2021/10/05 by Wenchuan Guo, Guo, Wenchuan, Yongcheng Qi +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Random Matrices and Applications #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2110.02384
openalex publication_date 2021/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider k independent random samples from p-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of k covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when k and p are fixed integers. Jiang and Yang~\citeJY13 and Jiang and Qi~\citeJQ15 have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimensionality p goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either p or k goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for p and k. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation.