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ORIGINAL ARTICLE

Tests of Zero Correlation Using Modified RV Coefficient

for High‑Dimensional Vectors

M. Rauf Ahmad1

© The Author(s) 2019

Abstract

Tests of zero correlation between two or more vectors with large dimension, pos-sibly larger than the sample size, are considered when the data may not necessarily follow a normal distribution. A single-sample case for several vectors is first pro-posed, which is then extended to the common covariance matrix under the assump-tion of homogeneity across several independent populaassump-tions. The test statistics are constructed using a recently proposed modification of the RV coefficient (a corre-lation coefficient for vector-valued random variables) for high-dimensional vectors. The accuracy of the tests is shown through simulations.

Keywords Block-diagonal structure · Cross-correlations · High-dimensional inference

1 Introduction

Let 𝐗k= (Xk1,… , Xkp)T , k = 1, … , n , be iid random vectors drawn from a

popula-tion with E (𝐗k) = 𝝁 ∈ ℝp and Cov (𝐗k) = 𝚺 ∈ ℝp×p , where 𝚺 > 0 can be expressed

as a partitioned matrix 𝚺 = (𝚺ij)i,j=1b with blocks 𝚺ij∈ ℝ pi×pj , 𝚺 ji= 𝚺 T ij , 𝚺 T ii= 𝚺ii . We

are interested to test

when the block dimensions, pi , may exceed the sample size, n , and the data may

not necessarily follow the multivariate normal distribution. Under H0b , 𝚺 reduces

to a block-diagonal structure, 𝚺 = diag(𝚺1,… , 𝚺bb) , 𝚺ii∈ ℝpi×pi . Obviously, under

normality, the test of H0b is equivalent to testing independence of the corresponding

vectors.

Now, consider g ≥ 2 independent populations with 𝐗lk= (Xlk1,… , Xlkp)T , k= 1, … , nl , as iid random vectors drawn from lth population with E (𝐗lk) = 𝝁l ,

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H0b∶ 𝚺ij= 𝟎 ∀ i < j vs. H1b∶ 𝚺ij𝟎 for at least one pairi < j

* M. Rauf Ahmad rauf.ahmad@statistik.uu.se

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Cov(𝐗lk) = 𝚺l> 0 , l = 1, … , g , defined similarly as for the single population above.

Assuming homogeneity of covariance matrices across g populations, i.e., 𝚺l= 𝚺 ∀ l ,

we are further interested to test hypotheses in (1) for common 𝚺 . To distinguish the two cases, we denote multi-sample hypotheses as H0g and H1g , respectively.

Tests of H0b or H0g are frequently needed in multivariate applications, where

the special case of pi = 1 ∀ i = 1, … , b leads to the well-known single- and

multi-sample tests of identity or sphericity hypotheses; see e.g., [1, 2] and the references therein. Tests of H0b or H0g for a classical case, n > p , have often been addressed

in the multivariate literature, including for multiple blocks; see [8, 16, 20, 21]. [14] give extensions to functional data, and [6] discuss permutation and bootstrap tests. Other recent extensions include nonparametric tests ([10, 13]) and a kernel-based approach ([22]); see also [9, 19] for general theory, in the context of canonical cor-relation analysis.

A potential advantage of H0b in (1) is a huge dimension reduction, which adds

further motivation to testing such hypotheses for high-dimensional data, p ≫ n . As the classical testing theory, mostly based on likelihood ratio approach, collapses in this case, several modifications have recently been introduced to cope with the prob-lem; see e.g., [26, 33, 34], where [32] provide a test based on rank correlations. Most of these tests assume normality and are constructed using a Frobenius norm measuring the distance between the null and alternative hypotheses. Use of a Frobe-nius norm as a distance measure is a common approach for estimation and testing of high-dimensional covariance matrices; for its use in other contexts, see e.g., [3] and the references cited therein.

Our construction of the test statistics for (1) is, however, based on a different approach. We use the RV coefficient (see Sect. 2) as the basic measure of orthogo-nality of vectors and extend it to test H0b and H0g . [5] recently proposed this

modi-fication to the RV coefficient for two high-dimensional vectors and used it to intro-duce a test of orthogonality. For details, including historical overview, comparison with its competitors and extensive bibliography, see the reference.

Apart from theoretical analysis of the modified coefficient and its competitors, extensive simulations are used to show the accuracy of the modified estimator and the significance test. The modified coefficient is composed of computationally very efficient estimators as simple functions of the empirical covariance matrix. In this paper, we first extend the theory to propose a test of no correlation between b ≥ 2 large vectors, possibly of different dimension.

We further extend this one-sample multi-block test to a multi-sample case with

g independent populations. Assuming homogeneity of covariance matrices across g

populations, we use pooled information to construct a test for the common covari-ance matrix. A particularly attractive aspect of the multi-sample test is its simplicity of pooling information across g independent samples, in such a way that the single-sample computations conveniently carry over to the multi-single-sample case, resulting in a highly efficient test statistic.

We begin in the nest section with the basic notational set up, where we also briefly recap the modified RV coefficient in [5] for further use and reference. We keep it as short as possible, referring for details to the original paper. Using these rudiments, we propose a test for H0b for multiple blocks in Sect. 3, with an extension

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to the test of H0g in Sect. 4. Accuracy of the tests through simulations is shown in

Sect. 5, where technical proofs are deferred to Appendix.

2 Notations and Preliminaries

Let the random vectors 𝐗k∈ ℝp , k = 1, … , n , with E (𝐗k) = 𝝁 ∈ ℝp ,

Cov(𝐗k) = 𝚺 ∈ ℝp×p , be partitioned as 𝐗k= (𝐗T1k,… , 𝐗Tbk) T , 𝐗 ik∈ ℝpi so that accordingly 𝝁= (𝝁T 1,… , 𝝁 T b) T , 𝝁 i∈ ℝpi , 𝚺= (𝚺ij)i,j=1b , 𝚺ij= E (𝐗ik− 𝝁i) ⊗ (𝐗jk− 𝝁j)T∈ ℝpi×pj , i ≠ j , 𝚺 ji= 𝚺 T ij , 𝚺 T ii = 𝚺ii , where ℝa×b is

the space of real matrices and ⊗ is the Kronecker product. We assume 𝚺 > 0 , 𝚺ii> 0

∀ i.

Let 𝐗 =n

k=1𝐗ik∕n and �𝚺 =

n

k=1( �𝐗k⊗ �𝐗k)∕(n − 1) be unbiased estimators of

𝝁 and 𝚺 , likewise partitioned as 𝐗 = (𝐗T1,… , 𝐗Tb)T , ̂𝚺 = (̂𝚺

ij)i,j=1b , so that

𝐗i=∑nk=1𝐗ik∕n and �𝚺ij=

n

k=1( �𝐗ik⊗ �𝐗jk)∕(n − 1) are unbiased estimators of 𝝁i

and 𝚺ij , where ̃𝐗k= 𝐗k− 𝐗 , ̃𝐗ik= 𝐗ik− 𝐗i . Denoting 𝐗 = (𝐗T1… 𝐗Tn) ∈ ℝ n×p as

the entire data matrix and 𝐗i= (𝐗Ti1,… 𝐗

T in)

