Testing Kronecker product covariance matrices for high-dimensional matrix-variate data

Long Yu, Jiahui Xie and Wang Zhou

Biometrika, 2023, vol. 110, issue 3, 799-814

Abstract: SummaryThe Kronecker product covariance structure provides an efficient way to model the inter-correlations of matrix-variate data. In this paper, we propose test statistics for the Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. A central limit theorem is proved for the linear spectral statistics, with explicit formulas for the mean and covariance functions, thereby filling a gap in the literature. We then show theoretically that the proposed test statistics have well-controlled size and high power. We further propose a bootstrap resampling algorithm to approximate the limiting distributions of the associated linear spectral statistics. Consistency of the bootstrap procedure is guaranteed under mild conditions. The proposed test procedure is also applicable to the Kronecker product covariance model with additional random noise. In our simulations, the empirical sizes of the proposed test procedure and its bootstrapped version are close to the corresponding theoretical values, while the power converges toquickly as the dimension and sample size increase.

Keywords: Bootstrap; Linear spectral statistic; Multivariate analysis; Random matrix theory; Separable covariance model (search for similar items in EconPapers)
Date: 2023
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