1}^{n_{1}} {(ξ_{ℓ i} - {\bar{ξ}}_{ℓ})}^{2} / {\bar{ξ}}_{ℓ}^{2} = \frac{E {(z_{ℓ 1}^{(1)})}^{4} - {(e_{ℓ}^{T} \sum_{*} e_{ℓ})}^{2}}{{(e_{ℓ}^{T} \sum_{*} e_{ℓ})}^{2}} + o_{p} (1) \\ = 2 + β_{1} \sum_{j = 1}^{p} {[e_{j}^{T} R_{*}^{1 / 2} e_{ℓ}]}^{4} + o_{p} (1), \end{matrix}

which yields ${\hat{β}}_{1} = β_{1} + o_{p} (1)$ . This completes the proof of Lemma 4.1.

5. Simulation studies.

In this section, we carried out simulation studies to evaluate the finite-sample performance of the proposed test statistics in terms of the empirical test size and power. We consider both one sample testing problem and two sample testing problem. For the one sample testing problem, we set the dimension p to be p = 50, 100, 200, 500, and 1000 and the sample size n₁ to be n₁ = 100, 120, 200, and 300. The data were generated according to $z_{i}^{(1)} = R^{1 / 2} w_{z i}^{(1)}$ for i = 1,…, n₁, where the elements of $w_{z i}^{(1)}$ were independently and identically generated from Gaussian population N (0, 1) or Gamma(4, 2) − 2. For the two sample testing problem, we set p = 50, 100, 200, 500, and 1000 and (n₁, n₂) = (100, 100), (150, 150), (200, 200). The data were generated according to $z_{i}^{(ℓ)} = R_{ℓ}^{1 / 2} w_{z i}^{(ℓ)}$ for i = 1,…, n_ℓ and ℓ = 1, 2, where the elements of $w_{z i}^{(ℓ)}$ were independently and identically generated from Gaussian population N (0, 1) or Gamma(4, 2) − 2. For the three sample testing problem, we set p = 50, 100, 200, 500, and 1000 and (n₁, n₂, n₃) = (100, 100, 100), (100, 100, 100), (100, 100, 200) and (100, 200, 200). The data were generated according to $z_{i}^{(ℓ)} = R_{ℓ}^{1 / 2} w_{z i}^{(ℓ)}$ for i = 1,…, n_ℓ and ℓ = 1, 2, 3, where the elements of $w_{z i}^{(ℓ)}$ were independently and identically generated from Gaussian population N (0, 1) or Gamma(4, 2) − 2. We set the nominal size to be 5%, run 2,000 replications for empirical sizes and 1000 replications for empirical powers for each setting.

We consider nine different sets of population correlation matrices for R_ℓ. For the two sample testing problem, we compare our tests denoted as “FDS” for M_n,2 and “MAX” for $M_{n, 2}^{'}$ with the extreme statistic test, denoted as “CZ” in Cai and Zhang (2016). However, for the one sample testing problem, we cannot find any competing method when R_∗ is not an identity matrix, so we do not include any alternative method. When R_∗ is an identity matrix, we compare our test “FDS” for M_n,1 and “MAX” for $M_{n, 1}^{'}$ with “GHPY” in Gao et al. (2017) and “LX” in Li and Xue (2015). For the three-sample testing problem, since we cannot find any competing method, we do not include any alternative method. For the sake of space, we selectively present some key results in Tables 1–3 and include additional results in the supplementary document. The first three models are designed for the one sample testing problem, whereas the middle four ones are for the two sample testing problem and the last three ones are for the three-sample testing problem. The ten different models of population correlation matrices are summarized as follows.

Table 1.

Empirical sizes in Model 1.1 and empirical powers in Models 1.2–1.3 for H₀₁ (in percentage)

			w_ij ~ N (0; 1)					w_ij ~ Gamma(4,2)–2

ϵ	n	Methods	p=50	100	200	500	1000	50	100	200	500	1000
			Empirical sizes in Model 1.1
0.0	100	FDS	6.20	4.75	5.90	5.75	6.25	5.45	6.25	6.60	6.55	5.90
		MAX	4.25	3.25	3.50	3.10	3.30	4.60	5.00	4.30	4.60	4.75
		GHPY	4.85	4.15	4.70	4.10	5.10	4.85	4.50	4.70	5.75	4.90
		LX	3.15	1.90	1.70	0.90	0.90	4.35	4.55	3.55	3.10	3.10
	200	FDS	6.15	5.15	5.90	6.05	5.95	5.70	5.55	6.35	5.55	5.85
		MAX	4.40	3.40	5.05	4.10	3.80	4.85	4.45	5.10	4.75	4.65
		GHPY	5.80	4.80	4.90	4.85	4.95	5.20	4.35	5.35	4.80	5.10
		LX	5.35	3.40	3.55	2.15	2.00	5.20	4.90	5.40	4.90	5.00
	300	FDS	5.15	4.60	5.15	5.65	5.05	6.25	6.05	6.50	5.85	6.40
		MAX	3.90	3.80	3.80	4.45	3.50	4.55	4.40	5.10	4.15	4.75
		GHPY	4.45	5.20	4.65	5.10	5.25	4.95	5.50	5.65	4.80	5.85
		LX	5.25	4.25	3.90	2.80	2.75	5.15	5.20	6.05	5.15	5.65
0.5	100	FDS	5.50	5.25	5.90	5.95	6.35	5.75	6.00	5.70	5.80	5.95
		MAX	5.20	4.80	5.50	5.65	5.15	5.10	4.45	4.55	4.75	4.10
	200	FDS	5.25	6.00	5.30	5.05	6.25	5.60	5.40	5.75	5.45	5.10
		MAX	4.60	4.65	4.45	4.45	4.90	4.65	4.55	4.20	3.85	4.35
	300	FDS	5.85	6.00	5.00	5.75	5.05	4.50	5.75	6.35	5.80	5.80
		MAX	5.05	4.25	4.60	4.20	4.00	3.70	4.65	5.10	4.30	4.30

			Empirical powers in Model 1.2
1.0	100	FDS	28.9	59.4	90.8	99.9	100.0	28.1	58.1	92.1	100.0	100.0
		MAX	21.0	49.8	86.8	99.8	100.0	20.4	49.6	88.3	99.9	100.0
		GHPY	19.6	49.5	86.9	100.0	100.0	20.4	46.9	87.4	99.3	100.0
		LX	7.10	7.6	8.9	15.7	23.3	10.7	13.4	16.1	24.4	34.3
	200	FDS	58.9	93.3	100.0	100.0	100.0	55.8	92.9	99.9	100.0	100.0
		MAX	48.9	90.8	100.0	100.0	100.0	48.8	90.2	99.8	100.0	100.0
		GHPY	51.6	90.4	99.9	100.0	100.0	50.9	89.8	100.0	100.0	100.0
		LX	17.8	28.3	49.7	86.90	97.55	22.9	34.0	54.0	85.9	96.8
	300	FDS	83.4	99.9	100.0	100.0	100.0	81.9	99.7	100.0	100.0	100.0
		MAX	77.4	99.5	100.0	100.0	100.0	75.6	99.5	100.0	100.0	100.0
		GHPY	78.8	99.6	100.0	100.0	100.0	77.3	99.0	100.0	100.0	100.0
		LX	33.6	59.0	90.6	99.8	100.0	37.2	62.8	89.6	99.4	100.0
0.0	100	FDS	75.0	79.0	85.2	90.0	94.6	74.5	81.6	85.5	92.1	95.3
		MAX	81.6	84.4	90.1	93.3	96.8	82.2	87.0	90.1	94.9	97.2
		GHPY	7.6	6.2	5.5	4.6	5.5	9.0	5.9	5.1	5.7	5.1
		LX	85.4	88.1	92.2	94.7	97.4	87.3	90.0	92.8	95.8	97.9
	200	FDS	71.4	78.7	79.9	84.6	88.2	72.0	76.4	80.3	84.5	88.4
		MAX	77.9	84.8	84.3	88.4	91.7	78.9	78.9	86.	89.3	91.6
		GHPY	10.4	6.2	5.5	5.3	4.9	10.1	5.6	6.4	5.1	5.5
		LX	83.3	88.3	88.1	90.6	93.7	83.7	86.3	88.7	91.8	93.0
	300	FDS	71.2	76.6	80.8	84.3	87.5	71.9	76.8	80.4	85.7	86.4
		MAX	77.2	81.6	86.0	88.8	90.5	77.5	82.7	85.4	88.9	90.4
		GHPY	8.4	7.0	5.2	5.3	5.1	9.2	6.7	6.8	4.7	5.1
		LX	83.2	85.0	88.2	90.7	92.7	82.6	86.6	88.4	91.6	92.5

			Empirical powers in Model 1.3
0.09	100	FDS	44.8	44.1	47.0	46.8	47.2	38.4	39.7	38.5	38.0	38.6
		MAX	53.8	53.4	55.0	54.2	54.5	46.3	47.3	46.1	44.3	44.4
	200	FDS	99.2	99.8	99.8	99.9	99.9	97.0	97.3	97.4	97.5	98.1
		MAX	99.4	99.6	99.9	99.9	100.0	98.3	98.5	98.3	98.2	98.7
	300	FDS	100.0	100.0	100.0	100.0	100.0	99.9	99.9	99.9	100.0	100.0
		MAX	100.0	100.0	100.0	100.0	100.0	99.9	99.9	99.9	100.0	100.0
0.12	100	FDS	93.6	94.6	96.0	96.0	96.3	84.9	84.5	82.6	78.8	78.5
		MAX	96.2	96.7	97.5	97.5	97.3	89.4	88.5	84.4	82.5	82.3
	200	FDS	100.0	100.0	100.0	100.0	100.0	99.8	99.8	99.7	99.3	99.3
		MAX	100.0	100.0	100.0	100.0	100.0	99.9	99.8	99.9	99.6	99.6
	300	FDS	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0
		MAX	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0

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Table 3.

Empirical test sizes in Model 3.1 and empirical powers in Models 3.2 and 3.3 for H₀₃ (in percentage)

		w_ij ~ N (0, 1)					w_ij ~ Gamma(4,2)–2

(n₁,n₂,n₃)		p=50	100	200	500	1000	50	100	200	500	1000
		Empirical test sizes in Model 3.1
(50,100, 100)	FDS	6.10	6.60	5.95	6.20	6.60	5.45	5.50	6.00	5.85	5.30
(100,100, 100)	FDS	6.30	5.90	5.20	5.85	5.35	5.90	4.85	5.15	5.90	5.60
(100,100, 200)	FDS	4.70	5.60	5.65	5.75	5.25	5.00	5.35	4.80	5.35	5.15
(100,200, 200)	FDS	5.85	5.45	5.60	5.70	4.55	5.75	4.85	5.40	4.90	5.25

		Empirical powers in Model 3.2
(50,100, 100)	FDS	18.5	44.9	86.4	100.0	100.0	14.4	36.3	81.6	100.0	100.0
(100,100, 100)	FDS	22.1	52.5	91.8	100.0	100.0	18.6	50.1	91.0	100.0	100.0
(100,100, 200)	FDS	28.8	74.1	99.6	100.0	100.0	26.8	71.2	98.9	100.0	100.0
(100,200, 200)	FDS	38.9	88.2	99.9	100.0	100.0	39.2	85.8	99.8	100.0	100.0

		Empirical powers in Model 3.3
(50,100, 100)	FDS	39.7	37.4	37.1	40.3	53.1	30.1	28.8	32.1	33.3	35.8
(100,100, 100)	FDS	47.2	45.9	47.7	50.4	50.1	40.6	43.2	41.5	45.1	46.8
(100,100, 200)	FDS	68.1	69.8	70.3	73.6	73.1	64.2	65.2	65.6	68.8	67.8
(100,200, 200)	FDS	84.5	87.5	87.2	89.3	88.8	83.2	85.0	85.1	86.3	86.1

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Model 1.1: The population correlation matrix is set as $R_{1} = R_{*} = {(ρ^{|j^{'} - j|})}_{j^{'}, j = 1}^{p}$ where ρ is taken as 0.0 and 0.5.
Model 1.2: The population correlation matrix is set as $R_{1} = R_{*} + ϵ 0.02 (1_{p} 1_{p}^{T} - I_{p}) + (1 - ϵ) 2.5 \sqrt{(\log p) / n} (e_{2} e_{1}^{T} + e_{1} e_{2}^{T})$ , where R_* = I_p, 1_p is a p × 1 vector of ones and e_k is the kth column of the p × p identity matrix. When ϵ = 1, the signal pattern of R₁ − R_∗ is dense. When ϵ = 0, the signal pattern of R₁ − R_∗ is sparse.
Model 1.3: The population correlation matrix is set as $R_{1} = R_{*} + ϵ \sqrt{\log p \log n} (e_{2} e_{1}^{T} + e_{1} e_{2}^{T})$ where $R_{*} = {({0.25}^{|i - j|})}_{i, j = 1}^{p}$ and ϵ = 0.09 and 0.12. In this case, the signal pattern of R₁ − R_∗ is sparse.
Model 2.1: The population correlation matrices are set as $R_{1} = R_{2} = {(ρ^{|j^{'} - j|})}_{j^{'}, j = 1}^{p}$ with ρ = 0.00, 0.25, 0.50, and 0.75. The simulation results for ρ = 0.25 and 0.75 are included in the supplementary file.
Model 2.2: The population correlation matrices are set as $R_{1} = {({0.5}^{|j^{'} - j|})}_{j^{'}, j = 1}^{p}$ and $R_{2} = R_{1} + ϵ (1_{p} 1_{p}^{T} - I_{p})$ with ϵ = 0.05 and 0.08. In this case, the signal pattern of R₂ − R₁ is dense.
Model 2.3: The population correlation matrices are set as $R_{1} = {(ρ^{|j^{'} - j|})}_{j^{'}, j = 1}^{p}$ and $R_{2} = R_{1} + ϵ_{p} (e_{2} e_{1}^{T} + e_{1} e_{2}^{T})$ with ρ = 0.05, 0.08, 0.10, and 0.12 and ϵ_p = exp(0.008p)/[1 + exp(0.008p)]. In this case, the signal pattern of R₂ − R₁ is sparse. The simulation results for ρ = 0.05, 0.10, and 0.12 are included in the supplementary file.
Model 2.4: The population correlation matrices are set as R₁=I_p and $R_{2} = {(ρ^{|j^{'} - j|})}_{j^{'}, j = 1}^{p}$ with ϵ = 0.2, 0.225, 0.25, and 0.275. In this case, the signal pattern of R₂ − R₁ is between dense case and sparse case. The simulation results are included in the supplementary file.
Model 3.1: The three population correlation matrices are taken as R₁ = R₂ = R₃ = I_p. The model is used for evaluating the empirical performance on Type I errors of the proposed test M_n,3.
Model 3.2: The three population correlation matrices are taken as R₁ = R₂ = I_p and $R_{3} = I_{p} + 0.03 {(11^{T} - I_{p})}_{i, j = 1}^{p}$ . Here R₃ − R₁ or R₃ − R₂ is dense.
Model 3.3: The three population correlation matrices are taken as R₁ = R₂ = I_p and $R_{3} = {(δ_{\{i = j\}} + 0.2 δ_{\{|i - j| = 1\}})}_{i, j = 1}^{p}$ . Here R₃ – R₁ or R₃ – R₂ is a little sparse.

