LIKELIHOOD RATIO TEST FOR THE HYPER- BLOCK MATRIX SPHERICITY COVARIANCE STRUCTURE CHARACTERIZATION OF THE EXACT

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1 LIKELIHOOD RATIO TEST FOR THE HYPER- BLOCK MATRIX SPHERICITY COVARIACE STRUCTURE CHARACTERIZATIO OF THE EXACT DISTRIBUTIO AD DEVELOPMET OF EAR-EXACT DISTRIBUTIOS FOR THE TEST STATISTIC Authors: Bárbara R. Correia CMA/UL - Centro de Matemática e Apicações, Universidade ova de Lisboa, Portuga regadas.correia@gmai.com Caros A. Coeho Departamento de Matemática, CMA/UL - Centro de Matemática e Apicações, Universidade ova de Lisboa, Portuga cmac@fct.un.pt Fiipe J. Marques Departamento de Matemática, CMA/UL - Centro de Matemática e Apicações, Universidade ova de Lisboa, Portuga fjm@fct.un.pt Abstract: In this paper the authors introduce the hyper-bock matrix sphericity test which is a generaization of both the bock-matrix and the bock-scaar sphericity tests and as such aso of the common sphericity test. This test is a too of crucia importance to verify eaborate assumptions on covariance matrix structures, namey on metaanaysis and error covariance structures in mixed modes and modes for ongitudina data. The authors show how by adequatey decomposing the nu hypothesis of the hyper-bock matrix sphericity test it is possibe to easiy obtain the expression for the ikeihood ratio test statistic as we as the expression for its moments. From the factorization of the exact characteristic function of the ogarithm of the statistic, induced by the decomposition of the nu hypothesis, and by adequatey repacing some of the factors with an asymptotic resut, it is possibe to obtain near-exact distributions that ie very cose to the exact distribution. The performance of these near-exact distributions is assessed through the use of a measure of proximity between distributions, based on the corresponding characteristic functions. Key-Words: Equaity of matrices test, Generaized Integer Gamma distribution, Generaized ear- Integer Gamma distribution, Independence test, ear-exact distributions, Mixtures of distributions AMS Subject Cassification: 6H15, 6H10, 6E10, 6E15, 6E0.

2 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques

3 The hyper-bock matrix sphericity test 3 1. ITRODUCTIO Likeihood ratio tests.r.t. s have a arge scope of appication in different fieds of research such as for exampe engineering, economics, medicine and ecoogy [7, 0, 5, 1]. However, in most cases, the exact distribution of the.r.t. statistics has a very compicated expression which makes difficut the practica use of the testing procedure. On the other hand the commony used asymptotic approximations [, 6] dispay ack of precision mainy in extreme situations such as for high number of variabes and/or sma sampe sizes [14, 9] and situations where the parameters of interest and/or nuisance parameters are on the boundary of the parameter space [11]. This is a we known and recognized probem in standard ikeihood ratio testing procedures which becomes even more serious when one wants to perform tests for more eaborate covariance structures. These eaborate structures have recenty become very important in different statistica techniques for the vaidation of assumptions required in different modes such as in hierarchica or mixed inear univariate and mutivariate modes. In this paper the authors introduce the hyper-bock matrix HBM sphericity test. This test is a usefu generaization of the bock-matrix and of the bockscaar sphericity tests and is of crucia importance to vaidate eaborate assumptions on covariance matrix structures, for exampe on meta-anaysis and error covariance structures in mixed modes and modes for ongitudina data. We wi say that a covariance matrix Σ has a HBM spherica structure if we can write I k Σ =....., unspecified, = 1,..., m. 0 I km m where denotes the Kronecker product, and for = 1,..., m, I k identity matrix of order and is a positive-definite matrix. denotes the The HBM spherica structure may arise in many situations and has as particuar cases many interesting and important structures which may be of interest not ony as covariance structures in mutivariate anaysis as we as covariance structures for the error in inear mixed and repeated measures modes. Let us consider a situation in which the same p random variabes r.v. s, X 1,..., X p, are measured in m ocas, in the -th of which = 1,..., m we take measurements, that is, a sampe of size, and et us suppose we organize such a meta-sampe in a matrix X of dimensions p n, with n = m, as in Figure 1. Here ocas is a genera designation for exampe for different ocas, factories, companies, hospitas, etc., and if we consider the p r.v. s X 1,..., X p organized in the random vector X = [X 1,..., X p ], with CovX =, then

4 4 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques X = p n { k 1 }} { { }} { X X 1k1... X 1,k X 1,k X 1... X k1... X,k X,k X p 1... X p k 1... X p,k X p,k k m {}}{... X 1,k1 + +k m X 1,k1 + +k m... X,k1 + +k m X,k1 + +k m..... X p,k 1 + +k m X p,k 1 + +k m Figure 1: Data matrix iustrating a situation of a meta-sampe from m ocas, with sampe size for the -th oca and p = p = 1,..., m. we have I k1 0 CovX = CovvecX = I km But, the HBM sphericity setup aows for more genera situations as for exampe those in which we may want to study or mode possibe differences in strength break in a set of p components manufactured by m different companies, or the measurements of p variabes thought to be possibe important indicators of some disorder or disease, measured across m hospitas, or measurements of p poutants in m different ocas, or measurements of p atmospheric variabes and indicators in m different cities, by taking a sampe of size in the -th oca, but that, furthermore, not in every oca, city, hospita or company, it was possibe to obtain measurements of a p variabes, athough we sti want to consider as many of these in each oca as possibe. Then we may want to consider a meta-sampe as the one iustrated in Figure, for p = 5. In this case, since in different ocas we may have different subsets of the p variabes being anayzed, we may end-up with a covariance setup for the matrix X as the one in 1.1, with different covariance matrices for each oca. Once we assume the HBM spherica structure for the covariance structure in our mode, we may then be interested in testing if that is indeed a pausibe mode for our covariances. The issues are thus: i how can we carry out a test for such an eaborate structure, and ii in case we find a way of doing so, how wi we then be abe to obtain p-vaues and/or quanties for our test statistic, since this may have a quite eaborate exact distribution.

