Testing Kronecker Product Covariance Matrices for High-dimensional Matrix-Variate Data

TL;DR

This paper develops new statistical tests for Kronecker product covariance matrices in high-dimensional matrix-variate data, providing theoretical guarantees and practical bootstrap methods to ensure accurate size control and high power.

Contribution

It introduces novel testing statistics based on linear spectral statistics, proves their asymptotic distribution, and proposes a bootstrap algorithm for practical implementation.

Findings

Test sizes are close to theoretical values in simulations.

Bootstrap method accurately approximates limiting distributions.

Test power approaches one as dimensions increase.

Abstract

Kronecker product covariance structure provides an efficient way to modeling the inter-correlations of matrix-variate data. In this paper, we propose testing statistics for Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. Central limit theorem is proved for the linear spectral statistics with explicit formulas for mean and covariance functions, which fills the gap in the literature. We then theoretically justify that the proposed testing statistics have well-controlled sizes and strong powers. To facilitate practical usefulness, we further propose a bootstrap resampling algorithm to approximate the limiting distributions of associated linear spectral statistics. Consistency of the bootstrap procedure is guaranteed under mild conditions. A more general model which allows the existence of noises will also be discussed. In the simulations, the empirical sizes of the proposed testing procedure and its bootstrapped version are close to corresponding theoretical values, while the powers converge to one quickly as the dimension and sample size grow.