On the sphericity test with large-dimensional observations. Citation Electronic Journal of Statistics, 2013, v. 7, p

Transcription

1 Title On the shericity test with large-dimensional observations Authors Wang, Q; Yao, J Citation Electronic Journal of Statistics, 203, v. 7, Issued Date 203 URL htt://hdl.handle.net/0722/89449 Rights This work is licensed under a Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 International License.

2 Electronic Journal of Statistics Vol ISSN: DOI: 0.24/3-EJS842 On the shericity test with large-dimensional observations Qinwen Wang and Jianfeng Yao Qinwen Wang Deartment of Mathematics Zhejiang University wqw883@gmail.com Jianfeng Yao Deartment of Statistics and Actuarial Science The University of Hong Kong Pokfulam, Hong Kong jeffyao@hku.hk Abstract: In this aer, we roose corrections to the likelihood ratio test and John s test for shericity in large-dimensions. New formulas for the limiting arameters in the CLT for linear sectral statistics of samle covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymtotic distribution of the two roosed test statistics under the null. These asymtotics are valid for general oulation, i.e. not necessarily Gaussian, rovided a finite fourth-moment. Extensive Monte-Carlo exeriments are conducted to assess the quality of these tests with a comarison to several existing methods from the literature. Moreover, we also obtain their asymtotic ower functions under the alternative of a siked oulation model as a secific alternative. MSC 200 subject classifications: Primary 62H5; secondary 62H0. Keywords and hrases: Large-dimensional data, large-dimensional samle covariance matrix, shericity, likelihood ratio test, John s test, Nagao s test, CLT for linear sectral statistics, siked oulation model. Received January 203. Contents Introduction Large-dimensional corrections The corrected likelihood ratio test CLRT The corrected John s test CJ Monte Carlo study Asymtotic owers: under the siked oulation alternative Generalization to the case when the oulation mean is unknown Additional roofs Research of this author was artly suorted by the National Natural Science Foundation of China Grant No. 0723, the Natural Science Foundation of Zhejiang Province No. R , and the Doctoral Program Fund of Ministry of Education No. J Research of this author was artly suorted by a HKU start-u grant. 264

3 Large-dimensional shericity test Proof of Lemma Proof of Lemma Concluding remarks A Formula for limiting arametersin the CLT for eigenvaluesofasamle covariance matrix with general fourth moments Acknowledgements References Introduction Consider a samle Y,...,Y n from a -dimensional multivariate distribution with covariance matrix Σ. An imortant roblem in multivariate analysis is to test the shericity, namely the hyothesis H 0 : Σ = σ 2 I where σ 2 is unsecified. If the observations reresent a multivariate error with comonents, the null hyothesis exresses the fact that the error is cross-sectionally uncorrelated indeendent if in addition they are normal and have a same variance homoscedasticity. Much of the existing theory about this test has been exosed first in details in [7] about Gaussian likelihood ratio test and later in [0,, 27] and also in textbookslike[8,chater8]and[,chater0].assumeforamomentthatthe samle has a normal distribution with mean zero and covariance matrix Σ. Let S n = n i Y iyi be the samle covariance matrix and denote its eigenvalues by {l i } i. Two well established rocedures for testing the shericity are the likelihood ratio test LRT and a test devised in [0]. The likelihood ratio statistic is, see e.g. [, 0.7.2], L n = l l / l + +l 2 n, which is a ower of the ratio of the geometric mean of the samle eigenvalues to the arithmetic mean. It is here noticed that in this formula it is necessary to assume that n to avoid null eigenvalues in the numerator of L n. If we let n while keeing fixed, classical asymtotic theory indicates that under the null hyothesis, 2logL n = χ 2 f, a chi-square distribution with degree of freedom f = 2 +. This asymtotic distribution is further refined by the following Box-Bartlett correction referred as BBLRT: P 2ρlogL n x = P f x+ω 2 {P f+4 x P f x}+on 3,. where P k x = Pχ 2 k x and ρ = , ω 2 = n n 2 ρ 2. By observing that the asymtotic variance of 2logL n is roortional to tr{σtrσ I } 2, [0] roosed to use the statistic T 2 = 2 n 2 tr{ S n trs n I } 2

4 266 Q. Wang and J. Yao Table Emirical sizes of BBLRT and Nagao s test at 5% significance level based on 0000 indeendent relications using normal vectors N0,I for n = 64 and different values of, n 4,64 8,64 6,64 32,64 48,64 56,64 60,64 BBLRT Nagao s test for testing shericity. When is fixed and n, under the null hyothesis, it also holds that T 2 = χ 2 f, which we referred to as John s test. It is observed that T 2 is roortional to the square of the coefficient of variation of the samle eigenvalues, namely T 2 = n 2 li l 2 l 2, with l = n l i. Following the idea of the Box-Bartlett correction, [9] established an exansion for the distribution function of the statistics T 2 referred as Nagao s test, PT 2 x = P f x+ n {a P f+6 x+b P f+4 x+c P f+2 x+d P f x} where +On 2,.2 a = , b = , c = , d = It has been well known that classical multivariate rocedures are in general challenged by large-dimensional data. A small simulation exeriment is conducted to exlore the erformance of the BBLRT and Nagao s test two corrections with growing dimension. The samle size is set to n = 64 while dimension increases from 4 to 60 we have also run other exeriments with larger samle sizes n but conclusions are very similar, and the nominal level is set to be α = The samles come from normal vectors with mean zero and identity covariance matrix, and each air of,n is assessed with 0000 indeendent relications. Table gives the emirical sizes of BBLRT and Nagao s test. It is found here that when the dimension to samle size ratio /n is below /2, both tests have an emirical size close to the nominal test level Then when the ratio grows u, the BBLRT becomes quickly biased while Nagao s test still has a correct emirical size. It is striking that although Nagao s test is derived under classical fixed, n regime, it is remarkably robust against dimension inflation. Therefore, the goal of this aer is to roose novel corrections to both LRT and John s test to coe with the large-dimensional context. Similar works have already been done in [5], which confirms the robustness of John s test in large-dimensions; however, these results assume a Gaussian oulation. In this i

