Pseudo-Gaussian Inference in Heterokurtic Elliptical Common Principal Components Models

The so-called Common Principal Components (CPC) Model, in which the covariance matrices Σi of m populations are assumed to have identical eigenvectors, was introduced by Flury (1984), who develops Gaussian parametric inference methods for this model (Gaussian maximum likelihood estimation and Gaussian likelihood ratio testing). A key result in that context is the joint asymptotic normality of the Gaussian maximum likelihood estimators of the common eigenvectors and the corresponding eigenvalues. Flury’s derivation of that result is based on a soft argument and is valid under Gaussian (hence, also homokurtic) conditions only. In this paper, we provide a formal proof of the same result under more general assumptions of elliptical, possibly heterokurtic, densities with finite fourth-order moments. This allows for a pseudo-Gaussian solution to all inference problems about eigenvectors and eigenvalues in CPC models. As an application, we consider inference about the proportion of total variance explained by a given subset of common principal components. More precisely, we test the null hypothesis H0 that this proportion is smaller than some fixed value p0 ∈ (0, 1) in each population. Based on our result, we provide a pseudo-Gaussian test which, contrary to Flury’s Gaussian one, is valid under arbitrary m-tuples of elliptical densities with finite fourth-order moments, while remaining asymptotically equivalent to Flury’s under multinormal distributions.