A new test for the proportionality of two large-dimensional covariance matrices. Citation Journal of Multivariate Analysis, 2014, v. 131, p.

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1 Title A new test for the proportionality of two large-dimensional covariance matrices Authors) Liu, B; Xu, L; Zheng, S; Tian, G Citation Journal of Multivariate Analysis, 04, v. 3, p Issued Date 04 URL Rights NOTICE: this is the author s version of a work that was accepted for publication in Journal of Multivariate Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Multivariate Analysis, 04, v. 3, p DOI: 0.06/j.jmva ; This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International License.

2 A new test for the proportionality of two large-dimensional covariance matrices Baisen Liu a, Lin Xu b, Shurong Zheng b, Guo-Liang Tian c a School of Statistics, Dongbei University of Finance and Economics b KLAS and School of Mathematics and Statistics, Northeast Normal University c Department of Statistics and Actuarial Science, The University of Hong Kong Abstract Let x,..., x iid n + N p µ, Σ ) and y,..., y iid n + N p µ, Σ ) be two independent random samples, where n p < n. In this article, we propose a new test for the proportionality of two large p p covariance matrices Σ and Σ. By applying modern random matrix theory, we establish the asymptotic normality property for the proposed test statistic as p, n, n ) together with the ratios p/n y 0, ) and p/n y 0, ) under suitable conditions. We further showed that these conclusions are still valid if normal populations are replaced by general populations with finite fourth moments. Keywords: Covariance matrix, Hypothesis testing, Large-dimensional data, Limiting spectral distribution, Proportionality, Random F -matrices.. Introduction With the rapid development and wide applications of computer techniques, huge data can be collected and stored. This is called as high-dimensional data or large-dimensional data, see Bai and Silverstein []. Many traditional estimation and test tools are no more valid or perform badly for such large-dimensional data, since these traditional methods are often based on the classical central asymptotic theorems assuming a large sample size n and fixed dimension p. In this article, we consider testing proportionality of large-dimensional covariance matrices from two different populations. The proportionality of covariance Corresponding zhengsr@nenu.edu.cn Preprint submitted to Journal of Multivariate Analysis June 0, 04

3 matrices is the simplest form of heteroscedasticity between populations, which has extensive applications in economics, discriminations, etc. As an instance, consider a quantitative genetic experiment, called paternal half-sib design. This experiment is conducted under the hypothesis of equal heritabilities in the two populations, and it corresponds to the hypothesis of proportionality between population covariance matrices which we will discuss in this paper. The goal of this experiment is to model measurements of some quantitative traits in two independent populations of animal offsprings. More detailed description of this experiment can be found in Jensen and Madsen [0]. Other related examples are: Dargahi-Noubary [4] considered the applications of discrimination between two normal populations when covariance matrices are proportional. Nel and Groenewald [4] studied the multivariate Behrens-Fisher problem under the assumption of proportional covariance matrices. Later, Villa and Pérignon [0] investigated the sources of time variation in the covariance matrix of interest rates. In their work, they discussed the similarities among covariance matrices of bond yields including the cases of equality and proportionality. Let x,..., x iid n + N p µ, Σ ) and y,..., y iid n + N p µ, Σ ) be two independent random samples, where n p < n. To describe the proposed new test, we define N k n k + k, ) and consider the joint sufficient statistics: and x N N x i, ˆV x i x)x i x), N ȳ N N y j, ˆV y j ȳ)y j ȳ), ) N for µ, Σ ) and µ, Σ ), respectively, where A denotes the transpose of A. Thus, ˆΣ k ˆV k /n k is the maximum likelihood estimator of Σ i for k,. The null hypothesis of interest is H 0 : Σ σ Σ, ) where σ > 0 is an unknown constant. When σ, Wilks [] suggested a likelihood ratio LR) test statistic Λ N N log ˆV + N log ˆV N log ˆV + ˆV,

