An Invariant Test for Equality of Two Large Scale Covariance Matrices

11/14/2019 ∙ by Taehyeon Koo, et al. ∙ Seoul National University 0

In this work, we are motivated by the recent work of Zhang et al. (2019) and study a new invariant test for equality of two large scale covariance matrices. Two modified likelihood ratio tests (LRTs) by Zhang et al. (2019) are based on the sum of log of eigenvalues (or 1- eigenvalues) of the Beta-matrix. However, as the dimension increases, many eigenvalues of the Beta-matrix are close to 0 or 1 and the modified LRTs are greatly influenced by them. In this work, instead, we consider the simple sum of the eigenvalues (of the Beta-matrix) and compute its asymptotic normality when all n_1, n_2, p increase at the same rate. We numerically show that our test has higher power than two modified likelihood ratio tests by Zhang et al. (2019) in all cases both we and they consider.

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1 Introduction

We revisit the test of equality (homogeneity) of two covariance matrices, which often allows us simplified procedures for many multivariate problems. Suppose we have samples

from a distribution with a mean vector

and covariance matrix for . The hypothesis, which is of interest, is

(1)

As pointed out by Zhang et al. (2019), the history of the test draws back to 1930s and a huge number of works are followed in literature. In this paper, we do not aim to compete with all methods in the literature (see Chapter 10 of Anderson, T.W. (2003) and references therein). Instead, we focus on a specific invariant test as an alternative to the modified likelihood ratio test (mLRT), which is recently suggested by Zhang et al. (2019). In Section 2, we compute the asymptotic null distribution of the new test, when all increase at the same rate. In Section 3, we numerically show it has higher power than mLRT in all cases both we and they consider. In Section 4, we conclude the paper with some remarks.

2 An alternative invariant statistic

We find , for , where

are independent and identically distributed (IID) with mean zero and variance one, respectively, and satisfy

Let and be the sample covariance matrix from each population. To build our test, we focus on the limiting distribution of eigenvalues of , named as the limiting spectral distribution (LSD) of . With the notations

the limiting spectral distribution of is evaluated as (see Zhang et al. (2019))

where and as .

Let where is empirical spectral distribution (ESD) of and is the limit spectral distribution (LSD) of with parameters replacing . Our main interest is in the limit distribution of

(2)

where are analytic functions on complex domain.

Suppose are the ESD and LSD of the -matrix , and

Following Bai and Silverstein (2010), the linear spectral statistic (LSS) of the -matrix for functions that is

under some regular conditions, converges weakly to a Gaussian vector with means

and the covariance matrix whose element is

If is an eigenvalue of , the eigenvalue of corresponds to is . Using this, we find that

(3)

In addition, we obtain the LSS of from the above by substituting in (3).

The mLRT statistics in Zhang et al. (2019) are

where denotes the -th smallest eigenvalue of . In mLRT statistics, the eigenvalues or are excluded for defining valid statistics. However, if (or ) is close to , many eigenvalues are close to (or ). The eigenvalues either close to or explains most part of the statistics. The mLRT statistics and are sensitive to those and do not fully reflect the information from other eigenvalues of .

To resolve this difficulty, we consider

To make above statistic meaningful, we modify it to

(4)

To get the asymptotic null distribution of the proposed statistic, we can find the mean and variance of LSS of by setting , in the formula above (3). However, before we proceed, we remark that

(5)

Thus, the LSS of and are opposite in their sign, and the covariance between LSS of is the negative of asymptotic variance of (or ).

We now have our main results on the asymptotic null distributions of and .

Theorem 1.

Suppose we assume (i) (moment assumptions)

, , and , and (ii) (dimensionality assumption) , , and . Then we have

(6)

where ,

and

Proof.

The proof is given in Appendix. ∎

3 Numerical study

In this section, we numerically compare the powers of the proposed to the modified LRTs and by Zhang et al. (2019) under various choices of sample sizes, dimensions, and distributions. In the study, we assume for and set , where is a constant. We consider four cases following Zhang et al. (2019):

  • Case 1: and

    are from the standard normal distributed and

    ;

  • Case 2: and

    are from the uniform distribution

    and ;

  • Case 3: and are from the uniform distribution and ;

  • Case 4: and are from the uniform distribution and .