5 The hyper-bock matrix sphericity test 5 X = p n k 1 {}}{{}}{ X X 1k1... X 1,k X 1,k X 1... X k X X 3k1... X 3,k X 3,k X 4,k X 4,k X X 5k k m {}}{... X 1,k1 + +k m X 1,k1 + +k m... X,k1 + +k m X,k1 + +k m X 4,k1 + +k m X 4,k1 + +k m... X 5,k1 + +k m X 5,k1 + +k m Figure : Data matrix iustrating a situation of a meta-sampe from m ocas, with sampe size for the -th oca = 1,..., m, p 1 = 4, p = 3 and p m = 4, with different covariance matrices = 1,..., m. These are indeed the issues we propose to address in this paper, namey showing how one can quite easiy buid the.r.t. statistic for the test of the HBM Spherica structure and how we can then obtain the expression for the moments of the statistic and even for the characteristic function c.f. of its ogarithm, from which factorization we wi then be abe to obtain very sharp approximations for the exact distribution of the statistic. The HBM sphericity test is thus a test where the nu hypothesis is written as I k H 0 : Σ =....., unspecified, = 1,..., m 0 I km m where Σ is the covariance matrix of the random vector X and the matrices are p p, = 1,..., m, with p = p and p = m p. This test is a generaization of the standard sphericity test and it has as particuar cases a number of interesting and important tests: i the bock-matrix sphericity BM-Sph test, for m = 1 [4, 3, 15], ii the bock-scaar sphericity BS-Sph test, for p = 1, = 1,..., m [19, 18, 13], iii the bock independence BI test, for = 1, = 1,..., m [4, 5], [1, Chap. 9], [17, Sec. 11.], [6, 7],

6 6 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques iv the common independence Ind test, for p = = 1, = 1,..., m [, Sec ],[9, Secs. 1,], and v the sphericity Sph test, for m = p 1 = 1 [16],[1, Sec. 10.7],[17, Sec. 8.3],[9]. These particuar cases as we as their reations may be anayzed in Figure 3. p =1, BS-Sph [ ] k, σ 1 0 Ik1 H0: Σ=... 0 σm Ikm HBM-Sph [ ] Ik1 1 0 H0: Σ=... m=1 0 Ikm m p =1 k, k, Ind p [ ] BI =1, 1=... [ ] σ = m H0: Σ=... H0: Σ=... 0 σm 0 m BM-Sph [ ] 0 H0: Σ=... 0 σ 1 =...=σ m Sph [ ] m=1 σ 0 H0: Σ=... 0 σ 1=...= m p =1, p1=1 Figure 3: Particuar cases of the HBM Hyper-Bock Matrix test and their inter-reations: HBM-Sph: Hyper-bock matrix sphericity test; BM-Sph: Bock-matrix sphericity test; BS-Sph: Bock-scaar sphericity test; Sph: Sphericity test; Ind: Independence test; BI: Bock independence test. The exact distribution of the HBM sphericity test statistic is amost intractabe in practica terms, thus our propose is to deveop near-exact distributions for the test statistic and its ogarithm, based on an adequate factorization of the c.f. of the ogarithm of the test statistic. In Section we wi show how we may decompose the overa nu hypothesis in 1. into a set of three conditionay independent hypotheses and then how from this decomposition see [8] we may derive expressions for the.r.t. statistic and its h-th moment. In Section 3 we wi show how we may easiy obtain the expression for the c.f. of the ogarithm of the test statistic and how we may use the decomposition of the nu hypothesis in Section, to induce an adequate factorization of this c.f. in two factors, one that is the c.f. a Generaized Integer Gamma GIG distribution [6], and the other the c.f. of a sum of independent r.v. s whose exponentias have Beta distributions. Then, in Section 4 we wi use this factorization to buid very sharp near-exact distributions both for the test statistic and its ogarithm. ear-exact distributions are asymptotic distributions buit using a different approach. Usuay working from an adequate factorization of the c.f. of the ogarithm of the.r.t. statistic, we eave unchanged the set of factors that correspond to a manageabe distribution and approximate asymptoticay the remaining set of factors, in such a way that the resuting c.f., which we wi ca a near-exact c.f., corresponds to a known manageabe distribution, from which p-vaues and

7 The hyper-bock matrix sphericity test 7 quanties may be easiy computed. These near-exact distributions ie much coser to the exact distribution than any common asymptotic distribution and, when correcty buit for statistics used in Mutivariate Anaysis, wi show a marked asymptotic behavior not ony for increasing sampe sizes but aso for increasing number of variabes invoved. In Section 5 we wi use a measure of proximity between the exact distribution and the near-exact distributions, based on the corresponding characteristic functions in order to assess the quaity and the asymptotic properties of the near-exact distributions deveoped. Section 6 is dedicated to power studies, where the very good behavior of the test is reveaed, through studies based on pseudo-random sampes and carried out on severa scenarios of vioation of the nu hypothesis of HBM sphericity.. THE TEST STATISTICS AD ITS MOMETS In genera terms, a nu hypothesis H 0 may be decomposed into a sequence of three conditionay independent nu hypotheses, if H 0 admits the decomposition H 0 H 03 1, H 0 1 H 01 where is to be read as after, as ong as Ω H0 Ω H03 1, Ω H13 1, Ω H0 1 Ω H1 1 Ω H01 Ω H11 Ω H1 where Ω H0 is the parameter space under H 0 and Ω H1 the union of the parameter spaces under H 0 and H 1, and where H 1 represents the aternative hypothesis to H 0 where is used as a widcard. The nu hypothesis.1 H 01 : Σ = bdiag Σ ; = 1,..., m corresponds to the test of independence of m groups of variabes, the -th group having p = p variabes = 1,..., m. If we consider that the random vector X has a p-variate orma distribution with expected vaue vector µ and covariance matrix Σ, that is, if we consider the vector X p µ, Σ and suppose that we have a sampe of size > p from X, then the.r.t. statistic used to test H 01 and its h-th moment are respectivey given by see secs. 9. and 9.3. in [1]. Λ 1 = A A