4 where N N + N. However, in practice, it is often to use a modified LR test statistic Λ N n log ˆV + n log ˆV n + n log ˆV + ˆV, due to its unbiasedness and monotonicity of the power function see, Srivastava, Khatri and Carter [8]; Sugiura and Nagao [9]). As the sample sizes N, N, the classical central limiting theorem states that under H 0 see [], Sec. 8.) Λ N D χ and Λ D pp+) N χ pp+). 3) Note that the limiting results in 3) are true only when the variable dimension p is assumed to be fixed and p minn, N ). With computer simulations, Bai et al. [] showed that, employing the χ approximation 3) for dimensions like 30 or 40, increases dramatically the size of the test. For dimension and sample sizes p, N, N ) 40, 800, 400), the test size equals.% instead of the nominal 5% level. The result is even worse for the case of p, N, N ) 80, 600, 800) which leads to a 49.5% test size. For the general case of hypothesis ) with unknown constant σ, to our knowledge, the first research was done by Federer [6] who developed a maximum likelihood method for two groups of normal populations with dimension p 3. Kim [] extensively studied the problem of proportionality between covariance matrices, and showed that the solution of the likelihood equations was unique. This result was later published by Guttman et al. [9]. Independently, Rao [5] considered the likelihood ratio test for proportionality of covariance matrices from two normal populations. As an extension, under normality assumptions, Eriksen [5] and Flury [8], independently, discussed maximum likelihood estimation of proportional covariance matrices and provided likelihood ratio tests for testing the hypothesis of proportionality. Furthermore, Schott [6] considered the test problem of proportional covariance matrices from k different groups and obtained a Wald statistic under general conditions without normal population assumptions. However, all of the above work is based on the classical approximating theory which assumes the dimension of data, p to be held fixed. As it will be shown, this classical approximation leads to a test size much higher than the nominal test level in the case of largedimensional data. All such problems and limitations bring up the need for further study on proportionality test of covariance matrices. 3

5 Based on the modern random matrix theory RMT), we propose a new test and further prove that the distribution of the test statistic proposed approximates to a normal distribution as p, n, n ) together with the ratios p/n y 0, ) and p/n y 0, ) under suitable conditions. The proposed new test is valid for both Gaussian data and non-gaussian data with finite fourth moments. The rest of the article is organized as follows. Section briefly reviews some preliminary and useful RMT results. In Section 3, we propose a new test and study its asymptotic normality as p, n, n ). In Section 4, the efficiency of the proposed test is illustrated by simulation studies. Section 5 presents conclusions and some discussions. Some technical derivations are put into the Appendix.. Review of some useful results of random matrix theory We first review several results from RMT, which will be used for the proposed test procedure. For any p p matrix M with real eigenvalues {λ M i } p, the empirical spectral distribution ESD) of M, denoted by Fn M, is defined by F M n x) p [λ M i x], x R. We will consider a random matrix M whose ESD Fn M converges in a sense to be precisely) to a limiting spectral distribution LSD) F M. To make statistical inference about a parameter θ fx)df M x), it is natural to use an estimator ˆθ fx)df M n x) p fλ M i ), which is a so-called linear spectral statistic LSS) of the random matrix M. Let {ξ ki C: i, k,,...} and {η kj C: j, k,,...} be two independent double arrays of i.i.d. complex variables with mean 0 and variance. Writing ξ i ξ i, ξ i,..., ξ pi ) and η j η j, η j,..., η pj ). For any given positive integers n and n, the vectors {ξ,..., ξ n } and {η,..., η n } can be thought as independent samples of size n and n, respectively, from some p-dimensional distribution. Let S and S be the associated sample covariance matrices, i.e., S n ξ i ξ i and S n η j η n n j. 4

6 Then, the following so-called F -matrix generalizes the classical Fisher-statistic for the present p-dimensional case, where n n, n ) and p < n. Define V n S S, 4) y n p n y 0, + ), y n p n y 0, ), 5) y n y n, y n ) and y y, y ). Under suitable moment conditions, the ESD, F Vn n, of V n has a LSD, F y x) which is given by see [], P. 79 or []) F y dx) g y x) [a,b] x)dx + /y ) [y >]δ 0 dx), 6) where δ c ) denotes the Dirac point measure at c, and h y + y y y, a g y x) h) + h), b y ) y ), y ) b x)x a), a < x < b. πxy + y x) Define the empirical process G n p[fn Vn F yn ] and let à be the set of analytic functions in an open region in the complex plane containing the interval [a, b] which is the support of continuous part of the LSD F y defined in 6). The following central limit theorem CLT) of LSS for a high-dimensional F-matrix was established in Zheng [], which will be applied in our work. Theorem. For each p, assume that ξ ij : i,..., p; j,..., n ) and η ij : i,..., p; j,..., n ) are i.i.d. and satisfy Eξ ) Eη ) 0, E ξ E η, E ξ 4, E η 4 <, y n p/n y 0, + ), and y n p/n y 0, ). Furthermore, let f,..., f k Ã. Then, the random vector G n f ),..., G n f k )) weakly converges to a k-dimensional Gaussian 5