For each case, we consider choices of , four cases are for each , ,, . Four choices are considered for , , where

is the choice of the null hypothesis. In all cases, we assume that forth moments of

and are known. In case 1, and in case 2,3, and 4, . For each combination, we generate data sets and the powers (the sizes) are evaluated by counting the number of rejected data sets. The results are reported in Tables 1 to 4. In the tables, and stand for two modified LRTs by Zhang et al. (2019) and is our statistic in Theorem 1. Moreover, Figures 2 - 4 display (empirical) powers with respect to different choices of for the Cases 1 - 4. We find that, in all cases considered, the power of is higher than both and .

(i)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.062 0.056(0.046) 0.058(0.044) 0.070(0.060) 0.050 0.051(0.058) 0.059(0.063) 0.069(0.072)
0.061 0.109(0.094) 0.335(0.303) 0.555(0.506) 0.045 0.102(0.110) 0.336(0.350) 0.610(0.625)
0.062 0.150(0.141) 0.568(0.535) 0.835(0.821) 0.054 0.129(0.117) 0.603(0.590) 0.869(0.865)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.041 0.043(0.058) 0.063(0.085) 0.070(0.091) 0.056 0.041(0.036) 0.053(0.043) 0.080(0.074)
0.047 0.109(0.119) 0.348(0.356) 0.620(0.639) 0.059 0.105(0.092) 0.352(0.330) 0.632(0.606)
0.049 0.175(0.182) 0.589(0.597) 0.882(0.886) 0.051 0.188(0.188) 0.601(0.600) 0.891(0.891)
(ii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.066 0.071(0.056) 0.110(0.092) 0.227(0.188) 0.056 0.065(0.187) 0.131(0.122) 0.211(0.195)
0.049 0.169(0.170) 0.519(0.523) 0.806(0.809) 0.042 0.158(0.187) 0.602(0.639) 0.858(0.874)
0.060 0.216(0.187) 0.738(0.718) 0.957(0.952) 0.046 0.224(0.228) 0.807(0.814) 0.974(0.976)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.057 0.055(0.051) 0.130(0.117) 0.209(0.195) 0.043 0.060(0.070) 0.142(0.158) 0.225(0.242)
0.065 0.174(0.142) 0.614(0.577) 0.864(0.831) 0.046 0.167(0.175) 0.626(0.635) 0.897(0.901)
0.060 0.234(0.209) 0.819(0.795) 0.979(0.975) 0.055 0.251(0.240) 0.827(0.821) 0.982(0.982)
(iii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.061 0.056(0.044) 0.095(0.086) 0.116(0.103) 0.061 0.074(0.046) 0.084(0.056) 0.099(0.068)
0.060 0.083(0.063) 0.291(0.257) 0.511(0.461) 0.058 0.107(0.097) 0.293(0.276) 0.516(0.498)
0.060 0.140(0.123) 0.494(0.473) 0.780(0.758) 0.057 0.153(0.144) 0.505(0.489) 0.822(0.808)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.041 0.051(0.058) 0.083(0.096) 0.142(0.155) 0.053 0.063(0.062) 0.071(0.071) 0.123(0.119)
0.042 0.086(0.094) 0.286(0.305) 0.563(0.579) 0.054 0.095(0.093) 0.318(0.312) 0.535(0.525)
0.033 0.138(0.176) 0.505(0.565) 0.825(0.849) 0.038 0.145(0.171) 0.576(0.615) 0.822(0.848)
(iv)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.061 0.059(0.042) 0.047(0.036) 0.061(0.043) 0.063 0.051(0.046) 0.387(0.384) 0.053(0.047)
0.059 0.108(0.090) 0.345(0.314) 0.592(0.555) 0.053 0.116(0.113) 0.387(0.384) 0.636(0.632)
0.048 0.146(0.154) 0.617(0.624) 0.892(0.896) 0.055 0.202(0.189) 0.666(0.651) 0.907(0.902)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.042 0.060(0.069) 0.059(0.064) 0.047(0.056) 0.044 0.053(0.056) 0.058(0.064) 0.49(0.054)
0.049 0.112(0.114) 0.403(0.406) 0.656(0.658) 0.046 0.119(0.130) 0.391(0.408) 0.686(0.700)
0.049 0.187(0.187) 0.675(0.678) 0.917(0.917) 0.055 0.195(0.187) 0.694(0.684) 0.931(0.927)
Table 1:

Case 1: Empirical sizes and powers. The results are based on the 5% significance level. The power number in small bracket indicates the size corrected power based empirical 95% quantile not asymptotic. (i) indicates the case of