5 Large-dimensional shericity test 267 aer, we remove such a Gaussian restriction, and rove that the robustness of John s test is in fact general. Following the idea of [5], [9] roosed to use a family of well selected U-statistics to test the shericity; however, as showed in our simulation study in Section 3, the owers of our corrected John s test are slightly higher than this test in most cases. More recently, [26] examined the erformance of T a statistic first ut forward in [24] under non-normality, but with the moment condition γ = 3 + O ǫ, which essentially matches the Gaussian case γ = 3 asymtotically. We have also removed this moment restriction in our setting. In short, we have unveiled two corrections that have a better erformance and removed the Gaussian or nearly Gaussian restriction found in the existing literature. From the technical oint of view, our aroach differs from [5] and follows the one devised in [4] and [6]. The central tool is a CLT for linear sectral statistics of samle covariance matrices established in[2] and later refined in[2]. The aer also contains an original contribution on this CLT reorted in the Aendix: new formulas for the limiting arameters in the CLT. Since such CLT s are increasingly imortant in large-dimensional statistics, we believe that these new formulas will be of indeendent interest for alications other than those considered in this aer. The remaining of the aer is organized as follows. Large-dimensional corrections to LRT and John s test are introduced in Section 2. Section 3 reorts a detailed Monte-Carlo study to analyze finite-samle sizes and owers of these two corrections under both normal and non-normal distributed data. Next, Section 4 gives the theoretical analysis of their asymtotic ower under the alternative of a siked oulation model. Section 5 generalizes our test rocedures to oulations with an unknown mean. Technical roofs and calculations are relegated to Section 6. The last Section contains some concluding remarks. 2. Large-dimensional corrections From now on, we assume that the observations Y,...,Y n have the reresentation Y j = Σ /2 X j where the n table {X,...,X n } = {x ij } i, j n are made with an array of i.i.d. standardized random variables mean 0 and variance. This setting is motivated by the random matrix theory and it is generic enough for a recise analysis of the shericity test. Furthermore, under the null hyothesis H 0 : Σ = σ 2 I σ 2 is unsecified, we notice that both LRT and John s test are indeendent from the scale arameter σ 2 under the null. Therefore, we can assume w.l.o.g. σ 2 = when dealing with the null distributions of these test statistics. This will be assumed in all the sections. Throughout the aer we will use an indicator κ set to 2 when {x ij } are real and to when they are comlex as defined in [3]. Also, we define the kurtosis coefficient β = E x ij 4 κ for both cases and note that for normal variables, β = 0 recall that for a standard comlex-valued normal random variable, its real and imaginary arts are two iid. N0, 2 real random variables.

6 268 Q. Wang and J. Yao 2.. The corrected likelihood ratio test CLRT For the correction of LRT, let L n = 2n logl n be the test statistic for n. Our first main result is the following. Theorem 2.. Assume {x ij } are iid, satisfying Ex ij = 0,E x ij 2 =,E x ij 4 <. Then under H 0 and when n = y n y 0,, L n + n log { n = N κ log y+ } 2 2 βy, κlog y κy. 2. The test based on this asymtotic normal distribution will be hereafter referred as the corrected likelihood-ratio test CLRT. One may observe that the limiting distribution of the test crucially deends on the limiting dimension-tosamle ratio y through the factor log y. In articular, the asymtotic variance will blow u quickly when y aroaches, so it is exected that the ower will seriously break down. Monte-Carlo exeriments in Section 3 will rovide more details on this behavior. The roof of Theorem 2. is based on the following lemma. In all the following, F y denotes the Marčenko-Pastur distribution of index y > 0 which is introduced and discussed in the Aendix. And F y fx = fxf y dx denotes the integral of function fx with resect to F y. Lemma 2.. Let {l i } i be the eigenvalues of the samle covariance matrix S n = n i Y iyi. Then under H 0 and the conditions of Theorem 2., we have i= logl i F yn logx i= l = Nµ i F yn x,v, with and V = µ = κ 2 log y 2 βy 0, κlog y+βy β +κy. β +κy β +κy The roof of this lemma is ostoned to Section 6. Proof of Theorem 2.. Let A n i= logl i F yn logx and B n i= l i F yn x. By Lemma 2., An = Nµ B,V. n Consequently, A n +B n is asymtotically normal with mean κ 2 log y+ βy and variance 2 V,+V 2,2 2V,2 = κlog y κy.

7 Besides, Large-dimensional shericity test 269 L n = Σ i= logl i +log Σ i= l i = A n +F logx+log yn B n + = A n F yn logx+log + B n Since B n N0,yβ+κ, B n = O and log+b n / = B n /+O / 2. Therefore, L n = A n F yn logx+b n +O. The conclusion follows with the following well-known integrals w.r.t. the Marčenko-Pastur distribution F y y < in [2], F y logx = y y The roof of Theorem 2. is comlete.. log y, F y x = The corrected John s test CJ Earlier than the asymtotic exansion.2 given in [9], [0] roved that when the observations are normal, the shericity test based on T 2 is a locally most owerful invariant test. It is also established in [] that under these conditions, the limiting distribution of T 2 under H 0 is χ 2 f with degree of freedom f = 2 +, or equivalently, nu = 2 χ2 f, where for convenience, we have let U = 2n T 2. Clearly, this limit has been establishedfor n and a fixed dimension. However,if wenow let in theright-handsideofthe aboveresult, it isnothardto seethat 2 χ2 f willtend to the normal distribution N, 4. It then seems natural to conjecture that when both and n grow to infinity in some roer way, it may haen that nu = N, This is indeed the main result of [5] where this asymtotic distribution was established assuming that data are normal-distributed and and n grow to infinity in a roortional way i.e. /n y > 0. In this section, we rovide a more general result using our own aroach. In articular, the distribution of the observation is arbitrary rovided a finite fourth moment exists. Theorem2.2. Assume {x ij } are iid, satisfying Ex ij = 0,E x ij 2 =,E x ij 4 <, and let U = 2n T 2 be the test statistic. Then under H 0 and when

