A note on the test for the covariance matrix under normality

Abstract

In this study, we consider the likelihood ratio test for the covariance matrix of the multivariate normal data. For this, we propose a method for obtaining null distributions of the likelihood ratio statistics by the MonteCarlo approach when it is difficult to derive the exact null distributions theoretically. Then we compare the performance and precision of distributions obtained by the asymptotic normality and the Monte-Carlo method for the likelihood ratio test through a simulation study. Finally we discuss some interesting features related to the likelihood ratio test for the covariance matrix and the Monte-Carlo method for obtaining null distributions for the likelihood ratio statistics.

Tests of equality of several variances with the likelihood ratio principle

“…However it cannot be denied either that the distributions of the LR functions and related statistics have not been fully developed until now. This phenomenon already has been confirmed by Park (2018) for the study of testing procedure for the covariance matrix even for the one-sample problem. If we confess the difficulty for the derivation of the LR statistic, we have found during the preliminary simulation study that the limiting distribution of 2 log LR(σ 2 1 , .…”

“…Sometimes even the derivation of an asymptotic distribution would not be possible or the serious discrepancy of an asymptotic distribution from the unknown distribution of the LR statistic might be detected from a simulation study. Then one may apply the Monte-Carlo method (Park, 2018).…”