Overall, the Type I error rates for our tests “FDS” and “MAX” are relatively accurate for all sample sizes, for all dimensions, for all correlation matrices, and for the two different distributions of error terms. For the one sample testing problem, “FDS” and “MAX” can deal with an arbitrary correlation matrix R_∗, whereas other test statistics “GHPY” and “LX” cannot. It seems that both ρ and p have some minor impact on its Type I error rates. The proposed tests ‘FDS” and “MAX” perform very well for both sparse and dense alternatives. Consistent with our expectations, the statistical powers for rejecting the null hypothesis increase as ϵ, n, and p increase. It seems that “MAX” has a little better performance than “FDS”.

For the two- and three- sample testing problems, “FDS” and “MAX” also can deal with arbitrary correlation matrices. It seems that ρ, p, and the error distribution have little impact on its Type I error rates. The proposed tests “FDS” and “MAX” perform reasonably well for sparse alternatives, dense alternatives, and between sparse and dense alternatives. It seems that “MAX” is slightly better than “FDS”.

6. Real data analysis.

6.1. Alzheimer’s Disease Neuroimaging Initiative (ADNI) data.

“Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). For up-to-date information, see www.adni-info.org.”¹ We consider 749 T1 weighted images collected at the baseline of ADNI1, consisting of 206 normal subjects, 364 mild cognitive impairment (MCI) subjects and 179 Alzheimer’s disease (AD) subjects. These scans were performed on a 1.5T MRI scanners using a sagittal MPRAGE sequence and the typical protocol includes the following parameters: repetition time (TR) = 2400 ms, inversion time (TI) = 1000 ms, flip angle = 8°, and field of view (FOV) = 24 cm with a 256×256×170 mm³ acquisition matrix in the x, y, and z dimensions, which yields a voxel size of 1.25 × 1.26 × 1.2 mm³.

The T1-weighted images were processed using the Hierarchical Attribute Matching Mechanism for Elastic Registration (HAMMER) pipeline. The processing steps include anterior commissure and posterior commissure correction, skull-stripping, cerebellum removal, intensity inhomogeneity correction, and segmentation. We performed automatic regional labeling by labeling the template and by transferring the labels following the deformable registration of subject images. Finally, we labeled 93 regions of interest (ROIs) and computed their volumes for each subject.

6.2. Group comparisons.

We are interested in characterizing differences among the three correlation matrices of ROI volumes for normal subjects, MCI subjects and AD subjects, which are denoted as R_NC, R_MCI, and R_AD, respectively. Statistically, we test three two sample testing problems, including R_NC = R_MCI, R_NC = R_AD, and R_MCI = R_AD, and one three sample testing problem, that is, R_NC = R_MCI = R_AD.

We applied the test statistics M_n,2 and M_n,3 to carry out these tests as follows. First, for each ROI, we fitted a linear regression model with its ROI volume as response and age, gender and whole brain volume as covariates by using data obtained from all subjects. Second, for each group, we calculated its correlated matrix based on the residuals of all ROIs obtained from the first step. Figure 1 presents the correlation matrices corresponding to the three groups. Then, we clustered the 93 ROIs according to the correlation matrix of the normal control group. For example, Cluster 1 includes the large area of prefrontal cortex, and its functions span over the frontoparietal control network (orbitofrontal cortex, middle frontal gyrus), default node network and ventral attention network. This region has been implicated in decision making, complex cognitive behavior, processing of higher information, decision making, personal expression, social behavior moderating, attention, memory, recognizing faces, characters and etc. Third, we calculated the p–value of testing R_NC = R_MCI, that of R_NC = R_AD, and that of R_MCI = R_AD as 1.23 × 10⁻⁹, 0, and 1.78×10⁻¹¹, respectively. Fourth, we calculated the p–value of testing R_NC = R_MCI = R_AD as 0.

Fig 1. — Graphical display of correlation matrices of normal subjects, MCI subjects and AD subjects.

Supplementary Material

supplement

NIHMS1000738-supplement-supplement.pdf^{(394.1KB, pdf)}

Fig 2. — Graphical display of difference between correlation matrices of normal subjects, MCI subjects and AD subjects.

Table 2.

Empirical sizes in Model 2.1 and empirical powers in Models 2.2–2.3 for H₀₂ (in percentage)

			w_ij ~ N (0, 1)				w_ij ~ Gamma(4,2)–2

ϵ	(n₁,n₂)	Methods	p=50	100	200	500	50	100	200	500
			Empirical sizes in Model 2.1
0.0	(100, 100)	FDS	5.85	6.00	6.20	6.85	4.85	4.65	5.15	5.50
		MAX	4.65	5.15	4.90	5.05	3.50	3.45	3.85	4.10
		CZ	5.00	5.10	5.85	5.25	3.25	3.70	3.55	3.10
	(150, 150)	FDS	5.85	6.45	5.50	5.50	4.85	5.55	4.80	4.65
		MAX	4.70	5.55	5.10	5.15	4.00	4.35	4.25	3.55
		CZ	4.40	4.45	4.80	5.05	3.35	3.95	4.25	2.80
	(200, 200)	FDS	5.30	5.20	5.25	5.80	4.85	4.95	5.40	5.25
		MAX	4.25	4.55	4.45	4.75	3.60	3.70	4.00	4.20
		CZ	4.20	5.45	4.50	4.70	3.60	4.00	3.00	3.35
0.5	(100, 100)	FDS	6.15	6.75	5.60	6.25	5.30	4.20	5.50	5.40
		MAX	5.30	5.75	5.00	5.60	3.45	4.15	4.60	4.55
		CZ	5.55	5.45	5.50	5.55	4.00	4.40	4.15	3.95
	(150, 150)	FDS	6.10	5.95	6.70	6.25	5.40	5.75	5.70	5.05
		MAX	5.30	4.75	5.85	4.85	4.05	4.04	4.20	4.45
		CZ	5.60	5.55	4.75	5.00	3.75	3.60	3.85	4.00
	(200, 200)	FDS	5.20	6.30	6.30	6.90	5.45	4.90	6.10	5.15
		MAX	4.20	5.30	5.05	5.95	4.50	3.65	5.65	4.30
		CZ	4.20	4.55	4.95	5.30	4.15	3.70	4.30	4.05