T

∈ ℝn×pi as the data matrix for ith block, we can express the estimators more succinctly as

where 𝐂 = 𝐈 − 𝐉∕n is the n × n centering matrix, 𝐈 is the identity matrix, 𝐉 = 𝟏𝟏T

with 𝟏 a vector of 1s. The null hypotheses in (1) thus imply the nullity of all off-diagonal blocks 𝚺ij in 𝚺 . Consider first the simplest case of b = 2 , with only one

off-diagonal block 𝚺12= Cov (𝐗1k, 𝐗2k) . Obviously, a test of zero correlation (or,

under normality, of independence) between 𝐗1k and 𝐗2k can be based on an

empiri-cal estimator of ‖𝚺12‖2 = tr (𝚺12𝚺21) or an appropriately normed version of it. One

such normed measure is proposed in [5] as

where ̂‖𝚺ij‖2 is an unbiased (and consistent) estimator of ‖𝚺ij‖2 , i,  j = 1, 2; see

Sect. 3 for formal definition. The ̂𝜂 is a modified form of the original RV coeffi-cient, ̃𝜂 = ‖̂𝚺12‖2∕‖̂𝚺11‖‖̂𝚺22‖ , as an extension of the scalar correlation

coef-ficient, to measure correlation between vectors of possibly different dimension. Note that ̂𝜂 is constructed using unbiased estimators in the true RV coefficient

𝜂 =‖𝚺12‖2∕‖𝚺11‖‖𝚺22‖ . The RV coefficient extends the usual scalar correlation

coefficient to data vectors of possibly different dimension, is often used to study relationship between different data configurations, and has many attractive proper-ties; see also [25].

Based on the aforementioned modification and subsequent test of zero correla-tion, we here extend the same concept and construct tests of hypotheses in (1) and their multi-sample variants. The proposed tests will be valid for the

follow-(2) 𝐗= 1 n𝐗 T 𝟏, 𝚺̂ = 1 n− 1𝐗 T 𝐂𝐗, 𝐗i= 1 n𝐗 T i𝟏, 𝚺̂ij= 1 n− 1𝐗 T i𝐂𝐗j, (3) ̂ 𝜂 = ̂ ‖𝚺12‖2 � ̂ ‖𝚺11‖2‖𝚺̂22‖2 ,

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Define 𝐘ik= 𝐗ik− 𝝁i , i = 1, … , b , so that 𝐘k= (𝐘1kT,… , 𝐘T2k)T can replace 𝐗k .

We define the multivariate model as

with 𝐙k∼  , E (𝐙k) = 𝟎p , Cov (𝐙k) = 𝐈p , k = 1, … , n , where  denotes a

p-dimen-sional distribution function having fourth moment finite; see the assumptions below. Model (4) covers a wide class of distributions such as the elliptical class including multivariate normal. Further, it will help us enhance the validity of the proposed test to a variety of applications under fairly general conditions.

3 One‑Sample Test with Multiple Blocks

Consider any two data matrices 𝐗i= (𝐗Ti1,… 𝐗

T in) T ∈ ℝn×pi and 𝐗j= (𝐗T j1,… 𝐗 T jn) T ∈ ℝn×pj in 𝐗 ∈ ℝn×p , where 𝐗 ik∈ ℝpi , 𝐗ik∈ ℝpj , k = 1, … , n , i, j= 1, … , b , i < j . Following ̂𝜂 in Eq.  (3), the RV coefficient of correlation between 𝐗ik and 𝐗jk can be computed as

where ̂𝜂ij estimates 𝜂ij=‖𝚺ij‖2∕‖𝚺ii‖‖𝚺jj . Since 𝚺ii> 0 , 𝜂ij= 0 ⇔ ‖𝚺ij‖2= 0

∀ i > j , which implies that a test for 𝜂ij= 0 can be equivalently based on 𝚺ij . In fact,

as will be shown shortly, the denominator of ̂𝜂ij adjusts itself through Var (̂‖𝚺ij‖2)

so that it suffices to use ̂‖𝚺ij‖2 to construct a test statistic. We begin by defining

the estimators used to compose ̂𝜂ij and their moments, which will also be useful

in subsequent computations. For notational convenience, denote 𝚺1∕2

ii = 𝚫ii so that

‖𝚫ii‖2= tr (𝚺ii) and correspondingly ‖̂𝚫ii‖2= tr (̂𝚺ii) . For the proof of the

follow-ing theorem, see “Appendix 1.2.1”.

Theorem 1 The unbiased estimators of ‖𝚺ii‖2 , ‖𝚺ij‖2 , and ‖𝚫ii‖‖𝚫jj are defined as

(4) 𝐘k= 𝚺1∕2𝐙k, (5) ̂ 𝜂ij= ̂ ‖𝚺ij‖2 � ̂ ‖𝚺ii‖2‖𝚺̂jj‖2 , (6) ̂ ‖𝚺ii‖2=𝜈(n − 1)(n − 2)‖̂𝚺ii‖2+ � ‖̂𝚫ii‖2 �2 − nQii � (7) ̂ ‖𝚺ij‖2 =𝜈(n − 1)(n − 2)‖̂𝚺ij‖2+‖̂𝚫ii‖2‖̂𝚫jj‖2− nQij � (8) ̂‖𝚫ii‖2 �2 =𝜈�2‖̂𝚺ii‖2+ (n2− 3n + 1)‖̂𝚫ii‖2 �2 − nQii�, (9) ̂ ‖𝚫ii‖2‖𝚫jj‖2=𝜈 � 2‖̂𝚺ij‖2+ (n2− 3n + 1)‖̂𝚫ii‖2‖̂𝚫jj‖2− nQij � ,

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where 𝜈 = (n − 1)∕n(n − 2)(n − 3) , Qij=∑nk=1̃zik̃zjk∕(n − 1) and

Qii=∑nk=1̃z2

ik∕(n − 1) with ̃zik= ̃𝐗Tik𝐗̃ik and ̃𝐗ik= 𝐗ik− 𝐗i , i, j = 1, … , b , i < j.

Note that the terms Qij are needed to make the estimators, and hence the

sub-sequent test statistics, valid under Model (4) beyond the normality assumption. Essentially, these terms involve fourth-order elements of 𝐗ik due to the variances

of bilinear forms to be computed. This in turn calls for bounding such moments of 𝐗ik to be finite (see assumptions below). For this, define

with zik= 𝐘Tik𝐘ik . As 𝜅ij = 0 under normality, it serves as a measure of

non-nor-mality and, given finite fourth moment, helps extend the results to a wide class of distributions under Model (4). The results of Theorem 1 also help obtain unbiased estimators of 𝜅ij and 𝜅ii as (see “Appendix 1.2.1”)

The estimators in Theorem 1 are composed of ̂𝚺ij (see Eq. 2) and Qij , both of which

are simple functions of mean-deviated vectors ̃𝐗ik . It makes the estimators very

sim-ple and computationally highly efficient for practical use. For mathematical amena-bility, however, an alternative form of the same estimators in terms of U-statistics helps us study their properties due to the attractive projection and asymptotic theory of U-statistics.