7 vector with mean vector mf j ) lim r 4πi ζ + β x y y ) πi h + β y y y ) πi h [ f j zζ)) ζ ζ ζ r + ζ + r ζ + y /hr f j zζ)) ζ + y /h) dζ 3 ] dζ ζ + /h f j zζ)) dζ, 7) ζ + y /h) 3 and covariance function covf j, f l ) f j zζ ))f l zζ )) lim dζ r π ζ ζ ζ rζ ) dζ β xy + β y y ) y ) f j zζ )) 4π h ζ ζ + y /h) dζ f l zζ )) ζ ζ + y /h) dζ, 8) where zζ) y ) [+h +hreζ)], h y + y y y, β x E ξ 4 3, and β y E η 4 3. This CLT allows for both n n, n ) and p approaching infinity. In the next section, based on the above CLT, we will develop a test statistic T n and provide the limiting distribution of T n. 3. Formulation of the new test Let Σ / and Σ / be two p p positive definite matrices such that Σ Σ / ) and Σ Σ / ), respectively. Define ξ i Σ / x i µ ) and η j Σ / y j µ ). Note that all the random vectors {x i }, {y j } and the covariance matrices Σ and Σ depend on the dimension p. However, we do not signify this dependence in notation for ease of statements. After standardizing, the arrays {ξ i } n + and {η j } n + contain i.i.d. variables with mean 0 and variance, for which we can apply Theorem.. Testing the null hypothesis specified by ) is equivalent to testing H 0 : Σ / Σ Σ / σ I p. 9) 6

8 This is, in fact, a sphericity test. Srivastava [7] discussed this sphericity test for a single population and proposed a measure of this sphericity. As stated in Srivastava [7] also see [7]), the above test is invariant for an orthogonal transformation x Gx. This test is also invariant under a scalar transformation x cx. Thus, without loss of generality, we assume that Σ / Σ Σ / diagλ,..., λ p ). It follows from the Cauchy Schwarz inequality that ) λ i p with equality holding if and only if λ λ p λ, for all i,..., p and some constant λ. Thus, following the idea of Srivastava [7], we suggest testing H 0 : Ψ against H : Ψ > with λ i Ψ p λ i /p) p λ i/p). Note that trσ Σ ) p λ i and trσ Σ Σ Σ ) p λ i ; hence, we propose the following test statistic tr ) T n p tr ˆΣ ˆΣ ˆΣ ˆΣ ) [tr ˆΣ p, 0) ˆΣ )] where ˆΣ and ˆΣ are unbiased sample covariance matrices. On the other hand, a scaled distance measure between σ Σ / Σ Σ / and I p can be defined by ) σ Σ / Σ Σ / /p)trσ Σ / Σ Σ / ) I p, p trσ Σ Σ Σ ) [trσ Σ )] p. This scaled distance measure was also discussed by Nagao [3] for the sphericity test of a single population. Thus, it is reasonable to consider the test statistic 0). Define A Σ / CΣ / and B Σ / DΣ /, where C N x i µ )x i µ ) and D N y j µ )y j µ ). n n 7

9 Note that V n AB indeed forms a random F-matrix. Then, we define T n p trcd CD ) [trcd )] p. ) As T n and T n have the same asymptotic distribution and CLT under H 0, in the next, we will consider the CLT of T n. Theorem 3. Suppose that the conditions in Theorem. hold under H 0 and T n is defined in 0), then, under H 0 and as n, n ), with where h y + y y y. v / T n [T n µ Tn ph y ) D ] N0, ), ) µ Tn h + y ) y ) + β x y + β y y, 3) v Tn 4h h + y ) y ) 4, 4) PROOF. For convenience, we define g x) x and g x) x. Let {λ i } p be the eigenvalues of the F-matrix V n. Following Theorem., we have, as n, n ), p /p) p λ j/σ ) g x)f y dx) /p) p λ j/σ D Nµ CLT, Σ CLT ), g x)f y dx) where mg ) ) vg, g, Σ CLT ) vg, g ) vg, g ) vg, g ) ) µ CLT mg ), and F y dx) is defined in 6). Denote δξ) y ) + h + hreξ), then the components of µ CLT are 8

10 given by mg j ) lim r 4πi ξ + β x y y ) πi h + β y y y ) πih g j δξ)) ξ ξ ξ r + ξ + r ξ + y /h ) g j δξ)) ξ + y /h) 3 ξ + /h g j δξ)), j,, ξ + y /h) 3 and the elements of Σ CLT are given by g k δξ ))g l δξ )) vg k, g l ) lim r π ξ ξ ξ rξ ) β xy + β y y ) y ) g k δξ )) 4π h ξ ξ + y /h) g l δξ )) ξ + y /h), k, l {, }, ξ with β x E ξ 4 3 and β y E η 4 3. By the Residue theorem, we have mg ) h y + y + h ) + β xy + 3β y y + h β y y y ) 4 y ) y ), 5) 3 y mg ) y ) + β yy y ), 6) vg, g ) 4h h 4 + 5h + ) y ) 8 + 4β xy + β y y ) + h y ) y ) 6, 7) vg, g ) 4h + h ) y ) 6 + β xy + β y y ) + h y ) y ) 4, 8) vg, g ) h y ) 4 + β xy + β y y y ). 9) Moreover, let b 0 0 x F y dx) and b 0 xf y dx). Then, we have b 0 + h y y ) 3, b y. 0) 9

