Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics, R a , as multinomial goodness-of-fit statistics. Each R a has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of R a under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of R a under nonlocal alternatives can be improved. We apply the expansion to the power approximation for R a . The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.