Given {𝐗ik, 𝐗ir, 𝐗il, 𝐗is} , k ≠ r ≠ l ≠ s , from 𝐗i∈ ℝn×pi , define 𝐃ikr = 𝐗ik− 𝐗ir

with E (𝐃ikr) = 0, Cov (𝐃ikr) = 2 𝚺ii , E (𝐃ikr𝐃jkrT ) = 2 𝚺ij . For Aikr= 𝐃Tikr𝐃ikr ,

E(Aikr) = 2‖𝚫ii‖2 and E (A

iksAjlr) = 4‖𝚫ii‖2‖𝚫jj‖2 . With 𝐃ils defined similarly, let Aijlskr = AilskrAjkrls where Ailskr= 𝐃T

ils𝐃ikr, so that E (A2ilskr) = 4‖𝚺ii‖2 ,

E(Aijlskr) = 4‖𝚺ij‖2. Denoting further B

iikrls= A2ilskr+ A

2

ilksr+ A

2

ilrsk , Cijklrs= Aijlskr+ Aijlksr+ Aijlrks , Dijkrls= AikrAjls+ AiklAjrs+ AiksAjlr and P(n) = n(n − 1)(n − 2)(n − 3) , the U-statistics versions of the estimators of ‖𝚺ii‖2 ,

‖𝚺ij‖2 and ‖𝚫ii‖2‖𝚫jj‖2 in Theorem 2, denoted U1, U2, U3 , are defined,

respec-tively, as

where the sum is quadruple over {k, l, r, s} and 𝜋(⋅) = 𝜋(k, r, l, s) implies

k ≠ r ≠ l ≠ s . The moments in the following theorem follow conveniently using the

alternative form of estimators in (13) and will be very useful for further computa-tions in the sequel.

(10) 𝜅ij= E (zikzjk) − 2‖𝚺ij‖ 2 −‖𝚫ii‖ 2‖𝚫 jj‖ 2 𝜅 ii= E (z 2 ik) − 2‖𝚺ii‖ 2 −�‖𝚫ii‖ 2�2 (11) ̂ 𝜅ij= − n𝜈 n− 1 � 2(n − 1)2‖̂𝚺ij‖2+ (n − 1)2‖̂𝚫ii‖2‖̂𝚫jj‖2− n(n + 1)Qij � (12) ̂ 𝜅ii= − n𝜈 n− 1 � 2(n − 1)2‖̂𝚺ii‖2+ (n − 1)2 � ‖̂𝚫ii‖2 �2 − n(n + 1)Qii � . (13) U1= 1 12P(n) nk,r,ls 𝜋(⋅) Biikrls, U2= 1 12P(n) nk,r,ls 𝜋(⋅) C12lskr, U3= 1 12P(n) nk,r,ls 𝜋(⋅) D12lskr,

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Theorem 2 For the unbiased estimators ̂‖𝚺ij‖2 and ̂‖𝚺ii‖2 in Theorem 2, we have

where a(n) = 3n3− 24n2+ 44n + 20 , b(n) = 6n3− 40n2+ 22n + 181 ,

c(n) = 2n2− 12n + 21 , d(n) = 2n3− 9n2+ 9n − 16 , e(n) = n2− 3n + 8 and K

con-tains terms involving 𝜅ij.

We skip the proof of Theorem 2 which follows from that of Theorem 2 in [5]. The terms KO(⋅) sum up constants that eventually vanish under the assumptions. In particular, K involves 𝜅ij in (10) and terms involving Hadamard products such as

𝚺ii⊙ 𝚺jj which converge to zero. Now, we have the required tools to proceed with

the test of H0b in (1). As mentioned above,

Moreover, 𝜂ij= 0 ⇔

b

i<j𝜂ij= 0 , so that a test of H0b can be based on a sum of

Frobenius norms over all off-diagonal blocks, i.e., ∑b

i<j‖𝚺ij‖2 . We thus define the

test statistic for H0b as

Here, Tij is a statistic to test H0ij∶ 𝜂ij= 0 for any single off-diagonal block 𝚺ij .

Moreover, the scaling factor pipj will help us obtain the limit of Tb for pi→∞

along with n → ∞ , under the following assumptions. Recall 𝐘k∈ ℝp in Model (4).

Assumption 3 E (Y4 ks) = 𝛾s𝛾 < ∞ , ∀ s = 1, … , p. Assumption 4 limpi→∞ ‖𝚫ii‖2 pi = O(1) , ∀ i = 1, … , b. Assumption 5 limn,pi→∞ pi n𝜁0∈ (0, ∞) , ∀ i = 1, … , b.

Assumption 3 bounds fourth moment of 𝐘k so that, by Cauchy–Schwarz

inequal-ity, E (y2

k1sy 2

k2s) < ∞ which implies that 𝜅ij‖𝚺ii‖2‖𝚺jj‖2 →0 . This conforms to the

definition of 𝜅ij and helps us present the test under Model (4), and it also makes the

terms involving K vanish in Theorem 2. Assumption 4 bounds the average of the eigenvalues of diagonal blocks. It is a mild assumption, often used in high-dimen-sional testing, and as its consequence, ‖𝚺ij‖2∕pipj= O(1).

(14) Var( ̂‖𝚺ij‖2) = 2 P(n)4a(n)‖𝚺ij𝚺ji‖2+ 2b(n) tr (𝚺ii𝚺ij𝚺jj𝚺ji) + c(n)��‖𝚺ij‖2 �2 +‖𝚺ii‖2‖𝚺jj‖2+ KO(1) �� (15) Var( ̂‖𝚺ii‖2) = 4 P(n)d(n)‖𝚺2ii‖2+ e(n)‖𝚺2 ii‖ �2 + KO(n2)� (16) H0b∶ 𝚺ij= 𝐎 ⇔ H0b∶ 𝜂ij= 0 ∀ i < j, i, j = 1, … , b. (17) Tb= bi=1 bj=1 i<j Tij, where Tij= 1 pipj ‖𝚺ij‖2.

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Whereas Assumption 4 is needed for limits under H0b and H1b , its consequence

is only needed under H1b since it neither holds nor is required under H0b .

Assump-tion 5 controls joint growth of n and pi so that the limit holds under a

high-dimen-sional framework, and it is also needed only under H1b . Now, for Tb , we have

E( Tb) =

b

i<j‖𝚺ij‖2∕pipj = 0 under H0b and

Equation (18) gives Var ( Tij) and the covariance, say C1 , C2 , C3 , respectively, given

as below.

a1(n) = 3n3− 38n2+ 170n − 262 , b

1(n) = 6n3− 57n2+ 178n − 182 ,

a2(n) = 3n3− 39n2+ 176n − 269 , b

2(n) = 6n3− 70n2+ 286n − 199 . Under H0b ,

the covariances vanish and the variance reduces to

(18) Var( Tb) = bi=1 bj=1 i<j Var( Tij) + 2 bi=1 bj=1 bj=1 i<j<j� Cov( Tij, Tij�) + 2 bi=1 bi=1 bj=1 i<i<j Cov( Tij, Tij) + bi=1 bi=1 bj=1 bj=1 i<i<j<j� Cov( Tij, Tij�) . (19) C1= 4 P(n)p2 ipjpj� � 2a1(n) tr (𝚺ij𝚺ji𝚺ij𝚺ji) + b1(n) tr (𝚺ii𝚺ij𝚺jj𝚺ji) + (n − 4)�2‖𝚺ij‖2‖𝚺ij�‖2+ 5‖𝚺jj�‖2‖𝚺ii‖2 �� (20) C2= 4 P(n)p2 ipjpi� � 2a1(n) tr (𝚺ii𝚺ii𝚺ij𝚺ji) + b1(n) tr (𝚺ii𝚺ii𝚺ij𝚺ji) + (n − 4)�2‖𝚺ij‖2‖𝚺ij‖2+ 5‖𝚺ii�‖2‖𝚺jj‖2 �� (21) C3= 4 P(n)pipjpipj� � 2a2(n) tr (𝚺ij𝚺ji𝚺ij𝚺ji) + b2(n) tr (𝚺ij𝚺jj𝚺ji𝚺ii) + (4n − 11) tr (𝚺ii𝚺ij𝚺jj𝚺ji) + 3(n − 3)‖𝚺ij‖2‖𝚺ij�‖2 + (3n − 10)‖𝚺ii�‖2‖𝚺jj�‖2 � (22) Var( Tb) = 2(2n2− 12n + 21) P(n) bi=1 bj=1 i<j 1 p2 ip 2 j ‖𝚺ii‖2‖𝚺jj‖2 = O1 n2 �b i=1 bj=1 i<j Vij,