11 The details of the above derivations are postponed to the Appendix. Define gx, x ) x /x and a 0 /b, b 0 /b 3 ). Applying the Delta method see Casella and Berger [3]), we obtain, as n, n ), [ /p) p p λ ] j/σ ) /p ) p λ j/σ ) b 0 D N a 0µ CLT, a 0Σ ) CLT a b 0, that is T n pb 0 /b ) D N a 0µ CLT, a 0Σ ) CLT a 0. ) Furthermore, substituting the equations 5) 0) into ) and after some manipulations, we obtain T n ph y ) D Nµ Tn, v Tn ), where µ Tn and v Tn are given in 3) and 4). This is equivalent to the statement of ). The proof is completed. Theorem 3. Under the assumptions of Theorem 3. and when Σ and Σ are bounded, β x and β y have consistent estimators ˆβ x and ˆβ y given by ˆβ x and ˆβ y pn ) pn ) N N x ix i N N p k y i y i N N p x ix i ) pn N ) N k N [ N )] x ix i N N x jx j ) x ik N N x jk y i y i ) pn N ) N N where x i x i,..., x ip ) and y i y i,..., y ip ). [ N )] y i y i N N y jy j ) y ik N N y jk 0

12 PROOF. We have x X,..., X p ) Σ ξ, where Σ a ij ), Σ σ kk ) and ξ ξ,..., ξ p ). Hence, EX 4 k) Ea k ξ + + a kp ξ p ) 4 EX kx l ) EX + + X p) where a 4 kj Eξ 4 ) + akj), a kja lj Eξ 4 ) + EX kx l ) a kj EXi 4 ) + k l ) a kj Eξ 4 ) + k ) k a lj ), akj), EX + + X p) Ex x), σ kk k a kj, a kj k σ kk. k That is, p Ex x) p ) σkk Eξ 4 ) + σ kk. p When Σ is bounded, then we have E ˆσ kk σ kk E ˆσ kk + σ kk ˆσ kk σ kk Eˆσ kk + σ kk ) Eˆσ kk σ kk ) o) uniformly for k. Then we have E ˆσ kk p p k k σ kk p E ˆσ kk σkk o). k

13 That is p ˆσ kk p k Let the estimator of p Ex x) p N p N k σ kk P 0. p σ kk) be x ix i ) x ix i x jx j i<j p N ) pn ) x ix i ) i pn N ) ) x ix i, i where pn ) x ix i ) i ) x pn N ) ix i i pn ) x ix i trσ + trσ) i N pn ) trσ) + p trσ N + pn ) x ix i trσ), i x ix i trσ) ) x pn N ) ix i ntrσ + ntrσ i N pn ) trσ) + p trσ x N ix i trσ) i i + [ N x ix i trσ)]. pn N )

14 Then, we have P 0. pn ) x ix i ) i x pn ix i trσ) i ) ) Ex x x) pn N ) ix i trσ) p p i i j x ix i trσ) x jx j trσ)] pn N ) Ex x trσ) p x ix x pn ix i trσ) Ex x trσ) i trσ) x jx j trσ)] i j p pn N ) i Then we obtain the consistent estimator of Eξ 4 ) as follows N pn ) p N x ix i ) x ix i ) pn N ) p k N +, N x ik ) where x i x i,..., x ip ). When Ex) is unknown, we also obtain the consistent estimators of Eξ) as follows: pn ) N x ix i N N p k x ix i ) pn N ) N N [ N )] x ix i N N x jx j ) +. x ik N N x jk 3