			Empirical sizes in Model 2.2
0.05	(100, 100)	FDS	49.6	83.6	99.2	100.0	48.6	82.8	99.9	100.0
		MAX	42.5	78.8	98.7	100.0	41.1	77.6	99.9	100.0
		CZ	9.8	10.4	12.6	14.3	9.7	9.8	10.3	10.7
	(150, 150)	FDS	71.8	96.7	100.0	100.0	70.8	97.3	100.0	100.0
		MAX	64.7	94.9	99.9	100.0	62.2	95.5	100.0	100.0
		CZ	14.4	15.0	16.8	16.6	11.5	11.8	14.1	14.3
	(200, 200)	FDS	84.9	99.6	100.0	100.0	82.4	99.8	100.0	100.0
		MAX	79.5	99.2	100.0	100.0	77.3	99.5	100.0	100.0
		CZ	14.9	17.5	19.5	23.6	14.6	15.5	17.9	20.6
0.08	(100, 100)	FDS	87.7	99.8	100.0	100.0	90.1	99.4	100.0	100.0
		MAX	84.2	99.7	100.0	100.0	85.8	99.3	99.9	100.0
		CZ	18.4	21.8	24.1	30.1	19.3	20.5	21.6	26.1
	(150, 150)	FDS	98.5	100.0	100.0	100.0	97.9	100.0	100.0	100.0
		MAX	97.5	100.0	100.0	100.0	97.2	100.0	100.0	100.0
		CZ	31.4	32.8	40.2	43.4	26.0	31.0	36.6	40.1
	(200, 200)	FDS	99.7	100.0	100.0	100.0	99.8	100.0	100.0	100.0
		MAX	99.5	100.0	100.0	100.0	99.6	100.0	100.0	100.0
		CZ	37.6	45.4	53.8	62.3	37.3	42.8	49.6	58.7

			Empirical powers in Model 2.3
0.08	(100, 100)	FDS	65.5	86.8	99.6	99.6	61.8	78.5	97.6	96.5
		MAX2	72.8	90.6	99.8	99.8	68.6	83.3	98.8	97.5
		CZ	78.7	93.2	99.9	99.9	76.0	87.7	99.5	98.3
	(150, 150)	FDS	94.1	99.4	100.0	100.0	92.6	98.5	99.9	100.0
		MAX	96.5	99.6	100.0	100.0	95.2	99.3	99.9	100.0
		CZ	97.7	99.9	100.0	100.0	97.0	99.5	99.9	100.0
	(200, 200)	FDS	99.5	100.0	100.0	100.0	99.1	99.9	100.0	100.0
		MAX	99.9	100.0	100.0	100.0	99.5	100.0	100.0	100.0
		CZ	99.9	100.0	100.0	100.0	99.7	100.0	100.0	100.0

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ACKNOWLEDGEMENT

The authors thank the editor, the associate editor and the referees for their constructive comments and suggestions that led to a significant improvement of this article.

APPENDIX A: SOME EXPRESSIONS

Let r_∗khj be the (h, j) entry of $R_{*}^{k}$ and r_ℓkhj be the (h, j) entry of $R_{ℓ}^{k}$ for k = 1/2, 1, 3/2, 2, 3. Let e_j be the jth column of the p × p identity matrix.

A.1. Expressions of µ_zA, µ_z0 and $σ_{z A}^{2}$ for one population in Theorem 2.1 and 2.2. Expression of µ_zA and _µz0:

\begin{array}{l} µ_{z A} = [n_{1} (n_{1} - 1) + 2] {(n_{1} - 1)}^{- 2} tr (R_{1}^{2}) + (n_{1}^{2} - n_{1} - 1) p^{2} n_{1}^{- 1} {(n_{1} - 1)}^{- 2} \\ + β_{1} n_{1} p {(n_{1} - 1)}^{- 2} - 2 tr (R_{1} R_{*}) + tr (R_{*}^{2}) \\ - 4 n_{1} {(n_{1} - 1)}^{- 2} [2 tr (R_{1}^{2}) + β_{1} \sum_{h = 1}^{p} \sum_{j = 1}^{p} r_{1 \frac{3}{2} h j} {(r_{1 \frac{1}{2} h j})}^{3}] \\ + 2 n_{1} {(n_{1} - 1)}^{- 2} [2 tr (R_{1} R_{*}) + β_{1} \sum_{h = 1}^{p} \sum_{j = 1}^{p} e_{j}^{T} R_{1}^{1 / 2} R_{*} e_{h} {(r_{1 \frac{1}{2} h j})}^{3}] \\ + 2 n_{1} (n_{1} - 2) {(n_{1} - 1)}^{- 3} [2 tr (R_{1}^{2}) + β_{1} \sum_{h = 1}^{p} r_{12 h h} \sum_{j = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{4}] \\ - 1.5 n_{1} {(n_{1} - 1)}^{- 2} [2 tr (R_{1} R_{*}) + β_{1} \sum_{j = 1}^{p} e_{j}^{T} R_{1} R_{*} e_{j} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{4}] \\ + n_{1} (n_{1} - 2) {(n_{1} - 1)}^{- 3} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} {(r_{11 j j^{'}})}^{2} [2 {(r_{11 j j^{'}})}^{2} + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} {(r_{1 \frac{1}{2} h j^{'}})}^{2}] \\ - 0.5 n_{1} {(n_{1} - 1)}^{- 2} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} (r_{11 j j^{'}}) (r_{* 1 j j^{'}}) [2 {(r_{11 j j^{'}})}^{2} + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} {(r_{1 \frac{1}{2} h j^{'}})}^{2}] . \end{array}