8 8 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques and [.3 E Λ 1 h] m 1 = p k=1 Γ q k Γ + h q k Γ k Γ k + h, h> p pm 1 where the matrix A is the maximum ikeihood estimator m..e. of Σ, A its -th diagona bock of order p = 1,..., m and q = p p m. The nu hypothesis.4 H 0 1 = where for = 1,..., m.5 m H 0 1 H 0 1 : Σ = bdiag Σ vv, v = 1,..., assuming Σ = bdiag Σ, = 1,..., m that is, assuming H 01 is the nu hypothesis of a test of independence of groups of variabes, with p variabes each. The.r.t. statistic to test H0 1 in.5 and its h-th moment are respectivey given by see Secs. 9. and 9.3. in [1] and [ ] k h 1 E Λ 1 = p v=1 k=1 Λ 1 = A Γ q v k Γ A v v=1 + h q v k Γ k Γ k + h, h> p 1 where the matrix A is the maximum ikeihood estimator of Σ, A v its v-th v = 1,..., diagona bock of order p and q v = v p, v = 1,...,. The.r.t. statistic to test the nu hypothesis in.4 is thus m.6 Λ 1 = Λ 1 = A A v v=1 and, given the fact that the statistics Λ 1, = 1,..., m, form a set of m independent statistics, since under H 01 in.1 the m matrices A are independent and each statistic Λ 1 is buit ony from A, the h-th moment of Λ 1 is [ Λ 1 ] h [ ] h E = E Λ.7 = k 1 p 1 v=1 k=1 Γ q v k Γ + h q v k Γ k Γ k + h, h > max { p 1, = 1,..., m}.

9 The hyper-bock matrix sphericity test 9 Finay, the nu hypothesis H 03 1, may be written as.8 H 03 1, = where, for = 1,..., m, m H 03 1,.9 H 03 1, : Σ 11 = = Σ = unspecified assuming Σ = bdiag Σ = bdiag Σ vv, v = 1,..., that is, assuming H 0 1 and H 01 is the nu hypothesis corresponding to the test of equaity of covariance matrices each with dimensions p p. Since under H 0 1, for each = 1,..., m, the matrices A v v = 1,..., are independent, The.r.t. statistic to test each nu hypothesis H03 1, in.9 and its h-th moment are respectivey, see Secs. 10. and in [1] Λ 3 1, = A v v=1 A p k and [ ] h E Λ 3 1, = p k=1 v=1 Γ 1 k 1 + v 1 Γ 1 k 1 + v 1 + h Γ k + h Γ k where the matrix A is the maximum ikeihood estimator of Σ, A v diagona bock of order p v = 1,..., and A = A1 + + A. its v-th The.r.t. statistic to test.8 is thus.10 Λ 3 1, = m Λ 3 1, = A v v=1 A p k and, since under H 01 in.1, the m statistics Λ 3 1, are independent, given that each statistic Λ 3 1, is buit ony from A and under H 01 the m matrices A = 1,..., m are independent, the h-th moment of Λ 3 1, is given by [ Λ3 1, ] h E = = [ E Λ p k=1 v=1 3 1, h ] Γ 1 k 1 + v 1 Γ k Γ 1 k 1 + v 1 + h + h Γ, k h > max { p 1, = 1,..., m }.

10 10 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques From Lemma in [1], the.r.t. statistic to test 1. is thus the product of the.r.t. statistics used to test the nu hypotheses in.1,.4 and.8, that is, the product of the.r.t. statistics in.,.6 and.10, thus with.11 Λ = Λ 1 Λ 1 Λ 3 1, = = A A { k p } A A A v v=1 A A v v=1 A k p where the matrix A is the m..e. of Σ, A is the -th diagona bock of order p of A = 1,..., m, with p = m p and A = A1 + + A, where Av is the v-th v = 1,..., diagona bock of order p of A and p = p. We shoud note that the expression in.11 is identica to the one we obtain when we use the usua method of derivation of the.r.t. statistic, through its definition see Appendix A. The hypotheses H 03 1,, H 0 1 and H 01 are independent in the sense that under the overa nu hypothesis H 0 it is possibe to prove that the.r.t. statistics used to test these hypotheses are independent. Indeed, by Lemma in [1] or Theorem 5 in [10], the.r.t. statistic Λ 1 in. is independent of the m matrices A, = 1,..., m, so that since Λ 1 and Λ 3 1, are buit ony from the m matrices A = 1,..., m, these.r.t. statistics are independent of Λ 1. But then, since we may use the same two resuts to argue that each statistic Λ 1 is independent of the matrices A v v = 1,...,, the statistic Λ 1 is independent of a m matrices A v = 1,..., ; v = 1,..., m and as such independent of Λ 3 1, which is buit ony on these matrices. As such, we may obtain the expression of the h-th moment of Λ from the expressions for the h-th moment of each of the statistics Λ 3 1,, Λ 1 and Λ 1 writing it as [ E Λ h] [ =E Λ 1 h] [ Λ 1 ] [ h Λ3 1, ] h E E p = k 1 k=1 v=1 p k=1 m 1 p k=1 Γ 1 k 1 + v 1 Γ 1 k 1 + v 1 + h Γ 1 k Γ 1 Γ k 1 q k Γ + h q k h Γ k Γ k + h, Γ q v k Γ + h q v k Γ k + h Γ k

11 The hyper-bock matrix sphericity test 11 for h>max {p p m / 1, p / 1,..., m} and where p = m p, q = p p m, p = p and q v = v p = 1,..., m; v = 1,...,. Actuay we may note that the two first nu hypotheses may be condensed into a singe nu hypothesis to test the independence of q = m groups of variabes, the ν-th group with p ν variabes for [ ] p = p 1.1,..., p 1, p,..., p,..., p,..., p,..., p m,..., p m, }{{}}{{}}{{}}{{} k 1 k k m with ν = 1,..., m, that is, with 1 p ν = p, for 1 + k i ν i=1 k i. i=1 This nu hypothesis may be written as H 0,1 : Σ = bdiag 1,..., k1,..., k ,..., k1 + +,... }{{}}{{} of order p 1 of order p..., k1 + +k m 1 +1,..., k1 + +k m. }{{} of order p m The.r.t. statistic to test H 0,1 is given by.13 Λ 1, = A q ν=1 A νν where A νν is the ν-th diagona bock of order p ν ν = 1,..., q, and the expression of its h-th moment is given by see secs. 9. and 9.3. in [1] for [ E Λ 1, h] q 1 = ν=1 p ν k=1 Γ q ν k Γ + h q ν k q ν = p ν p q where q = Γ k Γ k + h m. ote that the.r.t. statistic in.13 may be aso given by the product of the.r.t. s in. and.6, used to test the nu hypotheses in.1 and.4 respectivey, and the expression of its h-th moment may aso be given by the product of the expressions of the h-th moments in.3 and.7 of the.r.t. s in. and.6 respectivey. Finay, the expression of the h-th moment of Λ may be re-written as

