Asymptotic approximations for the distributions of the multinomial goodness-of-fit statistics under local alternatives

N. Cressie and T. R. C. Read (1984, J. Roy. Statist. Soc. B 46, 440-464) introduced a class of multinomial goodness-of-fit statistics R-a based on power divergence. All R-a have the same chi-square limiting distribution under null hypothesis and have the same noncentral chi-square limiting distribution under local alternatives. In this paper, we investigate asymptotic approximations for the distributions of R-a under local alternatives. We obtain an expression of approximation for the distribution of R-a under local alternatives. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the power of R-a. We call the approximation AE approximation. By numerical investigation of the accuracy of the AE approximation, we present a range of sample size n that the omission of the discontinuous term exercises only slight influence on power approximation of R-a. We find that the AE approximation is effective for a much wider range of the value of a than the other power approximations, except for an approximation method which requires high computer performance. (C) 2001 Elsevier Science (USA).