When R₁ = R_∗, we have

\begin{array}{l} μ_{z 0} = μ_{z A} = (n_{1}^{2} - n_{1} - 1) p^{2} n_{1}^{- 1} {(n_{1} - 1)}^{- 2} - [2 n_{1}^{2} + n_{1} + 1] {(n_{1} - 1)}^{- 3} tr (R_{*}^{2}) \\ + \frac{(n_{1}^{2} - 3 n_{1})}{{(n_{1} - 1)}^{3}} \sum_{j^{'} = 1}^{p} \sum_{j = 1}^{p} {(r_{* 1 j j^{'}})}^{4} + β_{1} [n_{1} p {(n_{1} - 1)}^{- 2} \\ - 2 n_{1} {(n_{1} - 1)}^{- 2} \sum_{h = 1}^{p} \sum_{j = 1}^{p} (r_{* \frac{3}{2} h j}) {(r_{* \frac{1}{2} h j})}^{3} \\ + 0.5 (n_{1}^{2} - 5 n_{1}) {(n_{1} - 1)}^{- 3} \sum_{j = 1}^{p} (r_{* 2 j j}) \sum_{h = 1}^{p} {(r_{* \frac{1}{2} h j})}^{4} \\ + 0.5 (n_{1}^{2} - 3 n_{1}) {(n_{1} - 1)}^{- 3} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} {(r_{* 1 j j^{'}})}^{2} \sum_{h = 1}^{p} {(r_{* \frac{1}{2} h j})}^{2} {(r_{* \frac{1}{2} h j^{'}})}^{2}] . \end{array}

Expression of $σ_{z A}^{2}$

\begin{array}{l} σ_{z A}^{2} = 8 n_{1}^{- 1} tr [{(R_{1} R_{*})}^{2}] + 4 β_{1} n_{1}^{- 1} \sum_{j = 1}^{p} {(e_{j}^{T} R_{1}^{1 / 2} R_{*} R_{1}^{1 / 2} e_{j})}^{2} \\ + 4 n_{1}^{- 1} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} e_{j}^{T} R_{1} R_{*} e_{j} e_{j^{'}}^{T} R_{1} R_{*} e_{j^{'}} [2 {(r_{11 j j^{'}})}^{2} + 4 β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} {(r_{1 \frac{1}{2} h j^{'}})}^{2}] \\ + 4 n_{1}^{- 1} [2 tr (R_{1}^{4}) + β_{1} \sum_{j = 1}^{p} {(r_{12 j j})}^{2}] + 4 {[n_{1}^{- 1} tr (R_{1}^{2})]}^{2} \\ + 4 n_{1}^{- 1} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} (r_{12 j j}) (r_{12 j^{'} j^{'}}) [2 {(r_{11 j j^{'}})}^{2} + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} {(r_{1 \frac{1}{2} h j^{'}})}^{2}] \\ - 8 n_{1}^{- 1} \sum_{j = 1}^{p} e_{j}^{T} R_{1} R_{*} e_{j} [2 e_{j}^{T} R_{1} R_{*} R_{1} e_{j} + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} e_{h}^{T} R_{1}^{1 / 2} R_{*} R_{1}^{1 / 2} e_{h}]) \\ - 8 n_{1}^{- 1} [2 tr (R_{1}^{3} R_{*}) + β_{1} \sum_{j = 1}^{p} e_{j}^{T} R_{1}^{1 / 2} R_{*} R_{1}^{1 / 2} e_{j} (r_{12 j j})] \\ + 8 n_{1}^{- 1} \sum_{j = 1}^{p} (r_{12 j j}) [2 e_{j}^{T} R_{1} R_{*} R_{1} e_{j} + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} e_{h}^{T} R_{1}^{1 / 2} R_{*} R_{1}^{1 / 2} e_{h}] \\ + 8 n_{1}^{- 1} \sum_{j = 1}^{p} e_{j}^{T} R_{1} R_{*} e_{j} [2 (r_{13 j j}) + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} (r_{12 h h})] \\ - 8 n_{1}^{- 1} \sum_{j = 1}^{p} \sum_{j^{'} = 1}^{p} e_{j}^{T} R_{1} R_{*} e_{j} (r_{12 j^{'} j^{'}}) [2 {(r_{11 j j^{'}})}^{2} + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} {(r_{1 \frac{1}{2} h j^{'}})}^{2}] \\ - 8 n_{1}^{- 1} \sum_{j = 1}^{p} (r_{12 j j}) [2 r_{13 j j} + β_{1} \sum_{h = 1}^{p} {(r_{1 \frac{1}{2} h j})}^{2} (r_{12 h h})] . \end{array}

When R₁ = R_∗, we have $σ_{z A}^{2} = 4 {[n_{1}^{- 1} tr (R_{*}^{2})]}^{2}$ .

A.2. Expressions of $μ_{A ℓ_{1} ℓ_{2}}$ , $μ_{z ℓ_{1} ℓ_{2}}$ , $σ_{A ℓ_{1} ℓ_{2}}^{2}$ _{and $σ_{A ℓ_{1} ℓ_{2} ℓ_{2} ℓ_{3}}$} Theorems 3.1–3.4.

\begin{array}{l} μ_{A ℓ_{1} ℓ_{2}} \\ = - 2 tr (R_{ℓ_{1}} R_{ℓ_{2}}) + \sum_{i = 1}^{2} a_{ℓ_{i}} \{(n_{ℓ_{i}}^{2} - n_{ℓ_{i}} + 2) n_{ℓ_{i}}^{- 1} tr (R_{ℓ_{i}}^{2}) + (n_{ℓ_{i}}^{2} - n_{ℓ_{i}} - 1) p^{2} n_{ℓ_{i}}^{- 2} + β_{ℓ_{i}} p \\ - 4 [2 tr (R_{ℓ_{i}}^{2}) + β_{ℓ_{i}} \sum_{h = 1}^{p} \sum_{j = 1}^{p} (r_{ℓ_{i} \frac{3}{2} h j}) {(r_{ℓ_{i} \frac{1}{2} h j})}^{3}] \\ + 2 [2 tr (R_{ℓ_{1}} R_{ℓ_{2}}) + β_{ℓ_{i}} \sum_{h = 1}^{p} \sum_{j = 1}^{p} e_{h}^{T} R_{ℓ_{1}}^{1 / 2} R_{ℓ_{2}} e_{j} {(r_{ℓ_{i} \frac{1}{2} h j})}^{3}] \\ + 2 (n_{ℓ_{i}} - 2) {(n_{ℓ_{i}} - 1)}^{- 1} [2 tr (R_{ℓ_{i}}^{2}) + β_{ℓ_{i}} \sum_{j = 1}^{p} (r_{ℓ_{i} 2 j j}) \sum_{h = 1}^{p} {(r_{ℓ_{i} \frac{1}{2} h j})}^{4}] \\ - 1.5 \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} [2 + β_{ℓ_{i}} \sum_{j = 1}^{p} {(r_{ℓ_{i} \frac{1}{2} h j})}^{4}] \\ + (n_{ℓ_{i}} - 2) {(n_{ℓ_{i}} - 1)}^{- 1} \sum_{h = 1}^{p} \sum_{j = 1}^{p} {(r_{ℓ_{i} 1 h j})}^{2} [2 {(r_{ℓ_{i} 1 h j})}^{2} + β_{ℓ_{i}} \sum_{j^{'} = 1}^{p} {(r_{ℓ_{i} \frac{1}{2} h j^{'}})}^{2} {(r_{ℓ_{i} \frac{1}{2} j j^{'}})}^{2}] \\ - 0.5 \sum_{h = 1}^{p} \sum_{j = 1}^{p} (r_{ℓ_{1} 1 h j}) (r_{ℓ_{2} 1 h j}) [2 {(r_{ℓ_{i} 1 h j})}^{2} + β_{ℓ_{i}} \sum_{j^{'} = 1}^{p} {(r_{ℓ_{i} \frac{1}{2} h j^{'}})}^{2} {(r_{ℓ_{i} \frac{1}{2} j j^{'}})}^{2}]\} \end{array}

where $a_{ℓ_{i}} = n_{ℓ_{i}} / [{(n_{ℓ_{i}} - 1)}^{2}]$ and $r_{ℓ_{i} k h j}$ is the (h, j) entry of ${(R_{ℓ_{i}})}^{k}$ for k = 1/2, 1, 3/2, 2, 3.

