Testing Kronecker product covariance matrices for high-dimensional matrix-variate data
针对矩阵型数据的Kronecker积协方差结构，提出基于线性谱统计量的检验方法，证明中心极限定理并给出均值协方差显式公式，同时引入bootstrap算法逼近极限分布，模拟显示检验效果良好。
Summary The 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 to $1$ quickly as the dimension and sample size increase.