AN ASYMPTOTIC APPROXIMATION FOR THE DISTRIBUTION OF φ-DIVERGENCE MULTINOMIAL GOODNESS-OF-FIT STATISTIC UNDER LOCAL ALTERNATIVES

Abstract

On the goodness-of-fit test for multinomial distribution, Zografos et al.(1990)proposed the φ-divergence family of statistics, which includes the power divergence family of statistics as a special case. They showed that under null hypothesis, the members of the φ-divergence family of statistics all have an asymptotically equivalent chi-square distribution. Furthermore, Menendez et al.(1997)derived an asymptotic expansion for the null distribution of the φ-divergence statistic. In this paper, we derive an approximation for the distribution of the φ-divergence statistic under local alternatives. The approximation is based on the continuous term of the asymptotic expansion for the distribution of the φ-divergence statistic. By using the approximation, we propose a new approximation for the power of the statistic. The results are generalizations of those derived by Taneichi et al.which discussed the power divergence statistic. We numerically investigate the accuracy of the approximation when two types of concrete φ-divergence statistics are applied. By the numerical investigation, we find that the present approximation performs better than the other approximations.