When $R_{ℓ_{1}} = R_{ℓ_{2}} = R$ , we have

μ_{z k_{1} ℓ_{2}} = μ_{A z_{1} z_{2}} = \sum_{i = 1}^{2} \frac{n_{ℓ_{i}}}{{(n_{ℓ_{i}} - 1)}^{3}} [\frac{(n_{ℓ_{i}}^{2} - n_{ℓ_{i}} - 1) p^{2} (n_{ℓ_{i}} - 1)}{n_{ℓ_{i}}^{2}} - \frac{(2 n_{ℓ_{i}}^{2} + n_{ℓ_{i}} + 1)}{n_{ℓ_{i}}} tr (R^{2}) + (n_{ℓ_{i}} - 3) \sum_{h = 1}^{p} \sum_{j = 1}^{p} {(r_{01 h j})}^{4} + β_{ℓ_{i}} (n_{ℓ_{i}} - 1) - 2 β_{ℓ_{i}} (n_{ℓ_{i}} - 1) \sum_{h = 1}^{p} \sum_{j = 1}^{p} (r_{0 \frac{3}{2} h j}) {(r_{0 \frac{1}{2} h j})}^{3} + 0.5 (n_{ℓ_{i}} - 5) β_{ℓ_{i}} \sum_{h = 1}^{p} (r_{02 h h}) \sum_{j = 1}^{p} {(r_{0 \frac{1}{2} h j})}^{4} + 0.5 (n_{ℓ_{i}} - 3) β_{ℓ_{i}} \sum_{h = 1}^{p} \sum_{j = 1}^{p} {(r_{01 h j})}^{2} \sum_{j^{'} = 1}^{p} {(r_{0 \frac{1}{2} h j^{'}})}^{2} {(r_{0 \frac{1}{2} j j^{'}})}^{2}]

where $r_{0 k h j}$ is the (h, j) entry of R_k for k= 1/2, 1, 3/2. 2, 3. We have $σ_{A ℓ_{1} ℓ_{2}}^{2} = a_{A ℓ_{1} ℓ_{2}} + b_{A ℓ_{1} ℓ_{2}}$ where

a_{A ℓ_{1} ℓ_{2}} = 8 n_{ℓ_{1}}^{- 1} tr [{(R_{ℓ_{1}} R_{ℓ_{2}})}^{2}] + 4 β_{ℓ_{1}} n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} {(e_{h}^{T} R_{ℓ_{1}}^{1 / 2} R_{ℓ_{2}} R_{ℓ_{1}}^{1 / 2} e_{h})}^{2} + 4 n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} e_{h}^{T} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} e_{j^{'}}^{T} R_{ℓ_{1}} R_{ℓ_{2}} e_{j^{'}} [2 {(r_{ℓ_{1} 1 h j^{'}})}^{2} + 4 β_{ℓ_{1}} \sum_{j = 1}^{p} {(r_{ℓ_{1} \frac{1}{2} h j})}^{2} {(r_{ℓ_{1} \frac{1}{2} j j^{'}})}^{2}] + 4 n_{ℓ_{1}}^{- 1} [2 tr (R_{ℓ_{1}}^{4}) + β_{ℓ_{1}} \sum_{h = 1}^{p} {(r_{ℓ_{1} 2 h h})}^{2}] + 4 {[n_{ℓ_{1}}^{- 1} tr (R_{ℓ_{1}}^{2})]}^{2} + 4 n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} (r_{ℓ_{1} 2 h h}) (r_{ℓ_{1} 2 j^{'} j^{'}}) [2 {(r_{ℓ_{1} 1 h j^{'}})}^{2} + β_{ℓ_{1}} \sum_{j = 1}^{p} {(r_{ℓ_{1} \frac{1}{2} h j})}^{2} {(r_{ℓ_{1} \frac{1}{2} j j^{'}})}^{2}] - 8 n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} [2 e_{h}^{T} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} + β_{ℓ_{1}} \sum_{j = 1}^{p} {(r_{ℓ_{1} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{1}}^{1 / 2} R_{ℓ_{2}} R_{ℓ_{1}}^{1 / 2} e_{j}] - 8 n_{ℓ_{1}}^{- 1} [2 tr (R_{ℓ_{1}}^{3} R_{ℓ_{2}}) + β_{ℓ_{1}} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{1}}^{1 / 2} R_{ℓ_{2}} R_{ℓ_{1}}^{1 / 2} e_{h} (r_{ℓ_{1} 2 h h})] + 8 n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} (r_{ℓ_{1} 2 h h}) [2 e_{h}^{T} R_{ℓ_{1}} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} + β_{ℓ_{1}} \sum_{j = 1}^{p} {(r_{ℓ_{1} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{1}}^{1 / 2} R_{ℓ_{2}} R_{ℓ_{1}}^{1 / 2} e_{j}] + 8 n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} [2 (r_{ℓ_{1} 3 h h}) + β_{ℓ_{1}} \sum_{j = 1}^{p} {(r_{ℓ_{1} \frac{1}{2} h j})}^{2} (r_{ℓ_{1} 2 j j})] - 8 n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} e_{h}^{T} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} (r_{ℓ_{1} 2 j^{'} j^{'}}) [2 {(r_{ℓ_{1} 1 h j^{'}})}^{2} + β_{ℓ_{1}} {\sum_{j = 1}^{p} {(r_{ℓ_{1} \frac{1}{2} h j})}^{2} (r_{ℓ_{1} \frac{1}{2} j j^{'}})}^{2}] - 8 n_{ℓ_{1}}^{- 1} \sum_{h = 1}^{p} (r_{ℓ_{1} 2 h h}) [2 (r_{ℓ_{1} 3 h h}) + β_{ℓ_{1}} \sum_{j = 1}^{p} {(r_{ℓ_{1} \frac{1}{2} h j})}^{2} (r_{ℓ_{1} 2 j j})]

