Simultaneous Test Procedure for ANOVA with Possibly Unequal Variances

In this study, we propose simultaneous test procedures for the means and variances for the multi-sample problem with the normality assumption for the underlying distribution. For this, first of all, we construct the null hypothesis for the simultaneous test and derive the likelihood ratio function for the multi-sample problem according to the null hypothesis and its alternative. Then we propose the likelihood ratio test based on the likelihood ratio principle with the asymptotic null distribution of the 2 times of the log likelihood ratio function. Also we propose another likelihood ratio test with the applications of the permutation principle, which heavily relies on the computer and its softwares, to obtain the null distribution of the likelihood ratio function. In order to compare the performance of the proposed likelihood ratio tests with the F-test for the one-way ANOVA model, we conduct a simulation study and compare their efficiency through empirical p-values under the various scenarios for the means and variances for the several symmetric distributions. Then we discuss some interesting features related with the simultaneous tests and state briefly our future research topic as concluding remarks.