A Note on the Likelihood Ratio Simultaneous Multivariate Tests

In this study, we propose the simultaneous tests for the mean vector and covariance matrix for the multivariate normal data. For this, first of all, we drive the likelihood ratio function and obtain the asymptotic distribution for the two times of the log likelihood ratio function with the likelihood ratio arguments. Then we propose a likelihood ratio simultaneous test for the mean vector and covariance matrix. Also for obtaining the null distribution of the likelihood ratio function, we consider the Monte-Carlo method which depends completely upon the computer facility and its softwares. Then the Monte-Carlo method may yield the exact likelihood ratio simultaneous test. Then we illustrate our procedure with a numerical example and compare efficiency among proposed tests with the combination tests under the various scenarios for the mean vector and covariance matrix by obtaining empirical powers through a simulation study. Finally, we discuss some interesting features for the proposed simultaneous tests with related topics and a method for obtaining the likelihood ratio statistics.