A Simultaneous Inference for the Multivariate Data

In this paper, we consider to propose several simultaneous tests for the mean vector and covariance matrix under the multivariate normality assumption for the underlying distribution. For doing this purpose, we consider to apply the likelihood ratio principle to obtain optimal test statistics in an optimal sense. Then we factor out two types of statistics from the likelihood ratio statistic and identify them a product two individual likelihood ratio test statistics such that one is for the test of the mean vector, the other, that of the covariance matrix. Since it would be difficult to obtain the joint distribution of the two types of the individual statistics, we consider to use the combination functions which combine the -values from the individual partial hypothesis test results. Then we derive the distributions for the combination functions under the null hypothesis. Also we illustrate our test procedure with a numerical bivariate example which is obtained for the measurement of head sizes for the first and second sons and compare their results based on the p-values. Then we discuss some interesting features our proposed simultaneous tests with some related topics. Finally we allude some future topics with simultaneous test procedure with covariance matrix.