15 Then, the consistent estimator of β x Eξ 4 ) 3 is as follows ˆβ x pn ) N x ix i N N p k x ix i ) pn N ) N N [ N )] x ix i N N x jx j ). x ik N N x jk The derivation of ˆβ y is similar. The proof is completed. 4. Simulation studies 4.. Large-dimensional Gaussian data In this section, we perform some simulation studies to evaluate the efficiency of the new test LZ test) proposed in Section 3 for testing hypothesis ) with large-dimensional Gaussian data. For comparison, we also conduct the Bartlett test Bartlett adjusted likelihood ratio test, see Eriksen [5] or Flury [8]) and the Wald test see Schott [6]). We generate x,..., x n + from N p 0, σ I p ) and y,..., y n + from N p 0, Σ ) with Σ ρ i j ) for σ 0.5,.0,.0 and ρ 0.0, 0.4, 0.8, respectively. We intend to test the null hypothesis: H 0 : Σ σ Σ. Notice that ρ 0 corresponds to the realized size Type-I error) of the test. The nominal test level is set as α We simulate 0,000 independent replicates for different values of p, n, n ) under the assumption of p < n and compare the above three test. The simulation results are summarized in Table. It needs to point out that the proposed new test allows for p n, which means that the observations from one of two groups can be fewer than the dimension. Table 4 displays the summary of Gaussian data as well as non-gaussian data for the case of p > n. From Table, we can see that, as the dimension p increases, both Wald test and Bartlett test lead to an increasing test size while the proposed LZ test is so stable. Especially, when the dimension p > 600, both Wald test and Bartlett test are so worse which almost always reject the null hypothesis; in contrast, our proposed test remains even accurate. Figure shows the QQ-plots for the,000 observed values of the LZ test T n with ρ 0.0. The values of p, n, n ) are chosen as 30, 640, 640) and 500, 50, 000). In both cases, the normality result appears to be satisfied by the 4

16 Normal Q Q Plot Normal Q Q Plot Sample Quantiles Sample Quantiles Theoretical Quantiles Theoretical Quantiles Figure : Normal QQ-Plot for T n as in 0) under H 0 based on 000 replicates. The left is for the case of p < n ; the right is for the case of p n. QQ-plots for large p, n, n ) which validates our theoretical asymptotic normality results. We also check the QQ-plots of T n under the alternative hypothesis i.e. ρ > 0). The similar normality property appears to be satisfied, but for the restriction of space, we ignore the QQ-plots at here. 4.. Large-dimensional non-gaussian data In fact, Theorem 3. is valid for general population distributions with finite fourth moments. To see this, two distributions are considered. One is the gamma distribution. We generate x i σz ) i, i,..., n + and y j Γz ) j, j,..., n + with z ) i Z ) i,..., Z) ip ) and z ) j Z ) j,..., Z) jp ) consisting of identical and independent standardized gamma4, 0.5) random variables so that they have mean 0 and variance, and σ σ ) / and Γ p p ρ i j ) / for σ 0.5,.0,.0 and ρ 0.0, 0.4, 0.8, respectively. The nominal test level is set as α We simulate 0,000 independent replications for different values of p, n, n ) and compute the estimated significance level of the LZ test. The simulation Results are summarized in Table. Another multivariate distribution considered is a mixture of multivariate normal distributions. We generate z ) i, i,..., n + and z ) j, j,..., n + with each component is i.i.d. from 0.9N0, ) + 0.N0, 3 ). Let x i σz ) i and 5

17 y j Γz ) j with the same values of σ and Γ as in the case of gamma distribution. The simulation results are summarized in Table 3. From Tables and 3, we can see that, our proposed test is still valid for non- Gaussian data. We also studied the QQ-plots of T n based on,000 replicates under the null and alternative hypotheses for the gamma data and mixture of multivariate normal data. The normality property appears to be satisfied by the QQ-plots for large p, n, n ) More simulations on the LZ test In this section, we continue the simulations of Section 4. and 4. but with another different set of values for p, n, n ). We set p 0, 5, 50, 00, n p, and n p + 5, p + 0,.... The results are displayed in Table 5 and Table 6. Obviously, when p is too small or when p/n is close to, the LZ test is not so efficient. Therefore, we recommend the practitioners to be careful to use the LZ test when p is small or when p/n is close to. 5. Conclusions and future work In this article, we developed a new test for comparing the proportionality of large-dimensional covariance matrices from two different population with finite fourth-moments. Its asymptotic normality is established based on modern random matrix theory as the dimension increases proportionally with the sample sizes. Simulation studies show that our proposed method is valid for both Gaussian data and non-gaussian data. Our proposed method requires that p < n but allows for p n. Generalizing our proposed method to the case of p maxn, n ) will be the future work. 6. Appendix Derivations of 4) Notice that 0 < y, y h given r >, we have ξ <. Applying Cauchy integration formula, for any ) ξ r + ξ + r 0. ) ξ + y /h 6

