Results 1 
2 of
2
Optimal tests for homogeneity of covariance, scale, and shape
 J. Multivariate Anal
, 2008
"... The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic cont ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic context of multivariate Gaussian models. And the modified likelihood ratio tests (MLRT) that are used in everyday practice are known to be highly sensitive to violations of Gaussian assumptions. In this paper, we provide a complete and systematic study of the problem, and propose test statistics which, while preserving the optimality features of the MLRT under multinormal assumptions, remain valid under unspecified elliptical densities with finite fourthorder moments. As a first step, the Le Cam LAN approach is used for deriving locally and asymptotically optimal testing procedures φ (n) f for any specified mtuple of radial densities f = (f1,..., fm). Combined with an estimation of the m densities f1,..., fm, these procedures can be used to construct adaptive tests for the problem. Adaptive tests however typically require very large samples, and pseudoGaussian tests—namely, tests that are locally and asymptotically optimal at Gaussian densities while remaining valid under a much broader class of distributions—in general are preferable. We therefore construct two pseudoGaussian modifications of the Gaussian version φ (n) N of the optimal test φ (n) f. The first one, φ
srOCHASTIC MAJORIZATION OF THE LOGEIGENVALUES OF A BIVARIATE WISHART MATRIX1.2
"... Let 1 = (l1 ' l2) and A = (A1. A2). where A1;:f; A2> 0 are the ordered eigenvalues of Sand:E, respectively, and S.... Wz(n, E) is a bivariate Wishart matrix. Let m = (mlo m2) and JL = (f.L1. f.L2). where 1T'/i =log 4 and /4. =log Ai ' It is shown that P _fm jl ' B l is Schur ..."
Abstract
 Add to MetaCart
Let 1 = (l1 ' l2) and A = (A1. A2). where A1;:f; A2> 0 are the ordered eigenvalues of Sand:E, respectively, and S.... Wz(n, E) is a bivariate Wishart matrix. Let m = (mlo m2) and JL = (f.L1. f.L2). where 1T'/i =log 4 and /4. =log Ai ' It is shown that P _fm jl ' B l is Schurconvex in J,l whenever B is a Schurmonotone set, i.e. [x E B, x majorizes x"]::;> x · E B. This result implies the unbiasedness and powermonotonicity of a class of invariant tests for bivariate sphericity and other orthogonally invariant hypotheses.