On tests detecting difference in means and variances simultaneously under normality

In this paper, we propose several tests for detecting difference in means and variances simultaneously between two populations under normality. First of all, we propose a likelihood ratio test. Then we obtain an expression of the likelihood ratio statistic by a product of two functions of random quantities, which can be used to test the two individual partial hypotheses for differences in means and variances. With those individual partial tests, we propose a union-intersection test. Also we consider two optimal tests by combining the p-values of the two individual partial tests. For obtaining null distributions, we apply the permutation principle with the Monte Carlo approach. Then we compare efficiency among the proposed tests with well-known ones through a simulation study. Finally, we discuss some interesting features related to the simultaneous tests and resampling methods as concluding remarks.