, (ii) does , (iii) does , and (iv) does .
(i)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.060 0.055(0.041) 0.067(0.051) 0.061(0.047) 0.054 0.052(0.047) 0.084(0.075) 0.078(0.072)
0.061 0.136(0.120) 0.371(0.328) 0.602(0.572) 0.064 0.129(0.105) 0.396(0.342) 0.666(0.623)
0.054 0.203(0.190) 0.700(0.686) 0.917(0.913) 0.052 0.207(0.207) 0.732(0.730) 0.951(0.950)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.045 0.050(0.068) 0.064(0.081) 0.098(0.113) 0.067 0.054(0.040) 0.072(0.056) 0.083(0.062)
0.047 0.122(0.130) 0.399(0.410) 0.692(0.712) 0.061 0.114(0.091) 0.392(0.355) 0.711(0.683)
0.055 0.199(0.194) 0.742(0.731) 0.953(0.943) 0.050 0.212(0.214) 0.754(0.758) 0.964(0.965)
(ii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.060 0.059(0.043) 0.132(0.116) 0.160(0.133) 0.066 0.057(0.044) 0.100(0.078) 0.169(0.129)
0.047 0.177(0.184) 0.646(0.660) 0.857(0.858) 0.044 0.168(0.201) 0.681(0.713) 0.919(0.933)
0.051 0.283(0.283) 0.890(0.890) 0.991(0.991) 0.048 0.308(0.327) 0.903(0.911) 0.999(0.999)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.056 0.055(0.049) 0.110(0.096) 0.193(0.175) 0.063 0.062(0.059) 0.095(0.087) 0.164(0.148)
0.062 0.183(0.176) 0.684(0.665) 0.932(0.919) 0.037 0.165(0.191) 0.692(0.731) 0.937(0.944)
0.055 0.321(0.314) 0.914(0.907) 0.998(0.997) 0.050 0.310(0.310) 0.922(0.924) 0.999(0.999)
(iii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.057 0.066(0.057) 0.096(0.088) 0.119(0.104) 0.053 0.053(0.052) 0.104(0.103) 0.129(0.127)
0.057 0.106(0.099) 0.331(0.316) 0.559(0.543) 0.059 0.090(0.079) 0.352(0.314) 0.601(0.552)
0.048 0.162(0.176) 0.618(0.640) 0.899(0.909) 0.040 0.181(0.199) 0.667(0.688) 0.927(0.933)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.053 0.066(0.066) 0.089(0.088) 0.129(0.127) 0.052 0.057(0.057) 0.091(0.091) 0.128(0.128)
0.052 0.124(0.111) 0.350(0.334) 0.600(0.586) 0.046 0.104(0.109) 0.369(0.380) 0.605(0.618)
0.039 0.190(0.214) 0.685(0.718) 0.928(0.941) 0.049 0.199(0.199) 0.689(0.691) 0.926(0.928)
(iv)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.040 0.051(0.057) 0.049(0.057) 0.050(0.056) 0.048 0.053(0.054) 0.061(0.063) 0.056(0.059)
0.046 0.133(0.133) 0.409(0.435) 0.656(0.677) 0.046 0.108(0.113) 0.421(0.434) 0.707(0.713)
0.052 0.254(0.250) 0.807(0.806) 0.970(0.970) 0.043 0.222(0.240) 0.814(0.840) 0.981(0.983)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.051 0.052(0.050) 0.055(0.054) 0.050(0.050) 0.050 0.061(0.062) 0.050(0.050) 0.052(0.052)
0.050 0.130(0.133) 0.458(0.460) 0.720(0.722) 0.040 0.117(0.127) 0.448(0.462) 0.741(0.754)
0.047 0.259(0.268) 0.833(0.846) 0.984(0.986) 0.058 0.252(0.241) 0.836(0.827) 0.994(0.992)
Table 2: Case 2: Empirical sizes and powers. The results are based on the 5% significance level. The power number in small bracket indicates the size corrected power based empirical 95% quantile not asymptotic. (i) indicates the case of , (ii) does , (iii) does , and (iv) does .
(i)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.055 0.050(0.039) 0.067(0.061) 0.066(0.057) 0.053 0.061(0.059) 0.061(0.055) 0.093(0.088)
0.060 0.112(0.102) 0.368(0.354) 0.586(0.570) 0.050 0.124(0.131) 0.372(0.379) 0.645(0.651)
0.049 0.185(0.194) 0.717(0.740) 0.915(0.923) 0.054 0.220(0.215) 0.711(0.706) 0.947(0.945)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.035 0.075(0.090) 0.070(0.089) 0.087(0.105) 0.039 0.057(0.072) 0.066(0.076) 0.084(0.096)