12 1 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques.14 [ E Λ h] [ = E Λ 1, h] E = q 1 ν=1 p ν k=1 p Γ k=1 v=1 q ν k Γ [ Λ3 1, h ] + h q ν k Γ k Γ k + h Γ 1 k 1 + v 1 Γ 1 k 1 + v 1 + h Γ k + h Γ. k The factorization of the c.f. of W = og Λ deveoped in the next section wi have as a starting base this ast expression. 3. THE CHARACTERISTIC FUCTIO OF W = ogλ Since in.14 the Gamma functions remain vaid for any stricty compex h, if we take W 1, = og Λ 1, and W 3 = og Λ 3 1,, we may write the c.f. of W = og Λ as 3.1 Φ W t = E e it og Λ = E [ Λ it] [ = E = Φ W1, t Φ W3 t = q 1 p ν Γ q ν k it q ν k Λ it 1, ] Γ k [ ] E Λ it 3 1, ν=1 k=1 Γ Γ k it }{{} Φ W1, t p Γ 1 k 1 + v 1 Γ k it Γ 1 k 1 + v 1 it Γ, k }{{} Φ W3 t k=1 v=1 where t R, i = 1, qν = p ν p q, q = m, and p ν in.1. From this expression we may state that Λ d q 1 ν=1 p ν k=1 where q Y νk Beta ν k, q ν are independent r.v. s. p Y νk / Y kv / and k=1 v=1 k Y kv Beta, v 1 + k 1 are defined 1

13 The hyper-bock matrix sphericity test 13 In order to be abe to buid sharp near-exact distributions for W and Λ, we need to further factorize Φ W1, t and Φ W3 t, by writing each one of these c.f. s as the product of two factors, one that is the c.f. of a Generaized Integer Gamma GIG distribution and the other the c.f. of a sum of independent r.v. s whose exponentias have Beta distributions The factorization of the characteristic function of W 1, = og Λ 1, The resuts in [7, 14] may be used to show that Φ W1, t in 3.1 may be written as Φ W1, t = Φ W1,,a t Φ W1,,b t, where 3. Φ W1,,a t = p k r1,k k k=3 r1,k it is the c.f. of the sum of p independent integer Gamma r.v. s, that is a Generaized Integer Gamma distribution of depth p, with integer shape parameters r 1,k given by 3.3 r 1,k = { h1,k + 1 k k k = 3, 4 r 1,k + h 1,k, k = 5,..., p with k = m, where m is the number of sets of variabes with an odd number of variabes, among the q groups of variabes, the ν-th of which with p ν variabes, and and h 1;k = # of p ν ν = 1,.., q k 1, k = 1,..., p 3.4 Φ W1,,b t = Γ 1 Γ Γ it Γ 1 it is the c.f. of the sum of k independent r.v. s with Logbeta distributions mutipied by. We shoud note that when k = 0, Φ W1,,b t = 1. k 3.. The factorization of the characteristic function of W 3 = og Λ 3 1, as Based on the resuts in [14] we may re-write the c.f. of W 3, Φ W3 t in 3.1, Φ W3 t = Φ W3,a t Φ W3,b t

14 14 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques where 3.5 Φ W3,a t = is the c.f. of the sum of m k= p Integer Gamma distribution of depth m r3; given by B.1 in Appendix B, and p / Φ W3, t = k=1 v=1 3.6 with 3.7 a k = k, b kv and 3.8 a p = p is the c.f. of the sum of p k r 3,k 1 k r it 3,k 1 m independent Gamma r.v. s, that is, a Generaized p m with integer shape parameters Γa k +b kv Γa k +b it vk Γa k +b kv Γa k +b it vk Γ a p +b p v Γ a p +b p v it v=1 Γ a p +b p v Γ a p +b p v it v k = k 1 +, b vk = b vk, b p v = p p +v 1 m, b p v = b p v, p mod p independent Logbeta r.v. s mutipied by and m p mod independent Logbeta r.v. s mutipied by. As such, the c.f. of W may be written as 3.9 Φ W t = Φ 1 t Φ t where 3.10 Φ 1 t = Φ W1,,a t Φ W3,a t, with Φ W1,,a t and Φ W3,a t given by 3. and 3.5, respectivey, and 3.11 Φ t = Φ W1,,b t Φ W3,b t with Φ W1,,b t and Φ W3,b t in 3.4 and 3.6, respectivey. The c.f. Φ 1 t in 3.10 can be seen as the c.f. of a GIG distribution of depth p 1, and it may be written as 3.1 Φ 1 t = p k k= r + k k r + it k

15 The hyper-bock matrix sphericity test 15 where 3.13 r + k = r 1,k + m with r 3,k, 3.14 r 1,k = { 0 k = r 1,k k = 3,..., p and r 3,k = { r 3,k k =,..., p 0 k = p,..., p with r 1,k given by 3.3 and r3, given by B.1 in Appendix B, whie Φ t is the c.f. of a sum of k + m p independent Logbeta r.v. s mutipied by and m p mod independent Logbeta r.v. s mutipied by. From this aternative expression for the c.f. of W = og Λ given by 3.9, we may see that the exact distribution of Λ in.11 may be written as Λ d { p k= e Z k } { k } Y 1,k k=1 { k Y v=1 3,v p / k=1 v=1 } p mod Y 3,kv where d means equivaent in distribution and a the r.v. s invoved are independent, with Z k Γ r + k, k, k =,..., p Y 1,k Beta, 1, k = 1,..., k 3.15 Y3,kv Beta a k + b kv, b kv p b kv, k = 1,..., ; v = 1,..., ; Y3,v Beta a p + b p v, b p v b p v, = 1,..., m with r + k given by 3.13 and 3.14, a k, b kv and b p v given by 3.8. and b kv given by 3.7 and a p, b p v This representation wi enabe us to deveop we-fitting near-exact distributions, which bear an extreme coseness to the exact distribution of Λ.

