Abstract
It is well-known that asymptotic distribution of the log-likelihood ratio statistics based on the observations from a normal population under the null hypothesis is explained as the chi-square distribution with 2 degrees of freedom in the case that sample size n tends to infinity. However, when sample size n is not sufficiently large, the chi-square distribution with 2 degrees of freedom is not accurate as the approximate distribution of the log-likelihood ratio statistics. Furthermore, the distribution of the log-likelihood ratio statistics under the alternative hypothesis has never been investigated. By the way, when the statistics are distributed in the non-negative space, it is known that the Patnaik's approximation is effective for approximating of the distribution of the statistics. In this paper, we first give the moment generating functions and the cumulant generating functions of the log-likelihood ratio statistics under the null and alternative hypotheses, and investigate the statistical characteristics of the log-likelihood ratio statistics under the null and alternative hypotheses. Thereafter, the Weibull approximation of the distribution of the log-likelihood ratio statistics is proposed in addition to the Patnaik's approximation. The approximate characteristics of the Weibull approximation are also compared with those of the Patnaik's approximation.