18 We know that Reξ ξ + ξ)/ and ξ ξ for any ξ with ξ. Then for g x) x and δξ) y ) + h + hreξ), in the formula ) mg ) lim g δξ)) r 4πi ξ ξ r + ξ + r ξ + y /h + β x y y ) g πi h δξ)) ξ + y /h) 3 + β y y y ) πi h ξ ξ ξ + /h g δξ)) ξ + y /h) 3 we have lim g δξ)) r ξ { lim y ) 4 + ξ r ξ ξ r + ξ + r [h + + h + hξ) ] ξ [ h ξ + h + h ) ξ ξ + y /h ) ξ r + ξ + r ] ξ r + ξ + r ξ + y /h πi y ) [h y 4 + y + h )) ] 4πi y ) 4 [h y + y + h )], ) ξ + y /h } { g δξ)) ξ + y /h) h + + h + hξ) 3 y ) 4 [ ξ ξ + y /h) 3 h + ξ ξ + h + ] } h ) ξ ξ + y /h) [ ] 3 h πi y ) h πi y ), 4 ) 7

19 and ξ { ξ + /h g δξ)) ξ + y /h) [h + + h + hξ) ξ + /h) ] 3 y ) 4 ξ ξ + y /h) [ 3 h + ξ ξ + h + ] } h ) ξ + /h) ξ ξ + y /h) [ ] 3 h3 + h 3y ) πi + 0 y ) 4 Thus, we obtain πi h3 + h 3y ) y ) 4. mg ) h y + y + h ) y ) 4 + β xy + 3β y y y ) + h β y y y ) 3. Derivations of 5) For g x) x and δξ) y ) + h + hreξ), in the formula ) mg ) lim g δξ)) r 4πi ξ ξ r + ξ + r ξ + y /h + β x y y ) g πi h δξ)) ξ ξ + y /h) 3 + β y y y ) ξ + /h g δξ)) πi h ξ ξ + y /h) 3 lim + h + hξ + hξ ) r 4πi y ) ξ + β x y + h + hξ + hξ ) πi h ξ ξ + y /h) 3 β y y + + h + hξ + hξ ξ + /h ) πi h y ) ξ ξ + y /h) 3 4πi y ) πi y + 0) + β xy 0 + h y y ) + y β y y ). ξ r + ξ + r ξ + y /h β y y πi h + 0) πi h y ) ) 8

20 Derivations of 6) For g x) x and δξ) y ) + h + hreξ), in the formula, g δξ ))g δξ )) vg, g ) lim r π ξ ξ ξ rξ ) β [ ] xy + β y y ) y ) g δξ )) 4π h ξ + y /h). we have lim r ξ and ξ lim r y ) 8 lim r y ) 8 lim r y ) 8 ξ πi h y ) 8 g δξ ))g δξ )) ξ rξ ) ξ ξ ξ ξ 4π h h 4 + 5h + ) y ) 8, g δξ )) ξ + y /h) y ) 4 ξ [ y ) 4 ξ πi y ) [h + 4 h y ) + 0] πi y ) h + 4 h y ), ξ + h + hξ + hξ ) [ + h + hξ + hξ ) [ ξ ξ + h + hξ + hξ ] ) ξ rξ ) + h + hξ + hξ ) r ξ ξ /r) + h + hξ + hξ ) πi h + h + hξ /r)r [ + h + hξ ) 3 + h + h + hξ ) + h + ] h + hξ ) ξ ξ + h + hξ + hξ ) ξ + y /h) h + + h + hξ ) ξ + y /h) + ξ ] h + h )ξ + h ξ ] ξ + y /h) 9

21 So, we derived that vg, g ) 4h h 4 + 5h + ) y ) 8 + 4β xy + β y y ) + h y ) y ) 6. Derivations of 7) For g x) x and g x) x, we have vg, g ) [ lim g r π δξ )) ξ β xy + β y y ) y ) 4π h Now, we can derive that [ lim g δξ )) r ξ lim r y ) 6 πi lim r y ) 6 and ξ lim r πi h y ) 6 ξ ξ ξ ξ 4π h + h ) y ) 6, g δξ )) ξ + y /h) So, we derived that ξ ξ ] g δξ )) r ξ ξ /r) + h + hξ + hξ ) [ ] g δξ )) r ξ ξ /r) g δξ )) ξ + y /h) g δξ )) ξ + y /h). ξ + h + hξ + hξ ) h + 0) [ h ξ + h + h ) ξ y ) ξ ] ξ + h + hξ + hξ ] r ξ ξ /r) + h + hξ + hξ ξ + y /h) πi h y ). vg, g ) 4h + h ) y ) 6 + β xy + β y y ) + h y ) y ) 4. 0

