Abstract
The so-called common principal components (CPC) model, in which the covariance matrices ${\Sigma}$ 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.