8 270 Q. Wang and J. Yao,n, n = y n y 0,, nu = Nκ+β,2κ. 2.3 The test based on the asymtotic normal distribution given in equation 2.3 will be hereafter referred as the corrected John s test CJ. A striking fact in this theorem is that as in the normal case, the limiting distribution of CJ is indeendent of the dimension-to-samle ratio y = lim /n. In articular, the limiting distribution derived under classical scheme fixed, n, e.g. the distribution 2 χ2 f in the normal case, when used for large, stays very close to this limiting distribution derived for large-dimensional scheme,n,/n y 0,. In this sense, Theorem 2.2 gives a theoretic exlanation to the widely observed robustness of John s test against the dimension inflation. Moreover, CJ is also valid for the larger or much larger than n case in contrast to the CLRT where this ratio should be ket smaller than to avoid null eigenvalues. It is also worth noticing that for real normal data, we have κ = 2 and β = 0 so that the theorem above reduces to nu N,4. This is exactly the result discussed in [5]. Besides, if the data has a non-normal distribution but has the same first four moments as the normal distribution, we have again nu N,4, which turns out to have a universality roerty. The roof of Theorem 2.2 is based on the following lemma. Lemma 2.2. Let {l i } i be the eigenvalues of the samle covariance matrix S n = n i Y iyi. Then under H 0 and the conditions of Theorem 2.2, we have i= l2 i +y n i= l = Nµ i 2,V 2, with and κ +βy µ 2 =, 0 2κy V 2 = 2 +4κ+βy +2y 2 +y 3 2κ+βy +y 2 2κ+βy +y 2. κ+βy The roof of this lemma is ostoned to Section 6. Proof of Theorem 2.2. The result of Lemma 2.2 can be rewritten as: n i= l2 i n κ+β y 0 i= l = N, i 0 y 2 V 2. Define the function fx,y = x y 2, then U = f Σ i= l2 i, Σ i= l i. We have f + x n κ+β y +, =,

9 f y f + n + n By the delta method, n U f where f Large-dimensional shericity test 27 κ+β y +, = 2 + n κ+β y +, + n x C = + n + κ+β y, f y + n + κ+β y, 2κ. = n κ+β y +. κ+β y +, κ+β y +, = N0,limC, T f 2 y 2V x + n + κ+β y, f y + n + κ+β y, Therefore, that is, n U n κ+β y = N0,2κ, nu = Nκ+β,2κ. The roof of Theorem 2.2 is comlete. Remark 2.. Note that in Theorems 2. and 2.2 aears the arameter β, which is in ractice unknown with real data. So we may estimate the arameter β using the fourth-order samle moment: β = n i= j= n Y j i 4 κ. According to the law of large numbers, ˆβ = β +o, so substituting ˆβ for β in Theorems 2. and 2.2 does not modify the limiting distribution. 3. Monte Carlo study Monte Carlo simulations are conducted to find emirical sizes and owers of CLRT and CJ. In articular, here we want to examine the following questions: how robust are the tests against non-normal distributed data and what is the range of the dimension to samle ratio /n where the tests are alicable. For comarison, we show both the erformance of the LW test using the asymtotic N,4 distribution in 2.2 Notice however this is the CJ test under normal distribution and the Chen s test denoted as C for short using theasymtoticn0,4distributionderivedin[9].thenominaltest levelissetto be α = 0.05, and for each air of,n, we run 0000 indeendent relications. We consider two scenarios with resect to the random vectors Y i :

10 272 Q. Wang and J. Yao Table 2 Emirical sizes of LW, CJ, CLRT and C test at 5% significance level based on 0000 indeendent alications with real N0, random variables and with real Gamma4,2-2 random variables,n N0, Gamma4,2-2 LW/CJ CLRT C LW CLRT CJ C 4, , , , , , , , , , , , , , , , , , , , , , , , , , , , a Y i is -dimensional real random vector from the multivariate normal oulation N0,I. In this case, κ = 2 and β = 0. b Y i consists of iid real random variables with distribution Gamma4,2 2 so that y ij satisfies Ey ij = 0,Eyij 4 = 4.5. In this case, κ = 2 and β =.5. Table 2 reorts the sizes of the four tests in these two scenarios for different values of,n. We see that when {y ij } are normal, LW =CJ, CLRT and C all have similar emirical sizes tending to the nominal level 0.05 as either or n increases. But when {y ij } are Gamma-distributed, the sizes of LW are higher than 0. no matter how large the values of and n are while the sizes of CLRT and CJ all converge to the nominal level 0.05 as either or n gets larger. This emirically confirms that normal assumtions are needed for the result of [5] while our corrected statistics CLRT and CJ also the C test have no such distributional restriction. As for emirical owers, we consider two alternatives here, the limiting sectral distributions of Σ under these two alternatives differs from that under H 0 :

11 Large-dimensional shericity test 273 Table 3 Emirical owers of LW, CJ, CLRT and C test at 5% significance level based on 0000 indeendent alications with real N0, random variables and with real Gamma4,2-2 random variables under two alternatives Power and 2 see the text for details N0,,n Power Power 2 LW/CJ CLRT C LW/CJ CLRT C 4, , , , , , , , , , , , , , Gamma4,2 2,n Power Power 2 CJ CLRT C CJ CLRT C 4, , , , , , , , , , , , , , Σ is diagonal with half of its diagonal elements 0.5 and half. We denote its ower by Power ; 2 Σ is diagonal with /4 of the elements equal 0.5 and 3/4 equal. We denote its ower by Power 2. Table 3 reorts the owers of LW=CJ, CLRT and C when {y ij } are distributed as N0,, and of CJ, CLRT and C when {y ij } are distributed as Gamma4,2-2, for the situation when n equals 64 or 28, with varying values of and under the above mentioned two alternatives. For n = 256 and varying from 6 to 240, all the tests have owers around under both alternatives so that these values are omitted. And in order to find the trend of these owers, we also resent the results when n = 28 in Figure and Figure 2.

12 274 Q. Wang and J. Yao N0, n=28 N0, n=28 Power LW/CJ CLRT C Power LW/CJ CLRT C Fig. Emirical owers of LW/CJ, CLRT and C test at 5% significance level based on 0000 indeendent alications with real N0, random variables for fixed n = 28 under two alternatives Power and 2 see the text for details. Gamma4,2 2 n=28 Gamma4,2 2 n=28 Power CJ CLRT C Power CJ CLRT C Fig 2. Emirical owers of CJ, CLRT and C test at 5% significance level based on 0000 indeendent alications with real Gamma4,2-2 random variables for fixed n = 28 under two alternatives Power and 2 see the text for details. The behavior of Power and Power 2 in each figure related to the three statistics are similar, excet that Power is much higher comared with Power 2 for a given dimension design,n and any given test for the reason that the first alternative differs more from the null than the second one. The owers of

