Model-based principal components of covariance matrices

Abstract

A flexible class of models is proposed for principal component (PCs) of covariance matrices. The models allow constraints to be imposed on the eigenvalues and/or the eigenvectors and yield simplified PCs that retain their variance maximization and orthogonality properties. The models are fitted to sample covariance matrices by minimizing a discrepancy function. Asymptotic distributions of estimators are obtained under the assumption that fourth-order moments of the parent distribution are finite. Hypothesis tests are obtained by comparing discrepancy functions that are minimized under different constraints. An Edgeworth expansion is used to obtain second-order accurate confidence intervals for differentiable eigenfunctions. The techniques are illustrated on a real data set.