Abstract
In the multinomial goodness-of-fit test, it is well known that the log-likelihood ratio statistic T has a limiting chi-square distribution under a simple null hypothesis. Siotani and Fujikoshi (1984) derived an asymptotic expansion for the null distribution of the log-likelihood ratio statistic T under a simple null hypothesis. They proposed J1+Ĵ2 as an approximation of the lower probability of T, where J1 is a term of multivariate Edgeworth expansion and Ĵ2 is a first order approximation of a discontinuous term. In this paper, we introduce discrete Bartlett-type transformed statistics T* and T** of the log-likelihood ratio statistic T that are constructed from J1 and Ĵ2. By numerical comparison, we found that the accuracy of approximation of the log-likelihood ratio statistic T to a chi-square distribution is improved by T* or T**.
