Asymptotic distributions for likelihood ratio tests for the equality of covariance matrices

Wenchuan Guo and Yongcheng Qi ()
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Wenchuan Guo: Bristol Myers Squibb
Yongcheng Qi: University of Minnesota Duluth

Metrika: International Journal for Theoretical and Applied Statistics, 2024, vol. 87, issue 3, No 2, 247-279

Abstract: 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.

Keywords: Likelihood ratio test; Central limit theorem; Multivariate normal distribution; Multivariate gamma function (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00184-023-00912-6

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