13 Large-dimensional shericity test 275 Table 4 Emirical sizes and owers Power and 2 of CJ test and C test at 5% significance level based on 0000 indeendent alications with real Gamma4,2-2 random variables when n,n CJ C Size Power Power 2 Size Power Power 2 64, , , , , LW in the normal case, CJ in the Gamma case and C are all monotonically increasing in for a fixed value of n. But for CLRT, when n is fixed, the owers first increase in and then become decreasing when is getting close to n. This can be exlained by the fact that when is close to n, some of the eigenvalues of S n are getting close to zero, causing the CLRT nearly degenerate and losing ower. Besides, we find that in the normal case the trend of C s ower is very much alike of those of LW while in the Gamma case it is similar with those of CJ under both alternatives. And in most of the cases esecially in large case, the ower of C test is slightly lower than LW in the normal case and CJ in the Gamma case. Lastly, we examine the erformance of CJ and C when is larger than n. Emirical sizes and owers are resented in Table 4. We choose the variables to be distributed as Gamma4,2-2 since CJ reduces to LW in the normal case, and [5] has already reorted the erformance of LW when is larger than n. From the table, we see that when is larger than n, the size of CJ is still correct and it is always around the nominal level 0.05 as the dimension increases and the same henomenon exists for C test. When we evaluate the ower, the same two alternatives Power and Power2 as above are considered. The samle size is fixed to n = 64 and the ratio /n varies from to 20. We see that Power are much higher than Power 2 for the same reason that the first alternative is easier to be distinguished from H 0. Besides, the owers under both alternatives all increase monotonically for n 5. However, when /n is getting larger, say /n = 20, we can observe that its size is a little larger and owers a little dro comared with /n = 5 but overall, it still behaves well, which can be considered as free from the assumtion constraint /n y. Besides, the owers of CJ are always slightly higher than those of C in this large small n setting. Since the asymtotic distribution for the CLRT and CJ are both derived under the Marcenko-Pasture scheme i.e /n y 0,, if /n is getting too large n, it seems that the limiting results rovided in this aer will loose accuracy. It is worth noticing that [7] has extended the LW test to such a scheme n for multivariate normal distribution. Summarizing all these findings from this Monte-Carlo study, the overall figure is the following: when the ratio /n is much lower than say smaller than /2,

14 276 Q. Wang and J. Yao it is referable to emloy CLRT than CJ, LW or C; while this ratio is higher, CJ or LW for normal data becomes more owerful slightly more owerful than C. 4. Asymtotic owers: under the siked oulation alternative In this section, we give an analysis of the owers of the two corrections: CLRT and CJ. To this end, we consider an alternative model that has attracted lots of attention since its introduction by [3], namely, the siked oulation model. This model can be described as follows: the eigenvalues of Σ are all one excet for a few fixed number of them. Thus, we restrict our shericity testing roblem to the following: H 0 : Σ = I vs H : Σ = diaga,...,a }{{,...,a } k,...,a k,,...,, }{{}}{{} n n k M where the multilicity numbers n i s are fixed and satisfying n + +n k = M. We derive the exlicit exressions of the ower functions of CLRT and CJ in this section. Under H, the emirical sectral distribution of Σ is H n = k i= n i δ ai + M δ, 4. and it will convergeto δ, adirac massat, which is the same asthe limit under the null hyothesis H 0 : Σ = I. From this oint of view, anything related to the limiting sectral distribution remains the same whenever under H 0 or H. Then recall the CLT for LSS of the samle covariance matrix, as rovided in [2], is of the form: fxdf Sn x F yn x = Nµ,σ 2, where the right side of this equation is determined only by the limiting sectral distribution. So we can conclude that the limiting arameters µ and σ 2 remain the same under H 0 and H, only the centering term fxdf yn x ossibly makes a difference. Since there s a in front of fxdf yn x, which tends to infinity as assumed, so knowing the convergence H n δ is not enough and more details about the convergence are needed. In [28], we have established an asymtotic exansion for the centering arameter when the oulation has a siked structure. We will use these formulas like equations 4.2, 4.3 and 4.6 in the following to derive the owers of the CLRT and CJ. Lemma 2. remains the same under H, excet that this time the centering terms become see formulas 4. and 4.2 in [28]: F yn logx = k i= n i loga i + log y n +O y n 2, 4.2

15 F yn x = + Large-dimensional shericity test 277 k n i a i M. +O i= Reeating the roof of Theorem 2., we can get: L n + nlog k + n i logai a i + n i= = N κ log y+ 2 2 βy, κlog y κy 4.4 under H. As a result, the ower of CLRT for testing H 0 against H can be exressed as: k β α = Φ Φ i= α n i ai loga i 4.5 κlog y κy for a re-given significance level α. It is worth noticing here that if the alternative has only one simle sike, i.e. k =, n k =, and assuming the real Gaussian variable case, i.e. κ = 2, 4.5 reduces to a result rovided in [20]. However, our formula is valid for a general number of sikes with eventual multilicities. Besides, these authors use some more sohisticated tools of asymtotic contiguity and Le Cam s first and third lemmas, which are totally different from ours. In order to calculate the ower function of CJ, we restated Lemma 2.2 as follows: i= l2 i x 2 Fyn i= l = Nµ i F yn x 2,V 2, where F yn x 2 = 2 n k n i a i 2 n M ++y n M + i= F yn x = + k n i a i M. +O 2 i= k n i a 2 i, +O Using the delta method as in the roof of Theorem 2.2, this time, we have nu n i= k n i a i 2 = Nκ+β,2κ 4.7 i= under H. Power function of CJ can be exressed as β 2 α = Φ Φ α n for a re-given significance level α. k i= n ia i 2 2κ 4.8

16 278 Q. Wang and J. Yao Powers of CLRT and CJ Power β α β 2 α y Fig 3. Theoretical owers of CLRT β α and CJ β 2α under the siked oulation alternative. Now consider the functions a i loga i and a i 2 aearing in the exressions 4.5 and 4.8, they will achieve their minimum value 0 at a i =, which is to say, once a i s going away from, the owers β α and β 2 α will both increase. This henomenon agrees with our intuition, since the more a i s deviateawayfrom,theeasiertodistinguishh 0 fromh.therefore,theowers should naturally grow higher. Then, we consider the ower functions β α and β 2 α as functions of y, and see how they are going along with y s changing. we see that in exression 4.5, log y y is increasing when y 0,, so β α is decreasing as the function of y, which attains its maximum value when y 0 + and minimum value α when y. Also, exression 4.8 is obviously a decreasing function of y, attaining its maximum value when y 0 + and minimum value α when y +. We resent the trends of β α and β 2 α corresonding to the ower of CLRT and CJ in Figure 3 when only one sike a = 2.5 exists. It is however a little different from the non-siked case as showed in the simulation in Section 3 both Figure and Figure 2, where the ower of CLRT first increases then decreases while the ower of CJ is always increasing along with the increase of the value of. These ower dros are due to the fact that when increases, since only one sike eigenvalue is considered, it becomes harder to distinguish both hyotheses. Besides, an interesting finding here is that these ower functions give a new confirmation of the fact that CLRT behaves quite badly when y, while CJ test has a reasonable ower for a significant range of y >.

