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

摘要

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. (C) 2003 Elsevier Inc. All rights reserved.