0.043 0.113(0.122) 0.408(0.423) 0.693(0.705) 0.054 0.111(0.102) 0.394(0.378) 0.697(0.678)
0.050 0.186(0.186) 0.738(0.738) 0.955(0.955) 0.041 0.199(0.236) 0.741(0.765) 0.961(0.967)
(ii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.070 0.054(0.043) 0.104(0.081) 0.165(0.116) 0.057 0.048(0.043) 0.108(0.099) 0.175(0.163)
0.056 0.170(0.159) 0.631(0.619) 0.859(0.853) 0.050 0.185(0.189) 0.654(0.660) 0.921(0.921)
0.056 0.280(0.268) 0.880(0.869) 0.979(0.979) 0.052 0.309(0.298) 0.902(0.900) 0.992(0.992)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.054 0.062(0.054) 0.095(0.091) 0.179(0.169) 0.054 0.058(0.055) 0.105(0.103) 0.160(0.156)
0.044 0.183(0.192) 0.691(0.700) 0.927(0.928) 0.039 0.182(0.199) 0.669(0.693) 0.934(0.947)
0.046 0.295(0.307) 0.902(0.915) 0.999(0.999) 0.047 0.328(0.346) 0.923(0.935) 0.998(0.998)
(iii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.060 0.056(0.051) 0.088(0.079) 0.116(0.107) 0.050 0.076(0.076) 0.109(0.110) 0.116(0.116)
0.054 0.092(0.074) 0.324(0.301) 0.558(0.541) 0.045 0.112(0.132) 0.354(0.382) 0.592(0.625)
0.050 0.155(0.155) 0.645(0.645) 0.903(0.903) 0.044 0.167(0.187) 0.659(0.687) 0.922(0.935)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.061 0.053(0.046) 0.079(0.065) 0.126(0.113) 0.043 0.057(0.060) 0.082(0.091) 0.136(0.147)
0.065 0.107(0.085) 0.334(0.299) 0.605(0.565) 0.043 0.096(0.109) 0.327(0.353) 0.626(0.642)
0.047 0.176(0.182) 0.676(0.692) 0.923(0.927) 0.045 0.175(0.185) 0.673(0.680) 0.949(0.949)
(iv)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.065 0.043(0.035) 0.056(0.046) 0.066(0.056) 0.060 0.046(0.042) 0.058(0.053) 0.049(0.046)
0.049 0.110(0.112) 0.420(0.425) 0.658(0.665) 0.059 0.135(0.113) 0.419(0.386) 0.700(0.676)
0.044 0.223(0.236) 0.780(0.793) 0.967(0.974) 0.050 0.239(0.242) 0.814(0.817) 0.979(0.979)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.049 0.045(0.046) 0.046(0.046) 0.054(0.057) 0.041 0.057(0.068) 0.047(0.064) 0.061(0.073)
0.052 0.117(0.107) 0.419(0.413) 0.744(0.736) 0.041 0.126(0.132) 0.473(0.491) 0.752(0.760)
0.050 0.257(0.265) 0.854(0.857) 0.989(0.989) 0.047 0.240(0.250) 0.864(0.868) 0.993(0.993)
Table 3: Case 3: Empirical sizes and powers. The results are based on the 5% significance level. The power number in small bracket indicates the size corrected power based empirical 95% quantile not asymptotic. (i) indicates the case of , (ii) does , (iii) does , and (iv) does .
(i)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.055 0.069(0.060) 0.074(0.071) 0.083(0.078) 0.058 0.061(0.054) 0.064(0.055) 0.075(0.065)
0.057 0.104(0.096) 0.359(0.347) 0.599(0.581) 0.054 0.111(0.109) 0.400(0.396) 0.663(0.659)
0.055 0.175(0.166) 0.705(0.686) 0.937(0.933) 0.058 0.179(0.166) 0.743(0.728) 0.956(0.951)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.054 0.061(0.055) 0.058(0.054) 0.091(0.083) 0.042 0.044(0.048) 0.081(0.099) 0.085(0.097)
0.059 0.118(0.115) 0.374(0.361) 0.665(0.652) 0.046 0.122(0.142) 0.387(0.423) 0.687(0.722)
0.066 0.195(0.150) 0.731(0.690) 0.953(0.943) 0.046 0.219(0.230) 0.735(0.745) 0.954(0.956)
(ii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.046 0.058(0.058) 0.119(0.125) 0.174(0.179) 0.064 0.062(0.044) 0.138(0.109) 0.162(0.128)
0.037 0.182(0.237) 0.619(0.742) 0.872(0.902) 0.046 0.178(0.189) 0.673(0.685) 0.897(0.904)