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where Vij=‖𝚺ii‖2‖𝚺jj‖2∕p2ip

2

j . It implies that Var (n Tb) is bounded and a

non-degenerate limit of n Tb may exist. That this indeed holds under the assumptions

follow from the limit of Tij given in Ahmad ([5], Theorem 3), by noting that the

covariance terms above essentially vanish under the same assumptions. The follow-ing theorem summarizes the result, the proof of which, along with that of Theo-rem 12 for the multi-sample case, is given in “Appendix 1.2.3”.

Theorem  6 Given Tb in (17) and Assumptions 3–5. Then,

( Tb− E ( Tb))∕𝜎T b  � ������→ N(0, 1) , as n, pi→∞ , with 𝜎2 Tb= Var ( Tb) in (18). Under H0 and Assumptions 3–4, n Tb∕𝜎T b ���→� N(0, 1) with 𝜎 2

Tb as in Eq. (22). Further, the same limits hold when Var ( Tb) is replaced with ̂Var( Tb).

The last part of Theorem 6 makes the statistic Tb applicable, when a consistent

estimator, i.e.,

is plugged in for the true variance, where ̂Vij= ̂‖𝚺ii‖2‖𝚺̂jj‖2∕p2ip

2

j . From the proof

of the theorem, we also note that the moments and the limit of Tb are functions of 𝜂ij=‖𝚺ij‖2‖𝚺

ii‖‖𝚺jj‖ . For the power of Tb , let 𝛽(𝜽) be the power function with

𝜽∈ 𝚯 , where

are the parameter subspaces under H0b and H1b , respectively. Let z𝛼 denote 100𝛼 %

quantile of N(0, 1) distribution, so that P( Tb∕𝜎Tb0 ≥z𝛼) = 𝛼 , by Theorem 6, where

𝜎2

T0b denotes Var ( Tb) under H0b . Also, let 𝛾 = 𝜎Tb0∕𝜎Tb , 𝛿 = E ( Tb)∕𝜎Tb with E( Tb) =∑bi<j‖𝚺ij‖2∕p

ipj , 𝜎2Tb= Var ( Tb) . Then, 𝛽(𝜽|H1) = P( Tb∕𝜎Tb𝛾z𝛼− 𝛿) = 1 - Φ(𝛾z𝛼− 𝛿) defines the power of Tb , where Φ(⋅) is the distribution function of N(0, 1). From Theorem 2, 𝛾 = O(1) and 𝛿 = O(n) under the assumptions, so that 1 - Φ(𝛾z𝛼− 𝛿) → 1 as n, pi→∞ . Finally, letting 𝚺ij= 𝐀ij∕√n for a fixed matrix 𝐀ij , i, j= 1, … , b , i < j , similar arguments imply that the local power also converges to 1 as n, pi→∞.

4 Multi‑Sample Extension with Multiple Blocks

Consider the multi-sample set up given after (1) with 𝐗lk= (𝐗Tl1k,… , 𝐗

T lbk)

T

∈ ℝp ,

𝐗lik∈ ℝpi , k = 1, … , n

l, as iid vectors from population l, where

E(𝐗lk) = 𝝁l= (𝝁Tl1,… , 𝝁 T lb) T ∈ ℝp , 𝝁 li∈ ℝpi , Cov (𝐗lk) = 𝚺l= (𝚺lij)i,j=1b ∈ ℝ p×p ,

l= 1, … , g , 𝚺lij= E (𝐗lik− 𝝁li) ⊗ (𝐗ljk− 𝝁lj)T∈ ℝpi×pj , i < j , 𝚺lji= 𝚺 T lij , 𝚺 T lii= 𝚺lii . Then 𝐗l= ∑nl k=1𝐗lk∕nl , �𝚺l= ∑nl

k=1( �𝐗lk⊗ �𝐗lk)∕(nl− 1) are unbiased estimators of 𝝁l ,

𝚺l, correspondingly partitioned as 𝐗l= (𝐗Tl1,… , 𝐗Tlb)T , ̂𝚺 l= (̂𝚺lij) b i,j=1 with (23) Var(n Tb) = bi=1 bj=1 i<j V ij, 𝜽0= {𝚺ii, i∈ {1, … , b}}, 𝚯1= {𝚺ii, 𝚺ij, i, j∈ {1, … , b}, i < j}

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𝐗li=∑nl

k=1𝐗lik∕nl , �𝚺lij=

nl

k=1( �𝐗lik⊗ �𝐗ljk)∕(nl− 1) as unbiased estimators of 𝝁li and

𝚺lij , where ̃𝐗k= 𝐗k− 𝐗 and ̃𝐗lik= 𝐗lik− 𝐗li . Denoting 𝐗l= (𝐗Tl1… 𝐗

T lnl)

T

∈ ℝnl×p

as the entire data matrix for lth sample, with 𝐗li= (𝐗Tli1,… , 𝐗

T linl)

T

∈ ℝnl×pi as ith data matrix in 𝐗l , we can rewrite the estimators, using 𝐂l= 𝐈nl− 𝐉nl∕nl , l = 1, … , g , as

The RV coefficient in Eq. (3) can now be computed from lth sample, l = 1, … , g , as

which estimates 𝜂lij=‖𝚺lij‖2∕‖𝚺lii‖‖𝚺ljj‖ . We are interested to test

hypothe-ses in (1) for common 𝚺 , assuming 𝚺l= 𝚺 ∀ l , i.e., to test if 𝚺 is block diagonal,

𝚺= diag(𝚺11,… , 𝚺bb) . Formally,

where the subscript g refers to g populations. A test of H0g can thus be constructed

by pooling information from g populations. For this, we state Assumptions 3–5 for the multi-sample case.

Assumption 7 E (Y4 lks) = 𝛾s𝛾 < ∞ , ∀ s = 1, … , p , l = 1, … , g. Assumption 8 limpi→∞ ‖𝚫lii‖2 pi = O(1) , ∀ i = 1, … , b , l = 1, … , g. Assumption 9 limnl,pi→∞ pi nl𝜁l𝜁0 ∈ (0, ∞) , ∀ i = 1, … , b , l = 1, … , g. Now, we begin by writing the estimators in Eqs. (6)–(9) for the lth sample as

where 𝜈l= (nl− 1)∕nl(nl− 2)(nl− 3) , Qlij=

n

k=1̃zlik̃zljk∕(nl− 1) and Qlii=∑n=1̃z2 ∕(n

l− 1) with ̃zlik= ̃𝐗T𝐗̃lik and ̃𝐗lik= 𝐗lik− 𝐗li , i, j = 1, … , b , i < j .

