Abstract
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 Qi (Scand J Stat 42:988–1009, 2015) and Jiang and Yang (Ann Stat 41(4):2029–2074, 2013) 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.
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Acknowledgements
The authors would like to thank the Editor, Associate Editor, and referees for reviewing the manuscript and providing valuable comments. The research of Yongcheng Qi was supported in part by NSF Grant DMS-1916014.
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Guo, W., Qi, Y. Asymptotic distributions for likelihood ratio tests for the equality of covariance matrices. Metrika 87, 247–279 (2024). https://doi.org/10.1007/s00184-023-00912-6
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DOI: https://doi.org/10.1007/s00184-023-00912-6