22 Derivations of 8) For g x) x, we have [ vg, g ) lim g r π δξ )) ξ β [ xy + β y y ) y ) 4π h Since that [ lim g δξ )) r ξ lim r y ) 4 lim r lim r πi y ) 4 πi h y ) 6 4π h y ) 4, ξ ξ ξ ξ ξ ξ ] g δξ )) r ξ ξ /r) [ + h + hξ + hξ ) ] g δξ )) r ξ ξ /r) ] g δξ )) ξ + y /h). ξ + h + hξ + hξ ) h + 0) [ h ξ + h + h ) ξ ] + h + hξ + hξ ] r ξ ξ /r) Then, we have vg, g ) h y ) 4 + β xy + β y y y ). Derivations of 9) Following the definition of F y dx) and using the substitution x y ) + h hcosθ), 0 < θ < π, we have b x)x a) hsinθ hsinθ, dx y ) y ) dθ, x + h hcosθ y ), y + y x h + y hy cosθ y ).

23 Therefore, b 0 b x y ) b x)x a) dx a πxy + y x) 4h π + h hcosθ)sin θ dθ π y ) 3 0 h + y hy cosθ 4h π + h hcosθ)sin θ dθ 4π y ) 3 0 h + y hy cosθ h π y ) 3 ih 4πy y ) 3 ih 4πy y ) 3 z + h hz + z ) h + y hy z + z ) z ) 4z z z z +h z + h )z ) dz z 3 z h +y hy z + ) z /h)z h)z ) z + ) dz z 3 z y /h)z h/y ) dz iz letting z eiθ ) Obviously, there are two poles inside the unit circle: 0 and y /h. Their corresponding residues are Hence, b 0 R0) + h + y)[h + y) y + h )] h y Ry /h) h y) y )y h ). h y ih 4πy y ) 3 πi [R0) + Ry /h)] + h y y ) 3.

24 On the other hand, b b x y ) b x)x a) dx a πxy + y x) 4h π sin θ π y ) 0 h + y hy cosθ dθ 4h π sin θ 4π y ) 0 h + y hy cosθ dθ h π y ) z h + y hy z + z ) z ) 4z h z ) 4πi y y ) z z z h +y hy z + ) dz h z ) 4πi y y ) z z h/y )z y /h) dz z dz iz letting z eiθ ) There are two poles inside the unit circle: 0 and y /h. Their corresponding residues are R0) y + h, Ry /h) y h. hy hy Hence, b Acknowledgments h 4πi y y ) πi [R0) + Ry /h)]. y The authors would like to thank the Editor, an Associate Editor and two referees for their helpful comments and useful suggestions. GL Tian s research was fully supported by a grant HKU 7790M) from the Research Grant Council of the Hong Kong Special Administrative Region. References [] Z.D. Bai, D.D. Jiang, J.F. Yao, S.R. Zheng, Corrections to LRT on largedimensional covariance matrix by RMT., Ann. Statist ) [] Z.D. Bai, J.W. Silverstein, Spectral analysis of large-dimensional random matrices, st ed., Science Press, Beijing, China.,

25 [3] G. Casella, R.L. Berger, Statistical Inference, nd ed., Duxbury Press., 00. [4] G.R. Dargahi-Noubary, An application of discrimination when covariance matrices are proportional, Austral. J. Statist. 3 98) [5] P.S. Eriksen, Proportionality of covariance matrices., Ann. Statist ) [6] W.T. Federer, Testing proportionality of covariance matrices, Ann. Math. Statist. 95) [7] T.J. Fisher, X.Q. Sun, C.M. Gallagher, A new test for sphericity of the covariance matrix for high dimensional data., J. Multivariate Anal. 0 00) [8] B.K. Flury, Proportionality of k covariance matrices., Statist. Prob. Lett ) [9] I. Guttman, D.Y. Kim, I. Olkin, Statistical inference for constants of proportionality, Technical Report 95, Department of Statistics, Stanford University, 983. [0] S.T. Jensen, J. Madsen, Estimation of proportional covariances in the presence of certain linear restrictions, Ann. Statist ) 9 3. [] D.Y. Kim, Statistical inference for constants of proportionality between covariance matrices, Technical Report 59, Department of Statistics, Stanford University, 97. [] R.J. Muirhead, Aspects of Multivariate Statistical Theory. st Edition., John Wiley & Son, New York, 98. [3] H. Nagao, On some test criteria for covariance matrix., Ann. Statist. 973) [4] D.G. Nel, P.C.N. Groenewald, A bayesian approach to the multivariate behrens-fisher problem under the assumption of proportional covariance matrices, Test 993) 4. [5] C.R. Rao, Likelihood ratio tests for relationships between two covariance matrices, in: S. Karlin, T. Amemiya, L.A. Goodman Eds.), Studies in Econometrics, Time Series and Multivariate Statistics, Academic Press, New York, 983, pp