and

\begin{array}{l} b_{A ℓ_{1} ℓ_{2}} = 8 n_{ℓ_{2}}^{- 1} tr [{(R_{ℓ_{1}} R_{ℓ_{2}})}^{2}] + \frac{8}{n_{ℓ_{2}} n_{ℓ_{1}}} {[tr (R_{ℓ_{1}} R_{ℓ_{2}})]}^{2} + 4 β_{ℓ_{2}} n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} {(e_{h}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{h})}^{2} \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} e_{j^{'}}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{j^{'}} [2 {(r_{ℓ_{2} 1 h j^{'}})}^{2} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} {(r_{ℓ_{2} \frac{1}{2} j j^{'}})}^{2}] \\ + 4 n_{ℓ_{2}}^{- 1} [2 tr (R_{ℓ_{2}}^{4}) + β_{ℓ_{2}} \sum_{h = 1}^{p} {(r_{ℓ_{2} 2 h h})}^{2}] + 4 {(n_{ℓ_{2}}^{- 1} tr R_{ℓ_{2}}^{2})}^{2} \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} (r_{ℓ_{2} 2 h h}) (r_{ℓ_{2} 2 j^{'} j^{'}}) [2 {(r_{ℓ_{2} 1 h j^{'}})}^{2} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} {(r_{ℓ_{2} \frac{1}{2} j j^{'}})}^{2}] \\ - 8 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} [2 e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{j}] \\ - 8 n_{ℓ_{2}}^{- 1} [2 tr (R_{ℓ_{2}}^{3} R_{ℓ_{1}}) + β_{ℓ_{2}} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{h} (r_{ℓ_{2} 2 h h})] \\ + 8 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} (r_{ℓ_{2} 2 h h}) [2 e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{j}] \\ + 8 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} [2 (r_{ℓ_{2} 3 h h}) + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} (r_{ℓ_{2} 2 j j})] \\ - 8 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} (r_{ℓ_{2} 2 j^{'} j^{'}}) [2 {(r_{ℓ_{2} 1 h j^{'}})}^{2} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} {(r_{ℓ_{2} \frac{1}{2} j j^{'}})}^{2}] \\ - 8 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} (r_{ℓ_{2} 2 h h}) [2 (r_{ℓ_{2} 3 h h}) + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} (r_{ℓ_{2} 2 j j})] . \end{array}

When $R_{ℓ_{1}} = R_{ℓ_{2}} = R$ , we have $a_{A ℓ_{1} ℓ_{2}} = 4 {[n_{ℓ_{1}}^{- 1} tr (R^{2})]}^{2}$ , $b_{A ℓ_{1} ℓ_{2}} = 8 n_{ℓ_{1}}^{- 1} n_{ℓ_{2}}^{- 1} {[tr (R^{2})]}^{2} + 4 {[n_{ℓ_{2}}^{- 1} tr (R^{2})]}^{2}$ and $σ_{A ℓ_{1} ℓ_{2}}^{2} = 4 {(n_{ℓ_{1}}^{- 1} + n_{ℓ_{2}}^{- 1})}^{2} {[tr (R^{2})]}^{2}$

For $ℓ_{1} \neq ℓ_{2} \neq k_{3}$ we have

\begin{array}{l} σ_{A ℓ_{1} ℓ_{2} ℓ_{2} ℓ_{3}} = \\ 4 n_{ℓ_{2}}^{- 1} [2 tr (R_{ℓ_{1}} R_{ℓ_{2}} R_{ℓ_{3}} R_{ℓ_{2}}) + β_{ℓ_{2}} \sum_{j = 1}^{p} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{j} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{3}} R_{ℓ_{2}}^{1 / 2} e_{j}] \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} e_{j^{'}}^{T} R_{ℓ_{2}} R_{ℓ_{3}} e_{j^{'}} [2 {(r_{ℓ_{2} 1 h j^{'}})}^{2} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} {(r_{ℓ_{2} \frac{1}{2} j j^{'}})}^{2}] \\ + 4 n_{ℓ_{2}}^{- 1} [2 tr (R_{ℓ_{2}}^{4}) + β_{ℓ_{2}} \sum_{h = 1}^{p} {(r_{ℓ_{2} 2 h h})}^{2}] + 4 {[n_{ℓ_{2}}^{- 1} tr (R_{ℓ_{2}}^{2})]}^{2} \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} (r_{ℓ_{2} 2 h h}) (r_{ℓ_{2} 2 j^{'} j^{'}}) [2 {(r_{ℓ_{2} 1 h j^{'}})}^{2} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} {(r_{ℓ_{2} \frac{1}{2} j j^{'}})}^{2}] \\ - 2 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{3}} e_{h} [2 e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{j}] \\ - 2 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} [2 e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{3}} R_{ℓ_{2}} e_{h} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{3}} R_{ℓ_{2}}^{1 / 2} e_{j}] \\ - 4 n_{ℓ_{2}}^{- 1} [2 tr (R_{ℓ_{2}}^{3} R_{ℓ_{1}}) + β_{ℓ_{2}} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{h} (r_{ℓ_{2} 2 h h})] \\ - 4 n_{ℓ_{2}}^{- 1} [2 tr (R_{ℓ_{2}}^{3} R_{ℓ_{3}}) + β_{ℓ_{2}} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{3}} R_{ℓ_{2}}^{1 / 2} e_{h} (r_{ℓ_{2} 2 h h})] \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} (r_{ℓ_{2} 2 h h}) [2 e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} R_{ℓ_{2}} e_{h} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{1}} R_{ℓ_{2}}^{1 / 2} e_{j}] \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} (r_{ℓ_{2} 2 h h}) [2 e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{3}} R_{ℓ_{2}} e_{h} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} e_{j}^{T} R_{ℓ_{2}}^{1 / 2} R_{ℓ_{3}} R_{ℓ_{2}}^{1 / 2} e_{j}] \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} [2 (r_{ℓ_{2} 3 h h}) + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} (r_{ℓ_{2} 2 j j})] \\ + 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{3}} e_{h} [2 (r_{ℓ_{2} 3 h h}) + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} (r_{ℓ_{2} 2 j j})] \\ - 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{1}} e_{h} (r_{ℓ_{2} 2 j^{'} j^{'}}) [2 {(r_{ℓ_{2} 1 h j^{'}})}^{2} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} {(r_{ℓ_{2} \frac{1}{2} j j^{'}})}^{2}] \\ - 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} \sum_{j^{'} = 1}^{p} e_{h}^{T} R_{ℓ_{2}} R_{ℓ_{3}} e_{h} (r_{ℓ_{2} 2 j^{'} j^{'}}) [2 {(r_{ℓ_{2} 1 h j^{'}})}^{2} + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} {(r_{ℓ_{2} \frac{1}{2} j j^{'}})}^{2}] \\ - 4 n_{ℓ_{2}}^{- 1} \sum_{h = 1}^{p} (r_{ℓ_{2} 2 h h}) [2 (r_{ℓ_{2} 3 h h}) + β_{ℓ_{2}} \sum_{j = 1}^{p} {(r_{ℓ_{2} \frac{1}{2} h j})}^{2} (r_{ℓ_{2} 2 j j})] . \end{array}

Where $R_{ℓ_{1}} = R_{ℓ_{2}} = R_{ℓ_{3}} = R$ we have $σ_{A ℓ_{1} ℓ_{2} ℓ_{2} ℓ_{3}} = 4 {[n_{ℓ_{2}}^{- 1} tr (R^{2})]}^{2}$ .

