Simultaneous multivariate tests under the normality assumption

In this study, we propose several simultaneous tests for the mean vector and covariance matrix of multivariate normal data. First of all, we consider the likelihood ratio (LR) test with a limiting distribution, and estimate one using the Monte-Carlo (MC) method for the null distribution. Next, we consider a union-intersection (UI) test by identifying the LR statistic as a product of two functions of random quantities. In addition, we propose simultaneous tests with combination functions. Then, we illustrate our procedure with an example, and compare the efficiency through a simulation study. Finally, we discuss interesting features for the proposed simultaneous tests with related topics and a method for obtaining the LR statistics.