Preliminary Multiple-Test Estimation, With Applications to k-Sample Covariance Estimation

Abstract

Multisample covariance estimation—that is, estimation of the covariance matrices associated with k distinct populations—is a classical problem in multivariate statistics. A common solution is to base estimation on the outcome of a test that these covariance matrices show some given pattern. Such a preliminary test may, for example, investigate whether or not the various covariance matrices are equal to each other (test of homogeneity), or whether or not they have common eigenvectors (test of common principal components), etc. Since it is usually unclear what the possible pattern might be, it is natural to consider a collection of such patterns, leading to a collection of preliminary tests, and to base estimation on the outcome of such a multiple testing rule. In the present work, we therefore study preliminary test estimation based on multiple tests. Since this is of interest also outside k-sample covariance estimation, we do so in a very general framework where it is only assumed that the sequence of models at hand is locally asymptotically normal. In this general setup, we define the proposed estimators and derive their asymptotic properties. We come back to k-sample covariance estimation to illustrate the asymptotic and finite-sample behaviors of our estimators. Finally, we treat a real data example that allows us to show their practical relevance in a supervised classification framework.

KEYWORDS:

• Covariance matrix estimation
• Local asymptotic normality
• Multisample problems
• Preliminary test estimation
• Valid post-selection inference

Supplementary Materials

In the supplementary materials, we perform finite sample comparisons of PMTEs with estimators resulting from a BIC-based model selection and provide the technical proofs of the results.