Testing for Common Principal Components under Heterokurticity

Abstract

The so-called common principal components (CPC) model, in which the covariance matrices Σ i of m populations are assumed to have identical eigenvectors, was introduced by Flury [Flury, B. (1984), ‘Common Principal Components in k Groups’, Journal of the American Statistical Association, 79, 892–898]. Gaussian parametric inference methods [Gaussian maximum-likelihood estimation and Gaussian likelihood ratio test (LRT)] have been fully developed for this model, but their validity does not extend beyond the case of elliptical densities with common Gaussian kurtosis. A non-Gaussian (but still homokurtic) extension of Flury's Gaussian LRT for the hypothesis of CPC [Flury, B. (1984), ‘Common Principal Components in k Groups’, Journal of the American Statistical Association, 79, 892–898] is proposed in Boik [Boik, J.R. (2002), ‘Spectral Models for Covariance Matrices’, Biometrika, 89, 159–182], see also Boente and Orellana [Boente, G., and Orellana, L. (2001), ‘A Robust Approach to Common Principal Components’, in Statistics in Genetics and in the Environmental Sciences, eds. Sciences Fernholz, S. Morgenthaler, and W. Stahel, Basel: Birkhauser, pp. 117–147] and Boente, Pires and Rodrigues [Boente, G., Pires, A.M., and Rodrigues I.M. (2009), ‘Robust Tests for the Common Principal Components Model’, Journal of Statistical Planning and Inference, 139, 1332–1347] for robust versions. In this paper, we show how Flury's LRT can be modified into a pseudo-Gaussian test which remains valid under arbitrary, hence possibly heterokurtic, elliptical densities with finite fourth-order moments, while retaining its optimality features at the Gaussian.