Approximation and Quantiles of the Distribution of the Modified Likelihood Ratio Criteria for Covariance Matrix Hypothesis Testing andMonitoring

Sugiura (1969) gives an asymptotic expansion of the modified likelihood ratio criteria for testing the hypothesis that a covariance matrix is equal to a given matrix. An improvement of this expansion is presented here. Numerical comparisons via simulation with the original Sugiura's approximation to the distribution of the criteria confirm the superiority of our expansion. This enable us to use the proposed method in usual hypotheses testing and in applications where extreme tail quantiles are necessary, as for instance, for monitoring dispersion in multivariate processes quality control charts.