Many-sample tests for the equality and the proportionality hypotheses between large covariance matrices

TL;DR

This paper develops asymptotically normal tests for comparing multiple large covariance matrices, with applications to transposable data and real datasets, demonstrating satisfactory finite sample performance.

Contribution

It introduces new procedures for testing equality and proportionality of many large covariance matrices under high-dimensional asymptotics.

Findings

Test statistics are asymptotically normal under the growth scheme.

Finite sample simulations show good performance.

Application to real datasets demonstrates practical utility.

Abstract

This paper proposes procedures for testing the equality hypothesis and the proportionality hypothesis involving a large number of $q$ covariance matrices of dimension $p\times p$. Under a limiting scheme where $p$, $q$ and the sample sizes from the $q$ populations grow to infinity in a proper manner, the proposed test statistics are shown to be asymptotically normal. Simulation results show that finite sample properties of the test procedures are satisfactory under both the null and alternatives. As an application, we derive a test procedure for the Kronecker product covariance specification for transposable data. Empirical analysis of datasets from the Mouse Aging Project and the 1000 Genomes Project (phase 3) is also conducted.