A Simultaneous Test for Mean and Variance based on the Likelihood Ratio Principle

In this study, we consider to propose a simultaneous test procedure for the mean and variance under the normality assumption for the one-sample problem. With formulating the null hypothesis and its alternative, we derive the jointly optimal test statistics for the mean and variance using the likelihood ratio principle and combine the individual tests into a single simultaneous test using the Tippett type of combining function. Also we consider a similar test procedure which uses two likelihood ratio tests for each individual parameter by combining them with Tippett function. Then we compare our proposed test with this individually optimal test by obtaining empirical powers through a simulation study. Also we include the individual tests for this comparison study under the scenario that only one parameter value varies. Then we discuss some interesting aspects for the proposed procedure and quantile points of the chi-square distributions. Also we review several types of combining functions as concluding remarks. Finally in Appendix, we derive the correlation coefficient between two optimal individual statistics briefly.