A Note on the Two-Sample Comparison Tests under Normality

We propose the likelihood ratio test for the two-sample problem with the assumption of equality of unknown variances. Also we consider a test without the assumption of equality of variances. For obtaining the null distributions of the proposed test statistics, we apply the permutation principle, which is a resampling method, with the Monte-Carlo approach since the derivations of the null distributions with a theoretical manner would be difficult. We state the procedure for the use of the permutation principle in some detailed fashion by describing step-by-step. Also we compare their efficiency and investigate their behaviors for some famous and widely used tests such as the two-sample T-test by getting empirical powers through a simulation study under various scenarios such as the means and/or variances may be different. Then we emphasize our outstanding observations from the simulation study. Furthermore we discuss some interesting aspects and state briefly peculiar phenomena related with the likelihood ratio tests and well-known ones.