0.045 0.302(0.315) 0.869(0.874) 0.993(0.993) 0.052 0.295(0.289) 0.899(0.898) 0.995(0.995)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.063 0.048(0.045) 0.108(0.096) 0.163(0.147) 0.061 0.063(0.059) 0.106(0.095) 0.165(0.144)
0.044 0.176(0.195) 0.672(0.694) 0.936(0.941) 0.053 0.215(0.211) 0.709(0.701) 0.933(0.931)
0.049 0.322(0.324) 0.922(0.922) 0.997(0.997) 0.045 0.349(0.378) 0.921(0.927) 0.996(0.997)
(iii)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.058 0.073(0.068) 0.085(0.078) 0.125(0.116) 0.051 0.061(0.061) 0.096(0.095) 0.125(0.124)
0.061 0.098(0.082) 0.328(0.299) 0.546(0.518) 0.056 0.096(0.088) 0.353(0.330) 0.606(0.579)
0.064 0.155(0.143) 0.626(0.599) 0.882(0.860) 0.054 0.166(0.161) 0.666(0.654) 0.914(0.910)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.058 0.078(0.073) 0.087(0.081) 0.114(0.102) 0.066 0.073(0.055) 0.094(0.079) 0.141(0.118)
0.046 0.111(0.121) 0.353(0.381) 0.584(0.606) 0.059 0.118(0.103) 0.363(0.324) 0.624(0.578)
0.044 0.193(0.213) 0.684(0.701) 0.935(0.939) 0.039 0.179(0.194) 0.720(0.744) 0.934(0.943)
(iv)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.055 0.046(0.044) 0.058(0.051) 0.068(0.066) 0.064 0.064(0.052) 0.059(0.051) 0.058(0.045)
0.059 0.122(0.103) 0.414(0.382) 0.649(0.624) 0.048 0.110(0.114) 0.451(0.456) 0.687(0.694)
0.051 0.257(0.256) 0.777(0.774) 0.980(0.979) 0.039 0.234(0.269) 0.826(0.843) 0.983(0.986)
a=0 a=3 a=7 a=10 a=0 a=3 a=7 a=10
0.049 0.053(0.054) 0.045(0.045) 0.057(0.058) 0.058 0.053(0.043) 0.043(0.038) 0.054(0.045)
0.050 0.126(0.128) 0.451(0.461) 0.758(0.761) 0.042 0.102(0.116) 0.455(0.477) 0.753(0.769)
0.034 0.256(0.313) 0.828(0.874) 0.992(0.995) 0.048 0.251(0.255) 0.864(0.867) 0.988(0.988)
Table 4: Case 4: Empirical sizes and powers. The results are based on the 5% significance level. The power number in small bracket indicates the size corrected power based empirical 95% quantile not asymptotic. (i) indicates the case of , (ii) does , (iii) does , and (iv) does .
Figure 2: Power divergence in Case 2. T1,T2 stand for and respectively. stands for .
Figure 1: Power divergence in Case 1.T1,T2 stand for and respectively. stands for .
Figure 4: Power divergence in Case 1.T1,T2 stand for and respectively. stands for .
Figure 3: Power divergence in Case 1.T1,T2 stand for and respectively. stands for .

4 Conclusion

In this paper, we suggest an alternative invariant test, named as , to two modified LRTs by Zhang et al. (2019) for the equality of two large scale covariance matrices. It is based on the sum of eigenvalues of . We find the asymptotic null distribution of , when all approach at the same rate. The numerical study shows the new invariant test is more powerful than the modified LRTs by Zhang et al. (2019) in all cases we consider. However, we do not claim the proposed is the most powerful for the problem because, as we learn from lower dimensional cases, there could be more powerful one than for some settings.

Acknowledgement

This research is supported by National Research Foundation of Korea (Grant Number: NRF-2017R1A2B 2012264).

Appendix: Proof of Theorem 1

We will use the Cauchy’s residue theorem and Lemma A.1. in Zhang et al. (2019).

Lemma 2 (Zhang et al. 2019, Lemma A.1).

In addition to the Moments Assumption and the Dimensions Assumption, we further assume that:

  • as , , , and .

  • let be the analytic functions on an open region containing the interval , where , , and is defined as