26 [6] J.R. Schott, A test for proportional covariance matrices., Comput. Statist. Data Anal ) [7] M.S. Srivastava, Some tests concerning the covariance matrix in high dimensional data., J. Japan Statist. Soc ) 5 7. [8] M.S. Srivastava, C.G. Khatri, E.M. Carter, On monotonicity of the modified likelihood ratio test for the equality of two covariance., J. Multivariate Anal ) [9] N. Sugiura, H. Nagao, Unbiasedness of some test criteria for the equality of one or two covariance matrices., Ann. Math. Statist ) [0] C. Villa, C. Pérignon, Sources of time variation in the covariance matrix of interest rates, Journal of Business ) [] S.S. Wilks, Sample criteria for testing equality of means, equality of variances, and equality of covariances in a normal multivariate distribution, Ann. Math. Statist ) [] S.R. Zheng, Central limit theorems for linear spectral statistics of large dimensional F-matrices., Annales de l Institut Henri Poincaré Probabilités et Statistiques 48 0)

27 Table : Sizes and powers of the Wald test, Bartlett test, and proposed LZ test based on 0,000 independent replications using real Gaussian variables with y 0.5 and y 0.5. Test sizes are calculated under the null hypothesis H 0 : Σ σ Σ with ρ 0; the powers are estimated under the alternative hypothesis H : Σ σ Σ with ρ > 0, where Σ σ I p and Σ ρ i j ) p p. ρ p, n, n ) Wald Test Bartlett Test LZ Test σ , 80, 80) , 60, 60) , 30, 30) , 640, 640) , 80, 80) , 80, 80) , 60, 60) , 30, 30) , 640, 640) , 80, 80) , 80, 80) , 60, 60) , 30, 30) , 640, 640) , 80, 80)

28 Table : Sizes and powers of the proposed LZ test based on 0,000 independent replications using real Gamma variables with y 0.5 and y 0.5. Test sizes are calculated under the null hypothesis H 0 : Σ σ Σ with ρ 0; the powers are estimated under the alternative hypothesis H : Σ σ Σ with ρ > 0, where Σ σ I p and Σ ρ i j ) p p. p, n, n ) ρ 0.0 ρ 0.4 ρ 0.8 σ , 80, 80) , 60, 60) , 30, 30) , 640, 640) , 80, 80) Table 3: Sizes and powers of the proposed LZ test based on 0,000 independent replications using real mixture of multivariate normal distributions variables with y 0.5 and y 0.5. Test sizes are calculated under the null hypothesis H 0 : Σ σ Σ with ρ 0; the powers are estimated under the alternative hypothesis H : Σ σ Σ with ρ > 0, where Σ σ I p and Σ ρ i j ) p p. p, n, n ) ρ 0.0 ρ 0.4 ρ 0.8 σ , 80, 80) , 60, 60) , 30, 30) , 640, 640) , 80, 80)

29 Table 4: Sizes and powers of the proposed LZ test based on 0,000 independent replications with y 0 and y 0.5. Test sizes are calculated under the null hypothesis H 0 : Σ σ Σ with ρ 0; the powers are estimated under the alternative hypothesis H : Σ σ Σ with ρ > 0, where Σ σ I p and Σ ρ i j ) p p. Multivariate Gaussian data p, n, n ) ρ 0.0 ρ 0.4 ρ 0.8 σ , 5, 00) , 0, 00) , 0, 400) , 50, 000) , 80, 600) Multivariate gamma data p, n, n ) ρ 0.0 ρ 0.4 ρ 0.8 σ , 5, 00) , 0, 00) , 0, 400) , 50, 000) , 80, 600) Mixture of multivariate Gaussian data p, n, n ) ρ 0.0 ρ 0.4 ρ 0.8 σ , 5, 00) , 0, 00) , 0, 400) , 50, 000) , 80, 600)

30 Table 5: Sizes and powers of the proposed LZ test based on 0,000 independent replications using real Gaussian variables with n p and n p + 5, p + 0,.... Test sizes are calculated under the null hypothesis H0: Σ σ Σ with ρ 0; the powers are estimated under the alternative hypothesis H: Σ σ Σ with ρ > 0, where Σ σ Ip and Σ ρ i j )p p. ρ p n n p + k k

31 Table 6: Sizes and powers of the proposed LZ test based on 0,000 independent replications using Gamma variables data with n p and n p + 5, p + 0,.... Test sizes are calculated under the null hypothesis H0: Σ σ Σ with ρ 0; the powers are estimated under the alternative hypothesis H: Σ σ Σ with ρ > 0, where Σ σ Ip and Σ ρ i j )p p. ρ p n n p + k k