17 Large-dimensional shericity test Generalization to the case when the oulation mean is unknown So far, we have assumed the observation Y i are centered. However, this is hardly true in ractical situations when µ = EY i is usually unknown. Therefore, the samle covariance matrix should be taken as S n = n n Y i YY i Y, i= where Y = n n i= Y i is the samle mean. Because YY is a rank one matrix, substitutings n forsn whenµisunknownwillnotaffectthelimitingdistribution in the CLT for LSS; while it is not the case for the centering arameter, for it has a in front. Recently, [22] showsthat if we use Sn in the CLT for LSS when µ is unknown, thelimiting varianceremainsthe sameasweuse S n ; whilethe limiting meanhas a shift which can be exressed as a comlex contour integral. Later, [30] looks into this shift and finally derives a concise conclusion on the CLT corresonding to Sn: the random vector Xnf,...,Xnf k converges weakly to a Gaussian vector with the same mean and covariance function as given in Theorem A., where this time, Xn f = fxdf S n F y n. It is here imortant to ay attention that the only difference is in the centering term, where we use the new ratio y n = n instead of the revious y n = n, while leaving all the other terms unchanged. Using this result, we can modify our Theorems 2. and 2.2 to get the CLT of CLRT and CJ under H 0 when µ is unknown only by considering the eigenvalues of Sn and substituting n for n in the centering terms. More recisely, now equations 2. and 2.3 in Theorems 2. and 2.2 become and L n + n+ log = N n { κ log y+ } 2 2 βy, κlog y κy nu n = Nκ+β,2κ. n The same rocedures can be alied to get the CLT of CLRT and CJ under H when µ is unknown. This time, equations 4.4 and 4.7 become and L n + n+log + k n i logai a i + n i= = N κ log y+ 2 2 βy, κlog y κy nu n n n k n i a i 2 = Nκ+β,2κ, i=

18 280 Q. Wang and J. Yao and therefore the owers of CLRT and CJ under the siked alternative remain unchanged as exressed in 4.5 and Additional roofs We recall these two imortant formulas which aear in the Aendix as A.2 and A.3 here for the convenience of reading: E[X f ] = κ I f+βi 2 f, CovX f,x g = κj f,g+βj 2 f,g. 6.. Proof of Lemma 2. Let for x > 0, fx = logx and gx = x. Define A n and B n by the decomositions logl i = fxdf n x F yn x+f yn f = A n +F yn f, i= i= l i = gxdf n x F yn x+f yn g = B n +F yn g. Alying Theorem A. given in the Aendix to the air f,g, we have An EXf CovXf,X = N, f CovX f,x g. B n EX g CovX g,x f CovX g,x g It remains to evaluate the limiting arameters and this results from the following calculations where h is denoted as y: I f,r = 2 log h 2 /r 2, 6. I g,r = 0, 6.2 I 2 f = 2 h2, 6.3 I 2 g = 0, 6.4 J f,g,r = h2 r2, 6.5 J f,f,r = r log h2 /r, 6.6 J g,g,r = h2 r2, 6.7 J 2 f,g = h 2, 6.8 J 2 f,f = h 2, 6.9 J 2 g,g = h

19 Large-dimensional shericity test 28 We now detail these calculations to comlete the roof. They are all based on the formula given in Proosition A. in the Aendix and reeated use of the residue theorem. Proof of 6.. We have I f,r = = = = [ + ξ = ξ = ξ = ξ = ξ = [ f +hξ 2 ξ ξ 2 r 2 ] dξ ξ [ log +hξ 2 ξ ξ 2 r 2 ] dξ ξ 2 log+hξ2 + 2 log+hξ [ 2 ξ log+hξ ξ 2 r 2dξ log+hξ ξ ξ 2 r 2dξ ξ = ξ = log+hξ ξ dξ ξ ξ 2 r 2 ξ log+hξ ξ dξ ] For the first integral, note that as r >, the oles are ± r and we have by the residue theorem, ξ log+hξ ξ = ξ 2 r 2dξ = log+hξ ξ ξ r + log+hξ ξ ξ +r = 2 log h2 r 2. ξ= r For the second integral, log+hξ ξ dξ = log+hξ ξ=0 = 0. ξ = z = ξ=r The third one is log+hξ ξ ξ = ξ 2 r 2dξ = z log+hz z = z 2 r 2 z 2 dz = log+hzr 2 log+hzr2 dz = = 0, zz +rz r z +rz r z=0 where the first equality results from the change of variable z = ξ, and the third equality holds because r >, so the only ole is z = 0. ] dξ

20 282 Q. Wang and J. Yao The fourth one equals log+hξ ξ dξ = ξ = = log+hz z=0 = 0. z = log+hz z z 2 dz Collecting the four integrals leads to the desired formula for I f,r. Proof of 6.2. We have I g,r = = = = ξ = ξ = ξ = ξ = ξ = [ g +hξ 2 [ +hξ 2 ξ +h+hξ 2 +h 2 ξ ξ ξ ξ 2 r 2 ξ ] dξ ξ ξ 2 r 2 ξ [ ξ +h+hξ 2 +h 2 ξ ξ 2 r 2 dξ ξ +h+hξ 2 +h 2 ξ ξ 2 dξ. These two integrals are calculated as follows: ξ +h+hξ 2 +h 2 ξ ξ 2 r 2 dξ ξ = = ξ +h+hξ2 +h 2 ξ ξ r = +h 2 ; ξ= r ] dξ ξ ξ 2 r 2 ξ ] dξ + ξ +h+hξ2 +h 2 ξ ξ=r ξ +r ξ +h+hξ 2 +h 2 ξ ξ = ξ 2 dξ = ξ ξ +h+hξ2 +h 2 ξ = +h 2. Collecting the two terms leads to I g,r = 0. Proof of 6.3. I 2 f = = [ ξ = ξ = log +hξ 2 ξ 3dξ log+hξ ξ 3 dξ + ξ = ξ=0 ] log+hξ ξ 3 dξ. We have log+hξ ξ = ξ 3 dξ = 2 2 ξ 2 log+hξ ξ=0 = 2 h2 ;