(24) 𝐗l= 1 nl𝐗 T l𝟏nl, ̂ 𝚺l= 1 nl− 1𝐗 T l𝐂l𝐗l, 𝐗li= 1 nl𝐗 T li𝟏nl, ̂ 𝚺lij= 1 nl− 1𝐗 T li𝐂l𝐗lj. (25) ̂ 𝜂lij= ̂ ‖𝚺lij‖2 � ̂ ‖𝚺lii‖2‖𝚺̂ljj‖2 (26)

H0g∶ 𝚺ij= 𝟎 ∀ i ≠ j vs. H1g∶ 𝚺ij𝟎 for at least one pair i ≠ j,

(27) ̂ ‖𝚺lii‖2=𝜈l(nl− 1)(nl− 2)‖̂𝚺lii‖2+ � ‖̂𝚫lii‖2 �2 − nlQlii � (28) ̂ ‖𝚺lij‖2=𝜈l

(nl− 1)(nl− 2)‖̂𝚺lij‖2+‖̂𝚫lii‖2‖̂𝚫ljj‖2− nlQlij

� (29) ̂‖𝚫lii‖2 �2 =𝜈�2‖̂𝚺ii‖2+ (n2l − 3nl+ 1) � ‖̂𝚫ii‖2 �2 − nlQii � , (30) ̂ ‖𝚫lii‖2‖𝚫ljj‖2=𝜈l

2‖̂𝚺lij‖2+ (n2l − 3nl+ 1)‖̂𝚫lii‖2‖̂𝚫ljj‖2− nlQlij

� ,

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Note that these are unbiased estimators of ‖𝚺ii‖2 , ‖𝚺ij‖2 ,

‖𝚫ii‖2

�2

and ‖𝚫ii‖‖𝚫jj‖ ,

respectively, obtained from sample l, so that they can be used to construct pooled estimators of the same parameters, as the following.

Theorem  10 Denote 𝜈0=

g l=1𝜈

−1

l with 𝜈l= (nl− 1)∕nl(nl− 2)(nl− 3) . The pooled unbiased estimators of ‖𝚺ii‖2 , ‖𝚺

ij‖2 ,

‖𝚫ii‖2

�2

and ‖𝚫ii‖‖𝚫jj are defined, respectively, as

In pooling information across g samples, Eqs. (31)–(34) use weights 1∕𝜈l ,

where the pooled estimators correspond to 𝚺 = (𝚺ij) b

i,j=1 for which H0g is defined.

Thus, an appropriate test statistic for H0g can be defined as

which extends Eq. (17) for the multi-sample case under homogeneity. Equivalently, we can write

In this form, Tlij , hence Tlb , are the same as Tij and Tb in Sect. 3 but now defined

for lth population. In either case, the main focus on the formulation of Tg is

simplic-ity, so that, by independence of the g samples, the computations for the one-sample case will mainly suffice for the multi-sample case, as for example the results of the following theorem; see “Appendix 1.2.2” for proof.

(31) ̂ ‖𝚺pii‖2= 1 𝜈0 gl=1 1 𝜈l‖𝚺̂lii‖2 (32) ̂ ‖𝚺pij‖2= 1 𝜈0 gl=1 1 𝜈l ̂ ‖𝚺lij‖2 (33) ̂‖𝚫pii‖2 �2 =1 𝜈0 gl=1 1 𝜈l ̂‖𝚫lii‖2 �2 , (34) ̂ ‖𝚫pii‖2‖𝚫pjj‖2= 1 𝜈0 gl=1 1 𝜈l‖𝚫liî‖2‖𝚫ljj‖2. (35) Tg= bi=1 bj=1 i<j

Tpij where Tpij=

1 pipj ‖𝚺pii‖2 (36) Tg= 1 𝜈0 gl=1 1 𝜈lTlb with Tlb= bi=1 bj=1 i<j

Tlij where Tlij=

1

pipj

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Theorem  11 The pooled estimators ̂‖𝚺pii‖2 and ̂‖𝚺pij‖2 in Eqs. (31)–(32) are

unbiased for ‖𝚺ii‖2 and ‖𝚺ij‖2 , respectively. Further, under Assumptions 7–9, as n, pi→∞, where 𝜈0= ∑g l=1𝜈 −1 l and 𝜂ij=‖𝚺ij‖2∕‖𝚺ii‖‖𝚺jj , i, j = 1, … , b , i < j.

Theorem 11 provides bounds on the variance ratios which are more important for our purpose, where the exact variances, which follow from those of single-sample case in Theorem 2, are given in “Appendix 1.2.2”. Moreover, Eq.  (38) also implies that

̂

‖𝚺pii‖2∕p2i is a consistent estimator of ‖𝚺pii‖2∕p2i , i = 1, … , b , as n, pi→∞ . Now,

from Theorem 11,

since Cov ( Tlb, Tlb) = 0 , l ≠ l′ , by independence, where Var ( Tlb) follows directly

from Eq. (18) as

l= 1, … , g . Equations (19)–(21) provide variances and covariances in Var ( Tlb) ,

using the results of Theorem 2 with Tij ’s replaced with Tlij and n with nl . We

there-fore skip unnecessary repetitions. Now, under H , E ( T ) = 0 and

(37) Var ⎛ ⎜ ⎜ ⎝ √𝜈 0 ̂ ‖𝚺pii‖2 ‖𝚺ii‖ ⎞ ⎟ ⎟ ⎠ = 4[1 + o(1)] (38) Var ⎛ ⎜ ⎜ ⎝ √𝜈 0 ̂ ‖𝚺pij‖2 ‖𝚺ij‖ ⎞ ⎟ ⎟ ⎠ = 4 � 1+ 1 𝜂2 ij[1 + o(1)], (39) E( Tg) = bi=1 bj=1 i<j 1 pipj‖𝚺ij‖ 2 (40) Var( Tg) = 1 𝜈2 0 gl=1 1 𝜈2 l Var( Tlb), (41) Var( Tlb) = bi=1 bj=1 i<j Var( Tlij) + 2 bi=1 bj=1 bj=1 i<j<j

Cov( Tlij, Tlij�)

+ 2 bi=1 bi=1 bj=1 i<i<j Cov( Tlij, Tlij) + bi=1 bi=1 bj=1 bj=1 i<i<j<j� Cov( Tlij, Tlij�)

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where 𝜈0=∑gl=1𝜈−1

l , P(nl) = nl(nl− 1)(nl− 2)(nl− 3) and

Vlij=‖𝚺lii‖2‖𝚺ljj‖2∕p2ip

2

j . It is obvious from the construction and moments of Tg

that its limit, by the independence of samples, follows along the same lines as that of Tb without much new computation, and likewise holds for the consistency of

̂

‖𝚺lii‖2∕p2i using that of ̂‖𝚺ii‖2∕p2i , so that, plugged in, they yield ̂Var( Tg) as a

con-sistent estimator of Var ( Tg) as the following, where ̂Vpij= ̂‖𝚺pii‖2‖𝚺̂pjj‖2∕p2ip

2

j:

The following theorem extends Theorem 6 to the multi-sample case; see

“Appen-dix 1.2.3“ for proof.

Theorem 12 For Tg in (35) and Assumptions 7–9, ( Tg− E ( Tg))∕𝜎Tg

������→ N(0, 1) ,

as nl, pi→∞ , with 𝜎2T

g= Var ( Tg) in Eq. (40). Under H0g and Assumptions 7–8, √𝜈

0Tg∕𝜎Tg

������→ N(0, 1) with 𝜎2Tg in Eq. (42). Further, the limits hold if Var ( Tg) is replaced with a consistent estimator ̂Var( Tg).