16 16 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques 4. EAR-EXACT DISTRIBUTIOS FOR W AD Λ To buid the near-exact distributions of W = og Λ and Λ we wi eave Φ 1 t in 3.9 and 3.1 unchanged and we wi repace Φ t in 3.9 and 3.11 by a sharp asymptotic approximation in such a way that the resuting c.f. corresponds to a known manageabe distribution. From the resuts in Section 5 of [3], which show that we may asymptoticay repace a Logbetaa, b distribution by an infinite mixture of Γb + j, a distributions, with j = 0, 1,..., using a somewhat heuristic approach, we wi repace Φ t by 4.1 Φ t = π j θ r+j θ it r+j, m + j=0 which is the c.f. of a finite mixture of Γr + j, θ distributions, where 4. r = k + m = k + m p / k=1 v=1 v k p +1 k 1 v k + m k v=1 v p 1 v p 1 p mod is the sum of a the second parameters of the Beta r.v. s in 3.15 and θ is obtained, together with s 1, s and π, as the numerica soution of the system of equations d h 4.3 dt h Φ t = dh t=0 dt h π θ s 1 θ it s π θ s θ it s, t=0 h = 1,..., 4, that is, as the rate parameter of a mixture of two Gamma distributions with a common rate, which matches the first 4 derivatives of Φ t, at t = 0, so that Φ 1 t π θ s 1 θ it s 1 +1 π θ s θ it s corresponds to a distribution that matches the first 4 exact moments of W. Then the weights π j, j = 0,..., m + 1 are determined in such a way that d h 4.4 dt h Φ t = dh t=0 dt h Φ t, h = 1,..., m +, t=0 with π m + = 1 m + 1 j=0 π j.

17 The hyper-bock matrix sphericity test 17 We wi thus take as near-exact c.f. of W the c.f. 4.5 Φ W t = Φ 1t Φ t { p k = = k= r + k { m + π j θ r+j θ it r+j j=0 k r + } it k m + π j θ r+j θ it r+j j=0 p k k= r + k k r + } it k with r and r + k respectivey given by 4. and 3.13, which is the c.f. of a mixture of m + +1 GIG distributions of depth p that matches the first m + exact moments of W. This c.f. yieds near-exact distributions for W with p.d.f. 4.6 f W w = m+ π j f w GIG r +, r+ 3,..., r+ p, r; and c.d.f. j=0 4.7 F W w = m+ π j F w GIG r +, r+ 3,..., r+ p, r; j=0 for w > 0, and near-exact distributions for Λ with p.d.f. 4.8 f Λ z= m+ π j f GIG og z r +, r+ 3,..., r+ p, r; and c.d.f. j=0, F Λ z= m+ π j 1 F GIG og z r +, r+ 3,..., r+ p, r; for 0 < z < 1. j=0, 3, 3 p,...,,, θ; p p,...,,, θ; p p,...,, θ; p 1 z,, 3 p,...,,, θ; p The modues for the GIG c.d.f. and p.d.f. are avaiabe in [1] and on the web-page Using these modues, the computation of the p.d.f. s and c.d.f. s of the near-exact distributions becomes easy and very manageabe, once the system of equations in 4.4 is inear and as such very simpe to sove, as it is aso the case with the system of equations in 4.3. The authors make avaiabe a set of Mathematica R modues to impement the computation of p.d.f, c.d.f., p-vaues and quanties for the near-exact distributions deveoped in the paper, as we as a modue to compute the vaue of the.r.t. statistic from a sampe, on the web-page In Appendix C the authors present a short manua for the use of these modues, aong with some exampes.

18 18 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques 5. UMERICAL STUDIES In order to assess the performance of the near-exact distributions obtained in the previous section we wi use the measure 5.1 = 1 + Φ W t Φ W t π t dt, with max w F W w FW w = max F z Λ z FΛz, where Φ W t and Φ W t represent respectivey the exact and near-exact c.f. s of W ; F W, FW, the exact and near-exact c.d.f. s of W and F Λ and F Λ those of Λ. Vaues for this measure, which is therefore an upper bound on the difference between the exact and near-exact c.d.f. s of both W and Λ for the near-exact distributions deveoped in the previous section, for different vaues of p and, may be anayzed in Tabes 1 and, with smaer vaues of the measure indicating even better agreements between the near-exact and the exact distributions. From Tabes 1 and we may see the cear asymptotic behavior of the nearexact distributions not ony for increasing sampe sizes but aso for increasing vaues of p and, as we as the very good performance of the near-exact distributions for very sma sampe sizes, which barey exceed the number of variabes in use. This asymptotic behavior is more marked for the near-exact distributions that match more exact moments. This may be seen from the more accentuated decrease in the vaues of the measure for these near-exact distributions, that is, e.g. increases either in sampe size or in the number of variabes make the vaues of the measure to decrease more for the near-exact distributions that match 10 exact moments than for those that match 6 exact moments. It is interesting to note that even near-exact distributions that match a very sma number of exact moments or even no exact moment, and that, as such, are much simper in their structure, and faster to compute, exhibit these asymptotic behaviors, with the behavior of the near-exact distribution that matches no exact moment being absoutey remarkabe. This atter one is a very simpe near-exact distribution, for the computation of which we do not even need to sove the system of equations in 4.4. In this case we wi have m + = 0, and from it is easy to see that the near-exact distribution is just a GIG or a GIG distribution, according to r in 4. being integer or not. 6. POWER STUDIES In order to try to assess the behavior of the test under the aternative hypothesis, some power studies, based on simuations, were carried out. These