21 Large-dimensional shericity test 283 log+hξ ξ = ξ 3 dξ = log+hz z = z 3 Combining the two leads to I 2 f = 2 h2. dz = 0. z2 Proof of 6.4. I 2 g = +hξ+hξ ξ = ξ 3 dξ = ξ +hξ 2 +h+h 2 ξ ξ = ξ 4 dξ = 0. Proof of 6.5. J f,g,r = We have, = The first term +hξ 2 2 log +hξ 2 ξ 2 = ξ = ξ rξ 2 2 dξ dξ 2. ξ = ξ = log +hξ 2 ξ rξ 2 2 dξ log+hξ ξ rξ 2 2 dξ + ξ = log+hξ ξ = ξ rξ 2 2 dξ = 0, log+hξ ξ rξ 2 2 dξ. because for fixed ξ 2 =, rξ 2 = r >, so rξ 2 is not a ole. The second term is log+hξ ξ = ξ rξ 2 2 dξ = log+hz z = z rξ 2 2 z 2 dz = log+hz rξ 2 2 z = z rξ 2 dz = 2 rξ 2 2 z log+hz z= rξ 2 h = rξ 2 rξ 2 +h, where the first equality results from the change of variable z = ξ, and the third equality holds because for fixed ξ 2 =, rξ 2 = r <, so rξ 2 is a ole of second order. Therefore, J f,g,r = h r 2 = h r 2 ξ 2 = ξ 2 = +hξ 2 +hξ 2 ξ 2 ξ 2 + h r dξ 2 ξ 2 +hξ 2 2 +h+h 2 ξ 2 ξ 2 2 ξ 2 + h r dξ 2

22 284 Q. Wang and J. Yao [ = h +h 2 h r 2 ξ 2 = ξ 2 ξ 2 + h r dξ 2 + ξ 2 = ξ 2 + h r Finally we find J f,g,r = h2 r since 2 h +h 2 h r 2 ξ 2 = ξ 2 ξ 2 + h r dξ 2 = 0, r 2 ξ 2 = h r 2 ξ 2 = h ξ 2 2 ξ 2 + h r dξ 2 = 0. dξ 2 + ξ 2 = h ξ 2 + h r dξ 2 = h2 r 2, ] h ξ2 2ξ 2 + h r dξ 2. Proof of 6.6. J f,f,r = = = h r 2 ξ 2 = ξ 2 = ξ 2 = f +hξ 2 2 f +hξ 2 ξ = ξ rξ 2 2 dξ dξ 2 f +hξ 2 2 h rξ 2 rξ 2 +h dξ 2 log+hξ 2 ξ 2 h r +ξ 2 dξ 2 + h r 2 ξ 2 = log+hξ2 ξ 2 h r +ξ dξ 2. 2 We have and h r 2 = r h log+hξ 2 r 2 ξ 2 = ξ 2 h r +ξ 2 dξ 2 [ = h log+hξ 2 r 2 h r +ξ + log+hξ 2 2 ξ 2 ξ2=0 = r log h2, r ξ 2 = z = log+hξ2 ξ 2 h r +ξ 2 log+hz z + r dz = 0, h ξ2= h r dξ 2 = h log+hz r 2 z = z h r + z z 2 dz where the first equality results from the change of variable z = ξ 2, and the third equality holds because r h >, so r h is not a ole. Finally, we find J f,f,r = h2 r log r. Proof of 6.7. J g,g,r = +hξ 2 2 ξ 2 = ξ = ] +hξ 2 ξ rξ 2 2dξ dξ 2.

23 Large-dimensional shericity test 285 We have since ξ = [ = + ξ = = h r 2 ξ2 2, +hξ 2 ξ rξ 2 2dξ = +h 2 ξ = ξ rξ 2 2dξ + ] h ξ ξ rξ 2 2dξ +h 2 ξ = ξ rξ 2 2dξ = 0, ξ = h ξ ξ rξ 2 2dξ = ξ = ξ = ξ +hξ 2 +h+h 2 ξ ξ ξ rξ 2 2 dξ hξ ξ rξ 2 2dξ ξ = h ξ rξ 2 2 hξ ξ rξ 2 2dξ = 0, ξ=0 = h r 2 ξ2 2, where the equality above holds because for fixed ξ 2 =, rξ 2 = r >, so rξ 2 is not a ole. Therefore, h ξ 2 +hξ2 2 +h+h 2 ξ 2 J g,g,r = r 2 ξ 2 = ξ2 3 dξ 2 [ h +h 2 h h = r 2 dξ 2 + dξ 2 + dξ 2 ], ξ 2 = ξ 2 ξ 2 = = h2 r 2. ξ = ξ 2 = ξ 2 2 Proof of 6.8, 6.9, 6.0. We have f +hξ 2 ξ = ξ 2 dξ = log +hξ 2 ξ = ξ 2 dξ = log+hξ +log+hξ dξ = h, since ξ = = ξ = z = ξ 2 log+hξ ξ 2 dξ = log+hξ ξ log+hξ ξ 2 dξ = log+hzdz = 0. z = ξ =0 = h, log+hz z 2 ξ 3 2 z 2dz

24 286 Q. Wang and J. Yao Similarly, ξ 2 = Therefore, J 2 f,g = J 2 f,f = J 2 g,g = g +hξ 2 2 ξ2 2 dξ 2 = ξ 2 = ξ = ξ = ξ = f +hξ 2 ξ 2 dξ f +hξ 2 ξ 2 dξ g +hξ 2 ξ 2 dξ The roof of Lemma 2. is comlete. ξ 2 +hξ2 2 +h+h2 ξ 2 ξ2 3 dξ 2 = h. ξ 2 = ξ 2 = ξ 2 = g +hξ 2 2 ξ2 2 dξ 2 = h 2, f +hξ 2 2 ξ2 2 dξ 2 = h 2, g +hξ 2 2 ξ2 2 dξ 2 = h Proof of Lemma 2.2 Let fx = x 2 and gx = x. Define C n and B n by the decomositions i= i= l 2 i = l i = fxdf n x F yn x+f yn f = C n +F yn f, gxdf n x F yn x+f yn g = B n +F yn g. Alying Theorem A. given in the Aendix to the air f,g, we have Cn EXf CovXf,X = N, f CovX f,x g. B n EX g CovX g,x f CovX g,x g It remains to evaluate the limiting arameters and this results from the following calculations: I f,r = h2 r2, 6. I g,r = 0, 6.2 I 2 f = h 2, 6.3 I 2 g = 0, 6.4 J f,g,r = 2h2 +2h 4 r 2, 6.5 J f,f,r = 2h4 +2h+2h 3 2 r r 3, 6.6 J g,g,r = h2 r2, 6.7