5 Simulations

We evaluate the performance of Tb and Tg through simulations. For the

one-sample multi-block statistic, Tb , we take b = 3 and sample random vectors of

sizes n ∈ {20, 50, 100} from multivariate normal, t and chi-square distribu-tions with 10 degrees of freedom each for the latter two, with block dimensions

p1∈ {10, 25, 50, 150, 300} , p2 = 2p1 , p3= 3p1. Two covariance patterns are assumed for the true covariance matrix, i.e., compound symmetry (CS) and AR(1), defined, respectively, as (1 − 𝜌)𝐈p+ 𝜌𝐉p and 𝚺 = 𝐁𝐀𝐁 with 𝐀 = 𝜌|k−l|

1∕5

and 𝐁 a diagonal matrix with entries square roots of 𝜌 + 𝐩1∕p , where 𝐩1 = 1 ∶ p , 𝐈 is the

identity matrix and 𝐉 is matrix of 1s. We set 𝜌 = 0.5 . Under H0b , the same

struc-tures are imposed on three diagonal blocks 𝚺ii of dimensions pi , so that 𝚺 = ⊕3i=1𝚺ii ,

where ⊕ denotes the Kronecker sum.

(42) Var( Tg) = 1 𝜈2 0 gl=1 1 𝜈2 l 2(2n2 l − 12nl+ 21) P(nl) bi=1 bj=1 i<j 1 p2 ip 2 j ‖𝚺lii‖2‖𝚺ljj‖2 = O � 1 𝜈0b i=1 bj=1 i<j Vlij, (43) Var(√𝜈0Tg) = bi=1 bj=1 i<j V pij.

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For the multi-sample statistic, Tg , we take g = 2 with b = 2 , and generate n1= {20, 40, 75} and n2= {30, 50, 100} iid vectors from respective populations with p1= {20, 50, 100, 300, 500} and p2= 2p1 . Under homoscedasticity, the

com-mon 𝚺 is assumed to follow the two covariance structures under the alternative, where under the null, the two diagonal blocks are assumed to follow the same structures with their respective dimensions. The estimated size and power of Tb

and Tg , reported in Tables 1, 2, 3 and 4, respectively, are an average of 1000

simulation runs.

We observe an accurate test size for both tests, Tb and Tg , for all parameter

settings under all distributions. For multi-sample case, the test performs relatively more accurately although the sample sizes reasonably differ. Note that, generally, both tests are very conservative for relatively small sample sizes, but improve with increasing sample size. In particular, the accuracy of the tests remains intact for increasing dimension. The results under the t and uniform distributions point to the evidence of robustness of the tests to the normality assumption, under Model (4).

Similar performance can be witnessed for the power of both test statistics. Like test size, the accuracy of which increases for increasing sample size, and the power also improves with increasing sample size, but also with increasing dimen-sion, even for seriously differing dimensions for different blocks. In particular, for the multi-sample case, we observe an accurate performance of Tg for unbalanced

design, and the accuracy is not disturbed for increasing dimension. The robust-ness character of the two tests also resembles that for the test size.

Table 1 Estimated size of Tb with 3 blocks for normal, t and uniform distributions

(ND,TD,UD) under compound symmetric and autoregressive (CS,AR) structures n p ND TD CD CS AR CS AR CS AR 20 60 0.931 0.936 0.920 0.939 0.933 0.945 150 0.931 0.938 0.928 0.938 0.941 0.947 300 0.932 0.940 0.935 0.941 0.941 0.949 900 0.936 0.944 0.937 0.945 0.947 0.950 1500 0.938 0.949 0.933 0.948 0.944 0.948 50 60 0.940 0.949 0.939 0.941 0.945 0.947 150 0.942 0.944 0.941 0.949 0.952 0.944 300 0.947 0.943 0.944 0.951 0.944 0.951 900 0.947 0.954 0.946 0.953 0.958 0.954 1500 0.958 0.951 0.949 0.951 0.948 0.952 100 60 0.948 0.940 0.944 0.947 0.946 0.951 150 0.944 0.952 0.942 0.945 0.951 0.950 300 0.947 0.954 0.945 0.948 0.955 0.954 900 0.948 0.950 0.948 0.955 0.951 0.949 1500 0.952 0.947 0.950 0.953 0.949 0.946

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6 Discussion and Conclusions

Test statistics for correlation between two or more vectors are presented when the dimensions of the vectors, possibly unequal, may exceed the number of vectors. The

Table 2 Estimated power of Tb with 3 blocks for normal, t and uniform distributions

(ND,TD,UD) under compound symmetric and autoregressive (CS,AR) structures n p ND TD CD CS AR CS AR CS AR 20 60 0.116 0.258 0.144 0.153 0.115 0.283 150 0.130 0.422 0.201 0.168 0.194 0.418 300 0.224 0.482 0.257 0.246 0.324 0.480 900 0.365 0.625 0.359 0.330 0.378 0.624 1500 0.596 0.726 0.571 0.453 0.508 0.738 50 60 0.213 0.389 0.243 0.254 0.228 0.531 150 0.305 0.547 0.289 0.315 0.295 0.798 300 0.572 0.849 0.559 0.539 0.564 0.875 900 0.761 0.958 0.759 0.736 0.783 0.957 1500 0.970 1.000 0.958 0.814 0.887 0.984 100 60 0.452 0.683 0.474 0.609 0.562 0.782 150 0.619 0.792 0.587 0.902 0.641 0.835 300 0.933 0.993 0.874 0.978 0.929 0.993 900 1.000 1.000 0.986 0.999 0.989 0.999 1500 1.000 1.000 0.999 1.000 0.993 1.000

Table 3 Estimated size of Tg with 3 blocks for normal, t and uniform distributions

(ND,TD,UD) under compound symmetric and autoregressive (CS,AR) structures (n1, n2) p ND TD CD CS AR CS AR CS AR (20, 30) 60 0.931 0.937 0.930 0.938 0.936 0.941 150 0.938 0.935 0.936 0.941 0.939 0.942 300 0.940 0.943 0.934 0.943 0.939 0.939 900 0.939 0.941 0.940 0.939 0.942 0.940 1500 0.942 0.945 0.941 0.940 0.947 0.944 (40, 50) 60 0.939 0.942 0.940 0.939 0.946 0.942 150 0.941 0.944 0.942 0.943 0.945 0.942 300 0.947 0.948 0.945 0.947 0.947 0.941 900 0.942 0.953 0.947 0.949 0.947 0.944 1500 0.944 0.945 0.952 0.949 0.951 0.947 (75, 100) 60 0.956 0.947 0.944 0.944 0.949 0.943 150 0.952 0.948 0.948 0.954 0.956 0.947 300 0.951 0.946 0.949 0.953 0.952 0.947 900 0.951 0.949 0.948 0.950 0.949 0.952 1500 0.955 0.948 0.953 0.954 0.951 0.954

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one-sample case is further extended to two or more independent samples coming from populations assumed to have common covariance matrix. Accuracy of the tests is shown through simulations with different parameters. Among potential advantages of the tests include their simple construction, particularly for the multi-sample case whence most computations follow conveniently from the one-sample results, and their wide practical applicability under fairly general conditions and for a larger class of mul-tivariate models including the mulmul-tivariate normal.

A particularly distinguishing feature of the tests is that they are composed of com-putationally very efficient estimators defined as simple functions of the empirical covariance matrix. From applications perspective, it may be mentioned that the tests are constructed using the RV coefficient so that they can only be used to assess linear independence of vectors. It distinguishes them from measures such as distance correla-tion or kernel methods which can also measure nonlinear dependence; see also [5] for more details.