19 The hyper-bock matrix sphericity test 19 Tabe 1: Vaues of the measure in 5.1 for different vaues of m + the number of exact moments matched by the near-exact distributions and for m = 4 and increasing vaues of, p and, for base vaues of = {3,, 3, 4}, p = {3, 5, 6, 4} and sampe sizes = p +, 00, 500, with p = m p , p + 0, p = 53 p p p , p + 0, p = 89 p p p , p + 0, p = 143 p p p , p +, p = 19 p p p , p + 5, p = 189 p p p , p +, p = 07 p p p , p + 5, p = 303 p p p p m + studies focused on two forms of vioation of the nu hypothesis: i the vioation of the equaity of the matrices inside each bock of of these matrices and ii the vioation of the bock-independence inside each group of p = p variabes see 1.. First of a we shoud bring to the attention of the reader the fact that we are working with a random vector where in turn, for = 1,..., m, X = [ X 1,..., X m], X = [ X 1,..., X ],

20 0 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques Tabe : Vaues of the measure in 5.1 for different vaues of m + the number of exact moments matched by the near-exact distributions and for m = 5 and increasing vaues of, p and, for base vaues of = {3,, 3, 4, 4}, p = {3, 5, 6, 4, 5} and sampe sizes = p +, 00, 500, with p = m p , p + 0, p = 73 p p p , p + 0, p = 119 p p p , p + 0, p = 188 p p p , p +, p = 171 p p p , p + 5, p = 49 p p p , p +, p = 70 p p p , p + 5, p = 393 p p p p m + with X j p µ j, j = 1,..., for some positive-definite matrix and some rea p 1 vector µ, and with j Cov X j, X j = 0p p for either = or, with j j if =. To keep things not too much invoved, mainy in terms of easiness of exposition and to restrain the number of possibe scenarios, whie at the same time being abe to give a view of a quite wide variety of situations under the aternative

21 The hyper-bock matrix sphericity test 1 hypothesis, we considered a case with m = and k 1 = and k = 3 with p 1 = 5 and p =, with 1 1/ 1/3 1/4 1/5 1/ /3 /4 /5 [ ] 1 1/ 1 = 1/3 /3 3 3/4 3/5 and =, 1/ 1/4 /4 3/4 4 4/5 1/5 /5 3/5 4/5 5 where the choice of 1 and did not obey to any other particuar criteria than that of being two positive-definite matrices. In the next two subsections we wi perform power studies for the cases of vioation of the hypothesis of equaity of the diagona bocks within each bock of matrices and the hypothesis of independence generating for each scenario pseudo random sampes of size Vioation of the equaity hypothesis In order to impement the vioation of the hypothesis of equaity of the diagona bocks inside each I k boc = 1,, we considered 40 different scenarios, with Σ covariance matrices of the form δ δ δ δ δ 3 with δ 11, δ 1, δ 1, δ and δ 3 assuming the vaues in Tabe 3. In this Tabe are aso defined the vaues for δ1 and δ. These new parameters summarize in a singe vaue, respectivey, the variabiity of the combinations of the vaues of δ 11, δ 1 and δ 1, δ, δ 3. The vaues of δ1 and δ in Tabe 3 are defined based on the rank of the vaues of, i=1 1/δi 1/ δ = 1, ; k =, 3, since this is for our purpose a more adequate measure of dispersion of the vaues of δ 1j and δ j j = 1,..., ; = 1, than the usua variance. We wi see that with this choice for the definition of the vaues of δ1 and δ, the power of the test wi be an increasing function of the vaues of both δ1 and δ. The vaues δ 11, δ 1 and the vaues δ 1, δ, δ 3 were indeed chosen in such a way that they woud generate a wide range of vaues of δ1 and δ that coud show how the power of the test behaves for this variety of vaues. We shoud remark that for δ1 = 1 and δ = 1 we are under the nu hypothesis in 1., whie for any other combination

22 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques of vaues of δ1 and δ we wi be under various forms of the aternative hypothesis due to the fact that for vaues of δ1 different from 1, the nu hypothesis H1 03 1, in.5 is vioated, since in these cases we have V arx 11 V arx 1, whie for δ 1 it is the hypothesis H 03 1, in.5 that is vioated, since for δ 1 we have at east two of V arx 1, V arx or V arx 3 different. Increasing vaues of either δ1 or δ indicate a arger departure from H 0 in 1.. Tabe 3: Definition of the vaues for δ 1 and δ. δ 1 δ 11 δ 1 δ δ 1 δ δ / 3 1/ 1 4 1/3 4 1/ 1/ 5 1/ / / /3 1/3 8 1/3 1/3 3 In Tabes 4 and 5 we may anayze the power vaues for different vaues of δ1 and δ, respectivey for α = 0.05 and α = We may note how for δ1 = δ = 1, situation in which we are under the nu hypothesis H 0 in 1., we obtain a vaue for power which coincides with the α vaue, showing the unbiasedness characteristic of the test. We may aso see how power has a good rate of convergence towards 1 for increasing vaues of δ1 and δ. Tabe 4: Power vaues, rounded to three decima paces, for α = 0.05 and different vaues of δ 1 and δ. δ δ Tabe 5: Power vaues, rounded to three decima paces, for α = 0.01 and different vaues of δ 1 and δ. δ δ

23 The hyper-bock matrix sphericity test 3 In Figure 4 we present smoothed surface and ine pots of the power vaues for the cases considered in Tabes 4 and 5. α = 0.05 a b α = 0.01 a b Figure 4: a Smoothed surface pots and b non-smoothed profie pots for different vaues of δ 1 and running vaue of δ, for the vaues of power for the vioation of the hypothesis of equaity of diagona bocks within each I k boc = 1,. 6.. Vioation of the independence hypothesis To impement the vioation of the independence hypothesis, we consider 65 different scenarios with covariance matrices of the form 1 γ 1 C γ 1 C γ 1 C γ C 0 0 γ 1 C γ 3 C 0 0 γ C γ 3 C