25 Large-dimensional shericity test 287 J 2 f,g = 2h 2 +2h 4, 6.8 J 2 f,f = 2h+2h 3 2, 6.9 J 2 g,g = h The results 6.2, 6.4, 6.7 and 6.20 are exactly the same as those found in 6.2, 6.4, 6.7 and 6.0 in the roof of Lemma 2.. The remaining results are found by similar calculations using Proosition A. in the Aendix and their details are omitted. 7. Concluding remarks Using recent central limit theorems for eigenvalues of large samle covariance matrices, we are able to find new asymtotic distributions for two major rocedures to test the shericity of a large-dimensional distribution. Although the theory is develoed under the scheme, n and /n y > 0, our Monte-Carlo study has roved that: on the one hand, both CLRT and CJ are alreadyvery efficient for middle dimension such as,n = 96,28both in size and ower, see Table 2 and Table 3; and on the other hand, CJ also behaves very well in most of large, small n situation, see Table 4. Three characteristic features emerge from our findings: a These asymtotic distributions are universal in the sense that they deend on the distribution of the observations only through its first four moments; b The new test rocedures imrove quickly when either the dimension or the samle size n gets large. In articular, for a given samle size n, within a wide range of values of /n, higher dimensions lead to better erformance of these corrected test statistics. c CJ is articularly robust against the dimension inflation. Our Monte-Carlo study shows that for a small samle size n = 64, the test is effective for 0 < /n 20. In a sense, these new rocedures have benefited from the blessings of dimensionality. Aendix A: Formula for limiting arameters in the CLT for eigenvalues of a samle covariance matrix with general fourth moments Givenasamlecovariancematrix S n ofdimension with eigenvaluesλ,...,λ, linear sectral statistics of the form F n g = i= gλ j for suitable functions g are of central imortance in multivariate analysis. Such CLT s have been successively develoed since the ioneering work of [2], see [2] and [6] for a recent account on the subject. The CLT in [2] see also an imroved version in [5] has been widely used in alications as this CLT also rovides, for the first time, exlicit formula for the mean and covariance arameters of the normal limiting distribution. In the

26 288 Q. Wang and J. Yao secial case with an array {x ij } of indeendent variables, this CLT assumes the following moment conditions: a For each n, x ij = x n ij,i,j n are indeendent. b Ex ij = 0,E x ij 2 =,max i,j,n E x ij 4 <. c If {x ij } s are real, then Ex 4 ij = 3; otherwise comlex variables, Ex2 ij = 0 and E x ij 4 = 2. In Condition c, the fourth moments of the entries are set to the values 3 or 2 matching the normal case. This is indeed a quite demanding and restrictive condition since in the real case for examle, it is incredibly hard to find a nonnormal distribution with mean 0, variance and fourth moment equaling 3. As a consequence, most of if not all alications ublished in the literature using this CLT assumes a normal distribution for the observations. Recently, effort have been made in [2, 6] and [29] to overcome these moment restrictions. We resent below such a CLT with general forth moments that will be used for the shericity test. In all the following, we use an indicator κ set to 2 when {x ij } are real and to when they are comlex. Define β = E x ij 4 κ for both cases and h = y. For the resentation of the results, let be the samle covariance matrix S n = n n i= X ix i where X i = x ki k is the i-th observed vector. It is then well-known that when, n and /n y > 0, the distribution of its eigenvalues converges to a nonrandom distribution, namely the Marčenko- Pastur distribution F y with suort [a,b] = [± y 2 ] an additional mass at the origin when y >. Moreover, the Stieltjes transform m of a comanion distribution defined by F y = yδ 0 + yf c satisfies an inverse equation for z C +, z = m + y +m. A. The following CLT is a articular instance of Theorem.4 in [2]. Theorem A. [2]. Assume that for each n, the variables x ij = x n ij,i,j n are indeendent and identically distributed satisfying Ex ij = 0, E x ij 2 =, E x ij 4 = β ++κ < and in case of they are comlex, Ex 2 ij = 0. Assume moreover,, n, /n y > 0. Let f,...f k be functions analytic on an oen region containing the suort of F y. The random vector {X n f,...x n f k } where X n f = {F n f F yn f} converges weakly to a normal vector X f,...x fk with mean function and covariance function: E[X f ] = κ I f+βi 2 f, CovX f,x g = κj f,g+βj 2 f,g, A.2 A.3

27 Large-dimensional shericity test 289 where I f = y{m/+m} 3 zfz [ y{m/+m} 2] 2 dz, I 2 f = y{m/+m} 3 zfz y{m/+m} 2 dz, J f,g = fz gz 2 4π 2 J 2 f,g = y 4π 2 mz mz 2 2m z m z 2 dz dz 2, { } m +m z dz fz z gz 2 z 2 { m +m z 2 } dz 2, where the integrals are along contours non overlaing in J enclosing the suort of F y. However, concrete alications of this CLT are not easy since the limiting arameters are given through those integrals on contours that are only vaguely defined. The urose of this aendix is to go a ste further by roviding alternative formula for these limiting arameters. These new formulas, resented in the following Proosition convert all these integral along the unit circle; they are much easier to use for concrete alications, see for instance the roofs of Lemma 2. and 2.2 in the aer. Furthermore, these formulas will be of indeendent interest for alications other than those develoed in this aer. Proosition A.. The limiting arameters in Theorem A. can be exressed as following: with I f = limi f,r, r I 2 f = f +hξ 2 ξ 3dξ, ξ = J f,g = limj f,g,r, r J 2 f,g = 4π 2 I f,r = J f,g,r = 4π 2 ξ = ξ = f +hξ 2 ξ 2 dξ ξ 2 = ξ = [ f +hξ 2 ξ 2 = ξ ξ 2 r 2 ξ A.4 A.5 A.6 g +hξ 2 2 ξ2 2 dξ 2, A.7 ] dξ, f +hξ 2 g +hξ 2 2 ξ rξ 2 2 dξ dξ 2. Proof. We start with the simlest formula I 2 f to exlain the main argument and indeed, the other formula are obtained similarly. The idea is to introduce the change of the variable z = + hrξ + hr ξ + h 2 with r > but close to and ξ = recall h = y. Note that this idea has been emloyed in [29].

