Optimal tests for homogeneity of covariance, scale, and shape

Abstract

a b s t r a c tThe 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 fourth-order 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 m-tuple of radial densities f = (f 1 , . . . , f m ). Combined with an estimation of the m densities f 1 , . . . , f m , these procedures can be used to construct adaptive tests for the problem. Adaptive tests however typically require very large samples, and pseudo-Gaussian 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 pseudo-Gaussian modifications of the Gaussian version φ (n) N of the optimal test φ (n) f . The first one, φ (n) N * , is valid under the class of homokurtic m-tuples f , while the validity of the second, φ (n) N Ď , extends to the heterokurtic ones, that is, to arbitrary m-tuples of elliptical distributions with finite fourth-order moments. We moreover show that these tests are asymptotically equivalent to modified Wald tests recently proposed by Schott [J.R. Schott, Some tests for the equality of covariance matrices, Journal of Statistical Planning and Inference 94 (2001) [25][26][27][28][29][30][31][32][33][34][35][36]. This settles the optimality properties of the latter. Our results however are much more informative than Schott's. They also allow for computing local powers, and for an ANOVAtype decomposition of the test statistics into two mutually independent parts providing tests against subalternatives of scale and shape heterogeneity, respectively, thus supplying additional insight into the reasons why rejection occurs. Reinforcing a result of Yanagihara et al. [H. Yanagihara, T. Tonda, C. Matsumoto, The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption, Journal of Multivariate Analysis 96 (2005) 237-264], we further show why another approach, based on bootstrapped critical values of the Gaussian MLRT statistic, although producing asymptotically valid pseudo-Gaussian tests, is highly unsatisfactory in this context. We also develop optimal pseudo-Gaussian tests for scale homogeneity and for * 423 shape homogeneity, based on the sa...

Intrinsic MANOVA for Riemannian Manifolds with an Application to Kendall's Space of Planar Shapes

“…For our data at hand, neither normality nor unequality could be rejected. In general under nonnormality, tests based on [16] could be employed.…”

Robust asymptotic tests for the equality of multivariate coefficients of variation

“…When the aim is to detect subtle differences in the covariance structure, it is preferable to use tests intended for this purpose, such as those proposed by Bartlett (1937), Schott (2001) or Hallin and Paindaveine (2009) as they are more powerful. However, when relative dispersion is the goal, the proposed tests were shown to be more appropriate.…”

“…Finally, the conceptual distinction between comparing MCVs and testing the homogeneity of covariance matrices is highlighted. Simulated data will be used to compare the results of the MCV tests with those obtained with the optimal tests of covariance homogeneity proposed by Hallin and Paindaveine (2009).…”

“…the well-known Bartlett (Bartlett , 1937) and Schott (Schott , 2001) tests under normality, or the optimal tests proposed by Hallin and Paindaveine (2009) which are valid under any K-tuple of elliptical distributions. In this subsection, we focus on the latter to briefly stress the differences and complementarities between these tests and the MCV ones.…”

“…In the Gaussian-Gaussian case, the tests under consideration are the classical Wald test based on MCV and its inversed version as well as the Gaussian test from Hallin and Paindaveine (2009). In the Student-Student case, the semiparametric Wald test (and its inverse-MCV version) is compared to Hallin and Paindaveine's semi-parametric homokurtic test.…”