Abstract
This article deals with the hypothesis test for the homogeneity of two normal populations. The likelihood ratio test is a usual method for such testing. In general theory, the asymptotic distribution of log-likelihood ratio statistics as testing statistics is well known under the null hypothesis, so we may design the test based on the asymptotic distribution. On one hand, the asymptotic distribution of log-likelihood ratio statistics is also considered under the alternative hypothesis. However, in the case that sample sizes are not so large, the asymptotic distributions are not so accurate under the arbitrary hypotheses. Recently, Arizono et al. have derived the cumulant generating function for the log-likelihood ratio statistics under the null hypothesis when the sample size is finite, and enabled to design accurately the null hypothesis test for homogeneity of two normal populations. However, the stochastic properties of log-likelihood ratio statistics under the alternative hypothesis and the power of testing have not been investigated in theory. Therefore, we consider the stochastic properties of log-likelihood ratio statistics under arbitrary hypotheses in this article. Concretely, the cumulant generating function for the log-likelihood ratio statistics and the cumulant of arbitrary order are derived under the arbitrary hypotheses when the sample size is finite. Further, by using the cumulants of log-likelihood ratio statistics derived in this article, we develop the approximation for the distribution of log-likelihood ratio statistics, and propose the theoretical evaluation of the power for the homogeneity hypothesis test.