Footnotes

^§

Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wpcontent/uploads/howtoapply/ADNI_Acknowledgement_List.pdf.

ADNI manuscript citation guidelines. https://adni.loni.usc.edu/wp-content/uploads//how to apply /ADNI DSP Policy.pdf

Contributor Information

Shurong Zheng, KLAS and School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, China zhengsr@nenu.edu.cn.

Guanghui Cheng, School of Economics and Statistics, Guangzhou University, Guangzhou, China chenggh845@nenu.edu.cn.

Jianhua Guo, KLAS and School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, China jhguo@nenu.edu.cn.

Hongtu Zhu, Department of Biostatistics, The University of North Carolina at Chapel Hill Chapel Hill, NC, USA htzhu@email.unc.edu.

REFERENCES

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

supplement

NIHMS1000738-supplement-supplement.pdf^{(394.1KB, pdf)}

[R1] Aitkin MA (1969), “Some tests for correlation matrices,” Biometrika, 56(2), 443–446. [Google Scholar]

[R2] Anderson TW (2003), An Introduction to Multivariate Statistical Analysis, 3rd Edition, New York: Wiley. [Google Scholar]

[R3] Bai ZD, and Silverstein JW (2004), “CLT for linear spectral statistics of large-dimensional sample covariance matrices,” The Annals of Probability, 32(1A), 553–605. [Google Scholar]

[R4] Bartlett MS, and Rajalakshman DV (1953), “Goodness of fit tests for simultaneous autoregressive series,” Journal of the Royal Statistical Society (Series B), 15, 107–124. [Google Scholar]

[R5] Bodwin K, Zhang K, and Nobel A (2016), “A testing-based approach to the discovery of differentially correlated variable sets,” arXiv:1509.08124v2,. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R6] Browne MW (1978), “The likelihood ratio test for the equality of correlation matrices,” The British Journal of Mathematical and Statistical Psychology, 31(2), 209–217. [Google Scholar]

[R7] Cai T, Liu WD, and Xia Y (2013), “Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings,” Journal of the American Statistical Association, 108(501), 265–277. [Google Scholar]

[R8] Cai TT (2017), “Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures,” Annual Review of Statistics and Its Application, 4, 423–446. [Google Scholar]

[R9] Cai TT, and Jiang TF (2011), “Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices,” The Annals of Statistics, 39(3), 1496–1525. [Google Scholar]

[R10] Cai TT, and Ma ZM (2013), “Optimal hypothesis testing for high dimensional covariance matrices,” Bernoulli, 19(5B), 2359–2388. [Google Scholar]

[R11] Cai TT, and Zhang A (2016), “Inference for high-dimensional differential correlation matrices,” Journal of multivariate analysis, 143, 107–126. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R12] Chen SX, Zhang LX, and Zhong PS (2010), “Tests for high-dimensional covariance matrices,” Journal of the American Statistical Association, 105(490), 810–819. [Google Scholar]

[R13] Cole N (1968), “The likelihood ratio test of the equality of correlation matrices. 1968–65: The L. L. Thurstone Psychometric Laboratory, University of North Carolina, Chapel Hill, North Carolina,”. [Google Scholar]

[R14] Debashis P, and Alexander A (2014), “Random matrix theory in statistics: a review,” Journal of Statistical Planning and Inference, 150, 1–29. [Google Scholar]

[R15] Gao JT, Han X, Pan GM, and Yang YR (2017), “High-dimensional correlation matrices: the central limit theorem and its application,” Journal of the Royal Statistical Society (Series B), 79(3), 677–693. [Google Scholar]

[R16] Gupta AK, Johnson BE, and Nagar DK (2013), “Testing equality of several correlation matrices,” Revista Colombiana de Estad´ıstica, 36(2), 237–258. [Google Scholar]

[R17] Jennrich RI (1970), “An asymptotic χ² test for the equality of two correlation matrices,” Journal of the American Statistical Association, 65(330), 904–912. [Google Scholar]

[R18] Kullback S (1967), “On testing correlation matrices,” Journal of the Royal Statistical Society (Series C), 16(1), 80–85. [Google Scholar]

[R19] Larntz K, and Perlman MD (1985), “A simple test for the equality of correlation matrices,” Technical Report, 63. [Google Scholar]

[R20] Li DN, and Xue LZ (2015), “Joint limiting laws for high-dimensional independence tests,” arXiv: 1512.08819V1,. [Google Scholar]

[R21] Qiu YM, and Chen SX (2012), “Test for bandedness of high-dimensional covariance matrices and bandwidth estimation,” The Annals of Statistics, 40(3), 1285–1314. [Google Scholar]

[R22] Schott JR (1996), “Testing for the equality of several correlation matrices,” Statistics and probability letters, 27, 85–89. [Google Scholar]

[R23] Schott JR (2005), “Testing for complete independence in high dimensions,” Biometrika, 92(4), 951–956. [Google Scholar]

[R24] Shao QM, and Zhou WX (2014), “Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices,” The Annals of Probability, 42(2), 623–648. [Google Scholar]

[R25] Zhou C, Han F, Zhang XS, and Liu H (2015), “An Extreme-value approach for testing the equality of large U-statistic based correlation matrices,” arXiv:1502.03211,. [Google Scholar]

PERMALINK

TEST FOR HIGH DIMENSIONAL CORRELATION MATRICES

Shurong Zheng

Guanghui Cheng

Jianhua Guo

Hongtu Zhu

Abstract

1. Introduction.

1.1. Existing Literature.

1.2. Our Contributions.

2. Test Statistics for One Sample Testing Problem.

2.1. Test statistics.

2.2. Null distribution.

2.3. Power properties and optimality.

3. Extensions to Two and Multiple Sample Testing Problems.

3.1. Extension to Two Sample Testing Problem.

3.1.1. Test statistics and their null distributions for two-sample testing problem.

3.1.2. Power properties and optimality.

3.2. Test statistic for multiple sample testing problem.

4. Estimation of the kurtosis β1.

5. Simulation studies.

Table 1.

Table 3.

6. Real data analysis.

6.1. Alzheimer’s Disease Neuroimaging Initiative (ADNI) data.

6.2. Group comparisons.

Fig 1.

Supplementary Material

Fig 2.

Table 2.

ACKNOWLEDGEMENT

APPENDIX A: SOME EXPRESSIONS

Footnotes

Contributor Information

REFERENCES

Associated Data

Supplementary Materials

ACTIONS

PERMALINK

RESOURCES

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Links to NCBI Databases

4. Estimation of the kurtosis β₁.