Acknowledgements The author is thankful to the editors and the two referees for their comments which helped improve the original draft of the article.

Compliance with Ethical Standards

Conflict of interest There is no conflict of interest concerning this manuscript.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-Table 4 Estimated power of

Tg with 3 blocks for normal, t and uniform distributions

(ND,TD,UD) under compound symmetric and autoregressive (CS,AR) structures n p ND TD CD CS AR CS AR CS AR (20, 30) 60 0.705 0.774 0.825 0.659 0.318 0.685 150 0.882 0.891 0.915 0.783 0.356 0.844 300 0.953 0.914 0.993 0.852 0.589 0.868 900 0.985 0.934 1.000 0.872 0.704 0.902 1500 1.000 0.951 1.000 0.895 0.795 0.934 (40, 50) 60 0.978 0.981 0.912 0.892 0.537 0.884 150 0.995 0.997 0.958 0.927 0.729 0.989 300 0.999 1.000 1.000 0.989 0.912 0.999 900 1.000 1.000 1.000 0.994 0.967 1.000 1500 1.000 1.000 1.000 0.999 0.993 1.000 (75, 100) 60 1.000 1.000 0.984 0.970 0.962 1.000 150 1.000 1.000 1.000 0.996 0.998 1.000 300 1.000 1.000 1.000 1.000 1.000 1.000 900 1.000 1.000 1.000 1.000 1.000 1.000 1500 1.000 1.000 1.000 1.000 1.000 1.000

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and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Appendix 1: Technical Results and Proofs

Some Basic Moments

Given Model (4) with 𝐘ik= 𝐗ik− 𝝁i , let Aik= 𝐘Tik𝐘ik , Aikr= 𝐘Tik𝐘ir , k ≠ r , and 𝜅ii , 𝜅ij

be as in Eq. (10) ; then, the moments in the following theorem hold under Model (4). Theorem  13 For Aik , Aikr defined above, E (Aik) = tr (𝚺ii) ,

E(Aikr) = 0 , E(A2ikr) =‖𝚺ii‖2 , E(AirkAiks) = 0 , E(AikAikr) = 0 ,

E(𝐘T

ik𝚺ij𝐘jk) =‖𝚺ij‖2 , E(𝐘Tik𝚺ij𝐘jr) = 0 , Var(𝐘Tik𝚺ij𝐘jk)

= K+ tr (𝚺ii𝚺ij𝚺jj𝚺ji) , Var(𝐘ikT𝚺ij𝐘jr) = tr (𝚺ii𝚺ij𝚺jj𝚺ji) ,

Var(AikrAjrk) = K +‖𝚺ii‖2‖𝚺jj‖2+ 2 tr (𝚺ii𝚺ij𝚺jj𝚺ji) , where K is a constant

involv-ing only 𝜅ij and 𝜅ii.

Main Proofs

Proof of Theorem 1

Since we can write ̂𝚺 =n

k=1𝐘k𝐘Tk∕n −

n

k≠r𝐘k𝐘Tr∕n(n − 1) , it implies ([2, 4])

Using ̂𝚺ij and Theorem 13, Eqs. (7) and (9) can be obtained from ‘Appendix B.1’ in

[4] by replacing 1 with i and 2 with j. They express ‖̂𝚺ij‖2 , ‖̂𝚫ii‖2‖̂𝚫jj‖2 and Qij in

terms of functions of Aik and Aikr , defined above, so that, taking expectation, it

fol-lows that ̂ 𝚺ij=1 n nk=1 𝐘ik𝐘Tjk− 1 n(n − 1) nk=1 nr=1 k≠r 𝐘ik𝐘Tjr, ̂ 𝚺ii=1 n nk=1 𝐘ik𝐘Tik− 1 n(n − 1) nk=1 nr=1 k≠r 𝐘ik𝐘Tir. (44) E�‖̂𝚺ij‖2 � =1 n𝜅ij+ 1 n− 1 � n‖𝚺ij‖2+‖𝚫 ii‖2‖𝚫jj‖2 � , (45) E � ‖̂𝚫ii‖2‖̂𝚫jj‖2 � =1 n𝜅ij+ 2 n− 1‖𝚺ij‖ 2+‖̂𝚫 ii‖2‖̂𝚫jj‖2, (46) E(Qij) =n2− 3n + 3 n2 𝜅ij+ n− 1 n � 2‖𝚺ij‖2+‖𝚫ii‖2‖𝚫jj‖2 � .

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Solving these equations simultaneously gives Eqs. (7) and (9), which can be used to show unbiasedness of ̂𝜅ij . Following the same lines, and using ̂𝚺ii above, we can

write

where, for simplicity, A01, A02, A03 contain terms the expectation of which vanishes,

so that, using Theorem 13, it follows after some simplification that

Solving simultaneously gives Eqs. (6) and (8), and also the unbiasedness of ̂𝜅ii in

Eq. (12).

Proof of Theorem 11

Under the assumption 𝚺l= 𝚺 ∀ l = 1, … , g , the unbiasedness follows immediately

since ‖̂𝚺ii‖2= 1 n2 ⎧ ⎪ ⎨ ⎪ ⎩ nk=1 A2ik+ nk=1 nr=1 k≠r �� 1+ 1 (n − 1)2 � A2ikr+ AikAir �⎫ ⎪ ⎬ ⎪ ⎭ + A01 {‖̂𝚫ii‖}2= 1 n2 ⎧ ⎪ ⎨ ⎪ ⎩ nk=1 A2ik+ nk=1 nr=1 k≠r � 2 (n − 1)2A 2 ikr+ AikAir �⎫ ⎬ ⎪ ⎭ + A02 Qii= 1 n3(n − 1)(n3− 3n + 3)(n − 1) nk=1 A2ik + (2n − 3) nk=1 nr=1 k≠r2A2ikr+ AikAir �⎫⎪ ⎬ ⎪ ⎭ + A03, (47) E�‖̂𝚺ii‖2 � =1 n𝜅ii+ 1 n− 1 � n‖𝚺ii‖2+�‖𝚫ii‖2 �2� , (48) E��‖̂𝚫ii‖2 �2� =1 n𝜅ii+ 2 n− 1‖𝚺ii‖ 2+‖𝚫 ii‖2 �2 , (49) E(Qii) =n2− 3n + 3 n2 𝜅ii+ n− 1 n � 2‖𝚺ii‖2+ � ‖𝚫ii‖2 �2� . E � ̂ ‖𝚺pii‖2 � = 1 𝜈0 gl=1 1 𝜈l‖𝚺ii‖ 2=‖𝚺 ii‖ 2, E‖𝚺̂ pij‖2 � = 1 𝜈0 gl=1 1 𝜈l‖𝚺ij‖ 2=‖𝚺 ij‖ 2