24 4 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques where C 1 = and C = 1 10 [ ] 1, 3 and where γ 1 assumes the vaues 0.0, 1.0, 1.5, 1.75 and 1.95 and γ 1, γ and γ 3 assume the vaues in Tabe 6. Whie for γ 1 = 0 we have the hypothesis H0 1 1, of independence between X 11 and X 1, confirmed, for vaues of γ 1 different from zero we wi be under the aternative hypothesis, since then the independence between these two sets of variabes wi be vioated, with increasing vaues of γ 1 indicating an increasing non-independence of these two sets of variabes, or equivaenty, decreasing vaues of the determinant of the matrix [ ] 1 γ 1 C 1 Σ 1 =. γ 1 C 1 1 In what concerns the vaues of γ 1, γ and γ 3, we wi have the hypothesis of independence among X 1, X and X 3 confirmed when a these three parameters are equa to zero, and we wi be under the aternative hypothesis if at east one of them is different from zero, with γ 1 0 = CovX 1, X = γ 1 C 0, γ 0 = CovX 1, X 3 = γ C 0, γ 3 0 = CovX, X = γ 3 C 0. In order to define a hierarchy of the tripets of vaues of these three parameters, we compute the determinant of the matrix γ 1 C γ C Σ = γ 1 C γ 3 C. γ C γ 3 C Vaues for Σ, as a function of the vaues of γ 1, γ and γ 3 are shown in Tabe 6. These vaues are isted in decreasing order of Σ and they are used to define the vaues of the new parameter γ, with increasing vaues of γ corresponding to decreasing vaues of Σ. The parameter γ is then used ahead in Tabes 7 and 8 and Figure 5. Then, whie for γ = 1 we wi be under the nu hypothesis H0 1 of independence among the sets of variabes X 1, X and X 3, for increasing vaues of γ we wi be increasingy further away from this nu hypothesis. We may see by ooking at Tabes 7 and 8 how the vaues for power give the vaue of α for γ 1 = 0 and γ = 1, situation in which we are under the nu

25 The hyper-bock matrix sphericity test 5 Tabe 6: Definition of the vaues of γ for the different vaues of the parameters γ 1, γ and γ 3. γ γ γ Σ γ Tabe 7: Power vaues, rounded to three decima paces, for α = 0.05 and different vaues of γ 1 and γ. γ 1 γ Tabe 8: Power vaues, rounded to three decima paces, for α = 0.01 and different vaues of γ 1 and γ. γ 1 γ hypothesis H 0 in 1., whie for a other combinations of vaues of these two parameters the vaue of power increases as the vaues of γ and/or γ 1 increase, easiy reaching 1. In Figure 5 we present smoothed surface and ine pots of the power vaues for the cases considered in Tabes 6 and 7. From the pots in this Figure and the vaues in Tabes 7 and 8 we may see how power attains the vaue 1 for the arger vaues of γ 1 and γ, as expected. We may aso note that as in the case of the previous subsection, for γ 1 = 0 and γ = 1, in which case we are under the nu hypothesis, the vaue of the power equas the α vaue considered, showing again the unbiasedness characteristic of the test.

26 6 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques α = 0.05 a b Γ Γ Γ Γ Γ Γ α = 0.01 a b Γ Γ Γ Γ Γ Γ Figure 5: a Smoothed surface pots; and b non-smoothed profie pots, for the different vaues of γ 1 and running vaue of γ, for the vaues of power in Tabes 7 and COCLUSIOS The procedure deveoped in this paper makes it possibe to test eaborate covariance structures such as the HBM spherica structure through the use of very precise near-exact approximations. The testing procedure is based on an adequate decomposition of the overa nu hypothesis into a sequence of subhypotheses, in our case the ones used to test the independence of severa groups of variabes and the equaity of severa covariance matrices in different sequences of covariance matrices. This decomposition of the nu hypothesis aows us to obtain the ikeihood ratio test statistic, the expression of its h-th moment and the expression of the characteristic function of its ogarithm. Furthermore, the suitabe decomposition of the nu hypothesis aso induces a factorization of this characteristic function which is the basis for the deveopment of the near-exact approximations. These approximations can be easiy impemented since there are aready computationa modues avaiabe in the internet for the two main distributions invoved, which are the GIG and GIG distributions.

27 The hyper-bock matrix sphericity test 7 The high precision of the near-exact distributions, which was assessed in the numerica studies section, makes them an efficient too to obtain p-vaues and quanties for the test statistic, even in cases where the sampe size is very sma and/or the number of variabes is arge. Power studies conducted through simuations show the unbiased nature of the test as we as its good power properties, reaching rapidy powers cose to 1 in the different scenarios considered. The procedure deveoped may be very usefu to address other, eventuay, more compex structures. A natura extension of this framework is to consider the same goba structure but for compex orma random variabes or even for quaternion random variabes. Other possibe extension is to consider specific structures for the bock-diagona covariance matrices such as the circuar or the compound symmetry structures. ACKOWLEDGMETS This research was supported by ationa Funds through FCT Fundação para a Ciência e a Tecnoogia, project UID/MAT/0097/013 CMA/UL. The authors woud aso ike to thank the referee and the Associate Editor for their comments which heped in improving the readabiity of the paper. APPEDICES APPEDIX A Obtaining the expression of the.r.t. statistc Λ, associated with the nu hypothesis H 0 in 1., by using the definition of ikeihood ratio statistic Let us consider the vector X p µ, Σ and et us suppose that we have a sampe of size from X. The.r.t statistic Λ associated with the HBM sphericity test is defined by A.1 Λ = sup L 0 sup L 1 where L 0 is the ikeihood function when the parameter space is under H 0 in 1. and L 1 is the ikeihood function under the aternative hypothesis. The ikeihood function associated with the sampe is A. L x 1,..., x ; µ; Σ = 1 e 1 tr X E1 µ T Σ 1 X E 1 µ T T, π p Σ

28 8 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques where E rs denotes a matrix of 1 s of dimension r s. Let then L 0 = L 0 x1,..., x ; µ; Σ Ho = og L0 x1,..., x ; Σ Ho. From A. we have L 0 = p og π og Σ Ho 1 X tr E1 x T Σ 1 H o X E1 x T T 1 tr E1 x T E 1 µ T Σ 1 H o E1 x T E 1 µ T T. As X tr E1 x T Σ 1 H o X E1 x T T = tr Σ 1 H o A where A = X E 1 x T T X E1 x T = X T X 1 X T E X, and E1 tr x T E 1 µ T Σ 1 H E1 o x T E 1 µ T T = tr the function L 0 can be written as Σ 1 H o x µ x µ T = x µ T Σ 1 H o x µ L 0 = p og π og Σ Ho 1 [ ] tr AΣ 1 H o 1 x µ T Σ 1 H o x µ. Given that Σ Ho = bdiag I k, = 1,..., m = m I k p = m = m I k where is a matrix of order p, and 1 Σ 1 H o = bdiag I k, = 1,..., m = bdiag I k 1, = 1,..., m = bdiag I k 1, = 1,..., m with tr AΣ 1 H o = tr A. bdiag I k 1, = 1,..., m = m tr A Ik 1 = m tr A v 1, v=1 we can write L 0 as L 0 = p og π og 1 m tr A v 1 v=1 1 x µ T Σ 1 H o x µ.

