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. B46, 440–464) introduced a class of multinomial goodness-of-fit statistics Ra based on power divergence. All Ra 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 Ra under local alternatives. We obtain an expression of approximation for the distribution of Ra 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 Ra. 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 Ra. 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.