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∀ i, j = 1, … , b , i < j . For variances, we note, following Theorem  2 for any

l∈ {1, … , g} , that

where d(nl) = 2n3l − 9n

2

l + 9nl− 16 , e(n) = n2l − 3nl+ 8 , so that we can write

The first and last terms vanish under the assumptions, so that, as n, pi→∞,

using 𝜈−1 l = O(n 2 l) ⇒ 𝜈0= O(g l=1n 2

l) , which also gives the consistency of ̂

‖𝚺pii‖2∕p2i . Similarly,

With 𝜂ij=‖𝚺ij‖2∕‖𝚺ii‖‖𝚺jj , it implies under the assumptions that, as pi→∞,

Var � ̂ ‖𝚺pii‖2 � =1 𝜈2 0 gl=1 1 𝜈2 l Var � ̂ ‖𝚺lii‖2 � = 1 𝜈2 0 gl=1 1 𝜈2 l 4 P(nl) � d(nl)‖𝚺2ii‖ 2+ e(n l) � ‖𝚺2 ii‖ �2 + KO(n2l) � , Var�‖𝚺̂pii‖2 � = 4 𝜈2 0 gl=1 1 𝜈2 lO � 1 nl‖𝚺2 ii‖ 2+ O � 1 n2l � � ‖𝚺2 ii‖ �2 + KO � 1 n2l �� . Var ⎛ ⎜ ⎜ ⎝ ̂ ‖𝚺pii‖2 ‖𝚺ii‖2 ⎞ ⎟ ⎟ ⎠ = 4 𝜈2 0 gl=1 1 𝜈2 l On−2l= 4O � 1 𝜈0 � , (50) Var( ̂‖𝚺pij‖2) = 1 𝜈2 0 gl=1 1 𝜈2 l Var�‖𝚺̂lii‖2 � (51) =2 𝜈2 0 gl=1 1 𝜈2 l �� 4‖𝚺ij𝚺ji‖2+ 2 tr (𝚺ii𝚺ij𝚺jj𝚺ji) � O � 1 nl � 2��‖𝚺ij‖2 �2 +‖𝚺ii‖2‖𝚺jj‖2+ KO(1)O � 1 n2 l �� . Var ⎛ ⎜ ⎜ ⎝ ̂ ‖𝚺pij‖2 ‖𝚺ij‖2 ⎞ ⎟ ⎟ ⎠ = 4 𝜈2 0 gl=1 1 𝜈2 l � 1+ 1 𝜂2 ijO � 1 n2 l � = 4 � 1+ 1 𝜂ijO � 1 𝜈0 � .

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Proofs of Theorems 6 and 12

Consider Tb=

b

i<jTij with Tij= ̂‖𝚺ij‖2∕pipj . In Ahmad ([5],  Theorem  3), it is

shown that

under Assumptions 3–5, as n, pi→∞ , where Var ( Tij) follows from Theorem 2. As

the limit holds for all Tij , i < j , and Tb is a sum of all such Tij , we basically need to

focus on the covariances in Eqs. (19)–(21) for the limit of Tb . Consider C1 , where

the first term, normed by ‖𝚺ii‖2‖𝚺jj‖‖𝚺jj′‖ , vanishes under Assumptions 3–5, as

pi , and the same holds for the second term. Repeating the same for C

2 and

C3 , and noting that the terms like ‖𝚺ii‖2∕p2

i are uniformly bounded under the same

assumptions, it follows that

where 𝜂ij=‖𝚺ij‖2∕‖𝚺ii‖‖𝚺jj‖ ; see Eq. (5). Combined with Eq. (22), it implies that

Var(n Tb) is bounded, but the covariances with the same order vanish. Now, denote

𝐓B= (𝐓1,… , 𝐓b−1)� with Tb= 𝟏𝐓B where 𝐓i= ( Ti,i+1,… , Tib)� , i = 1, … , b − 1 , B= b(b − 1)∕2 and 𝟏B is the vector of 1s. By the above arguments, as n, pi→∞ ,

Cov(𝐓B) = 𝚲 is a diagonal matrix with diagonal elements Var (Tij) , i, j = 1, … , b , i < j , i.e.,

where 𝚲i= Cov (𝐓i) = diag( Var ( Ti1,… , Var ( Tib)) , i = 1, … , b − 1 . Hence (see

Eq. 22),

This gives the limit Tb by a simple application of Cramér–Wold device ([31], p. 16),

including for the case under the null whence the covariances vanish exactly. For the last part of the theorem, we only need to prove that, we note, from Theorem 2, that

The first and last terms vanish under the assumptions, so that Var( ̂‖𝚺ii‖2∕‖𝚺ii‖2) → 4O(1∕n2) as pi→∞ . Thus, ̂‖𝚺ii‖2∕p2i

������→‖𝚺ii‖2∕p2

i , as n, pi→∞ . Plugging in ̂‖𝚺ii‖2 for ‖𝚺ii‖2 in Var ( Tb) gives ̂Var( Tb) as a consistent

estimator of Var ( Tb) . This completes the proof of Theorem 6.

The proof of Theorem  12 now essentially follows from above, by the independence of g samples. First, the covariance terms in Var ( T , i.e.,

Tij− E ( Tij) √ Var( Tij)  � ������→N(0, 1), C1=(2𝜂ij𝜂ij+ 5𝜂jj� ) O(n−3), C2 =(2𝜂ij𝜂ij+ 5𝜂ii� ) O(n−3), C3=(3𝜂ij𝜂ij+ 5𝜂ii𝜂jj� ) O(n−3), 𝚲 = diag(𝚲1,… , 𝚲b−1), E( Tb) = 𝟏𝐓B = E (Tb), Var( Tb) = 𝟏𝚲𝟏= Var ( Tb). Var �̂ ‖𝚺ii‖2 ‖𝚺ii‖2 � = 4 ⎡ ⎢ ⎢ ⎣ tr(𝚺4ii) � tr(𝚺2ii)�2 O�1 n+ [1 + o(1)]O�1 n2 �⎤ ⎥ ⎥ ⎦ .

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Cov( Tlij, Tlij�) , Cov ( Tlij, Tlij) and Cov ( Tlij, Tlij�), asymptotically

van-ish similarly as C1 , C2 , C3 did for Var ( Tb). Then, with 𝜈l= O(n2l) ⇒ 𝜈0= O(gl=1n2 l) , Var ( Tg) = 4O𝜈−1 0 � ∑ i

j Vlij , with Vlij uniformly bounded

under the assumptions, so that √𝜈0Tg has a non-degenerate limit, as n, pi→∞ .

Define 𝐓G= (𝐓1,… , 𝐓g)� with Tg= 𝟏𝐓G , where 𝐓l= (𝐓l1,… , 𝐓l,b−1)� and

𝐓li= ( Tli,i+1,… , Tlib)� , i = 1, … , b , l = 1, … , g . By the independence of g samples,

Cov(𝐓G) = 𝚫 reduces to

where, by the above arguments, 𝚫ll= Cov (𝐓li) is again a diagonal matrix with

diag-onal elements Var ( Tlij) , i, j = 1, … , b , i < j . Thus,

asymptotically coincide with Eqs. (39) and (40), respectively. Finally, the consist-ency of ̂Var( Tg) follows from that of 𝚺pii∕p2i as shown in “Appendix 1.2.2”. References

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Figure

Table 1   Estimated size of  T b  with 3 blocks for normal,  t and uniform distributions  (ND,TD,UD) under compound  symmetric and autoregressive  (CS,AR) structures n p ND TD CDCSARCSARCS AR20600.9310.9360.9200.9390.933 0.945 150 0.931 0.938 0.928 0.938 0
Table 3   Estimated size of  T g  with 3 blocks for normal,  t and uniform distributions  (ND,TD,UD) under compound  symmetric and autoregressive  (CS,AR) structures (n 1 , n 2 ) p ND TD CDCSARCSARCS AR(20, 30)60 0.931 0.937 0.930 0.938 0.936 0.941 150 0.9

References

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