29 The hyper-bock matrix sphericity test 9 By soving the system of ikeihood equations L 0 µ = 0 x µ Σ µ= µ 1 H o = 0 L 0 = 0, = 1,..., m = µ= µ µ = x k A v 1 = 0, v=1 = 1,..., m µ = x = 1, = 1,..., m A v v=1 we obtain the maximum ikeihood estimators of µ and Σ under H 0, which are, µ = X and Σ = I k I km m = I k1 1 k 1 A I km 1 k m A m where A = A v v=1 = 1,..., m. A.3 Then we have sup L 0 X1,..., X ; µ; Σ Ho = π p { = π p p = π p p = π p p = π p p = π p p = π p p k p k p k p k p k p k } e 1 tra.bdiagi p A e A e A e A e A e m m 1,,...,m tr A I k A 1 v=1 m m m A e p. p tr A v A 1 tr A A 1 tr I p

30 30 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques Under H 1, the ikeihood function is given by L 1 x1,..., x ; µ; Σ = π p Σ e 1 tr X E1 µ T T X E 1 µ T Σ 1 = π p Σ e 1 traσ 1 and L 1 = L 1 x1,..., x ; µ; Σ = og L 1 x1,..., x ; µ; Σ is given by L 1 = p og π + og Σ 1 tr [ AΣ 1] 1 tr x µ T x µ Σ 1. By soving the system of ikeihood equations L 1 µ = 0 µ= µ x µ Σ 1 = 0 L 1 Σ = 0 Σ Σ 1 A Σ 1 = 0 Σ= Σ µ= µ µ = x Σ = 1 A we concude that A.4 sup L 1 X1,..., X ; µ; Σ = π p Σ e 1 tra Σ 1 = π p 1 A e 1 traa 1 = π p p A e 1 trip = π p p A e 1 p A.5 Then, from A.1, A.3 and A.4 we have Λ = sup L 0 x ; µ; Σ Ho sup L 1 x ; µ; Σ = = π p p { k p } { k p } { A π p p A e p A A } e p where the matrix A is the maximum ikeihood estimator of Σ, A is the -th diagona bock of order p = p of A = 1,..., m, with p = m p and A = A1 + + A, where Av is the v-th v = 1,..., diagona bock of order p of A. We shoud note how expression A.5 is the same as expression.11.

31 The hyper-bock matrix sphericity test 31 APPEDIX B Shape parameters The shape parameters r3; in 3.5 are given by B.1 r3, = where and with r, k =1,..., p 1, k p 1 α 1 r +p mod α α 1 p 1 p +, k =p 1 α 1 α = p 1, α 1 = k 1 p 1, α = k 1 p +1, c k, k = 1,..., α + 1 p k r =, k = α +,..., min p α 1, p 1 and k = + p α 1,..., p 1, step p +1 k, k = 1 + p α 1,..., p 1, step c k = k k mod for k = 1,..., α, and p c α+1 = α k p + + mod α p k + k α + 1 k + k mod, α mod α + α. 4 4 For the derivation of the expressions for these parameters see [14] and references therein, making the necessary adjustments. APPEDIX C The Mathematica R modues The modues described in this Appendix are avaiabe at the web-page and may be downoaded from this web-page.

32 3 Bárbara R. Correia, Caros A. Coeho and Fiipe J. Marques C.1 - Computation of the p.d.f. and c.d.f. of the near-exact distributions The modues made avaiabe for the computation of the p.d.f. and c.d.f. of the near-exact distributions for Λ are caed, respectivey, Epdf and Ecdf. These modues have 4 mandatory arguments, which are: the sampe size, a ist with the vaues of p = 1,..., m, a ist with the vaues of = 1,..., m, the running vaue where the p.d.f. or c.d.f. is to be computed, and which have to be given in this order; and 5 optiona arguments, which are: nm: the number of exact moments to be matched by the near-exact distribution, that is, the vaue of m + in 4.1 and defaut vaue: 4, prec1: the number of digits used to print the vaue of the p.d.f. or c.d.f. defaut vaue: 10, prec: the number of precision digits used in the computation of the p.d.f. or c.d.f. defaut vaue: 00, prec3: the number of precision digits used to store the m + exact moments of W = og Λ computed defaut vaue: 00, prec4: number of precision digits used in the computation of the m + exact moments of W = og Λ by the modue that does this computation defaut vaue: These optiona arguments may be given in any order, but they wi have to be caed by their names, as it is exempified beow. If not used, they wi assume their defaut vaues. These modues use a number of other modues avaiabe on the same webpage which compute the weights π j and the rate parameter θ in 4.1, the shape and rate parameters in Φ 1 t as we as other shape and rate parameters invoved in the expressions of the near-exact p.d.f. and c.d.f.. The modue that computes the weights π j uses another modue which computes the exact moments of W = og Λ by appying a numerica method to the exact c.f. of W in 3.1. For exampe to compute the near-exact c.d.f. of Λ, on a vaue near the 0.05 quantie, for a case with the same parameters as those used for the exampes for which we computed power in Section 6, which was a case with m =, p 1 = 5, p =, k 1 = and k = 3, using the defaut vaues for a optiona arguments, we woud use the first command in Figure 6. The second command in that same figure uses the option prec1 in order to obtain an output with more digits. The options named prec, prec3, and prec4, wi usuay not be necessary, uness one suspects from ack of precision in the resut obtained, which may happen in cases where the number of variabes or the sampe size are very arge. This fact is iustrated with the third command in Figure 6, where athough 500 precision digits are requested for the interna representation of the exact moments of W, the resut obtained is exacty the same as the one obtained with the second