Many-sample tests for the dimensionality hypothesis for large covariance matrices among groups

TL;DR

This paper develops asymptotic tests for the dimension of the linear span of large covariance matrices across multiple groups, with applications to gene data analysis.

Contribution

It introduces new asymptotic normality results for tests on the span dimension of large covariance matrices among multiple groups.

Findings

Test procedures perform well in finite samples.

Application to gene data reveals new covariance structure insights.

Asymptotic normality established under increasing parameters.

Abstract

In this paper, we consider procedures for testing hypotheses on the dimension of the linear span generated by a growing number of $p\times p$ covariance matrices from independent $q$ populations. Under a proper limiting scheme where all the parameters, $q$, $p$, and the sample sizes from the $q$ populations, are allowed to increase to infinity, we derive the asymptotic normality of the proposed test statistics. The proposed test procedures show satisfactory performance in finite samples under both the null and the alternative. We also apply the proposed many-sample dimensionality test to investigate a matrix-valued gene dataset from the Mouse Aging Project and gain some new knowledge about its covariance structures.