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

Author

Listed:
  • Wenchuan Guo

    (Bristol Myers Squibb)

  • Yongcheng Qi

    (University of Minnesota Duluth)

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.

Suggested Citation

  • Wenchuan Guo & Yongcheng Qi, 2024. "Asymptotic distributions for likelihood ratio tests for the equality of covariance matrices," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 87(3), pages 247-279, April.
    • Handle: RePEc:spr:metrik:v:87:y:2024:i:3:d:10.1007_s00184-023-00912-6
    DOI: 10.1007/s00184-023-00912-6
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