28 290 Q. Wang and J. Yao It can be readily checked that when ξ runs counterclockwisely the unit circle, z will run a contour C that encloses closely the suort interval [a,b] = [±h 2 ] recall h = y. Moreover, by the Eq. A., we have on C m = +hrξ, and dz = hr r ξ 2 dξ. Alying this variable change to the formula of I 2 f given in Theorem A., we have I 2 f = lim fz rξ 2 r r ξ = ξ 3 rr 2 ξ 2 dξ = f +hξ 2 ξ 3dξ. ξ = This roves the formula A.5. For A.4, we have similarly I f = lim fz rξ 2 r r ξ = ξ 3 rr 2 ξ 2 r 2 ξ 2dξ = lim f +hξ 2 r ξξ 2 r 2 ξ = = lim r I f,r. Formula A.7 for J 2 f,g is calculated in a same fashion by observing that we have { } m z +m z dz = { } m ξ +m ξ dξ = { } dξ = ξ hrξ hrξ 2dξ. Finally for A.6, we use two non-overlaing contours defined by z j = + hr j ξ j +hr j ξ j +h 2, j =,2 where r 2 > r >. By observing that we find J f,g = m z j dz j = lim r 2 > r > r 2 = lim r 2 > r >, r 2 ξ j m dξ j = hr j +hr j ξ j 2dξ j, fz gz 2 4π 2 ξ = ξ 2 = {mz mz 2 } 2 hr +hr ξ 2 hr 2 +hr 2 ξ 2 2dξ dξ 2 4π 2 = lim r 4π 2 The roof is comlete. ξ = ξ 2 = ξ = ξ 2 = fz gz 2 {r ξ r 2 ξ 2 } 2dξ dξ 2 f +hξ 2 g +hξ 2 2 {ξ rξ 2 } 2 dξ dξ 2.

29 Large-dimensional shericity test 29 Acknowledgements We are grateful to the associate editor and two referees for their numerous comments that have lead to imortant modifications of the aer. References [] Anderson, T.W.984. An introduction to Multivariate Statistical Analysis 2nd edition. Wiley, New York. MR [2] Bai, Z.D. and Silverstein, J.W CLT for linear sectral statistics of large-dimensional samle covariance matrices. Ann. Probab., 32, MR [3] Bai, Z.D. and Yao, J.F On the convergence of the sectral emirical rocess of Wigner matrices. Bernoulli, 6, MR28908 [4] Bai, Z.D., Jiang, D.D., Yao, J.F., and Zheng, S.R Corrections to LRT on large-dimensional covariance matrix by RMT. Ann. Statist., 37, MR [5] Bai, Z.D. and Silverstein, J.W Sectral Analysis of Large Dimensional Random Matrices 2nd edition. Sringer, 20. MR [6] Bai, Z.D., Jiang, D.D., Yao, J.F., and Zheng, S.R Testing linear hyotheses in high-dimensional regressions. Statistics, doi:0.080/ [7] Birke, M. and Dette, H A note on testing the covariance matrix for large dimension. Statistics and Probability Letters, 74, MR [8] Chen, S.X. and Qin, Y.L.200. A two-samle test for high-dimensional data with alications to gene-set testing. Ann. Statist., 38, MR [9] Chen, S.X., Zhang, L.X., and Zhong, P.S Tests for highdimensional covariance matrices. J. Amer. Statist. Assoc., 05, MR [0] John, S.97. Some otimal multivariate tests. Biometrika, 58, MR [] John, S The distribution of a statistic used for testing shericity of normal distributions. Biometrika, 59, MR03269 [2] Jonsson, D Some limit theorems for the eigenvalues of a samle covariance matrix. J. Multivariate Anal., 2, 38. MR [3] Johnstone, I.M On the distribution of the largest eigenvalue in rincial comonents analysis. Ann. Statist., 292, MR86396 [4] Johnstone, I.M. and Titterington, D.M Statistical challenges of high-dimensional data. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367, MR [5] Ledoit, O. and Wolf, M Some hyothesis tests for the covariance matrix when the dimension is large comared to the samle size. Ann. Statist., 30, MR92669

30 292 Q. Wang and J. Yao [6] Lytova, A. and Pastur, L Central limit theorem for linear eigenvalue statistics of the Wigner and the samle covariance random matrices. Metrika, 69, MR24899 [7] Mauchly, J.W Test for shericity of a normal n-variate distribution. Ann. Mathe. Statist.,, MR [8] Muirhead, R.J Asects of Multivariate Statistical Theory. Wiley, New York. MR [9] Nagao, H On some test criteria for covariance matrix. Ann. Statist.,, MR [20] Onatski, A., Moreira, M.J., and Hallin, M Asymtotic ower of shericity tests for high-dimensional data. Ann. Statist., 43, [2] Pan, G.M. and Zhou, W Central limit theorem for signal-tointerference ratio of reduced rank linear receiver. Ann. Al. Probab., 8, MR [22] Pan, G.M Comarision between two tyes of large samle covariance matrices. To aear in Ann. Institut Henri Poincaré. [23] Schott, J.R Some high-dimensional tests for a one-way MANOVA. J. Multivariate Anal., 98, MR [24] Srivastava, M.S Some tests concerning the covariance matrix in high dimensional data. J. Jaan Statist. Soc., 35, MR [25] Srivastava, M.S. and Fujikoshi, Y Multivariate analysis of variance with fewer observations than the dimension. J. Multiv. Anal., 97, MR [26] Srivastava, M.S., Kollo, T., and Rosen, D. 20. Some tests for the covariance matrix with fewer observations than the dimension under non-normality. J. Multiv. Anal., 02, MR [27] Sugiura, N Locally best invariant test for shericity and the limiting distributions. Ann. Mathe. Statist., 43, MR03032 [28] Wang, Q.W., Silverstein, J.W., and Yao, J.F A note on the CLT of the LSS for samle covariance matrix from a siked oulation model. Prerint, available at arxiv: [29] Zheng, S.R Central limit theorem for linear sectral statistics of large dimensional F matrix. Ann. Institut Henri Poincaré Probab. Statist., 48, MR [30] Zheng, S.R. and Bai, Z.D A note on central limit theorems for linear sectral statistics of large dimensional F Matrix. Prerint, available at arxiv: