Abstract:

In this work we present a (small sample) asymptotic approximation for multinomial goodness-of-fit tests. We focus on the power divergence family described in Read, T. & Cressie, N.: Goodness-of-Fit Statistics for Discrete Multivariate Data. Springer New York (1988). The family includes as special instances Pearson's chi-square, the likelihood ratio and the Hellinger distance statistics and, under the null hypothesis, all its members have a chi-squared asymptotic distribution. We derive the (small sample) asymptotic distribution for the whole family under a general framework that includes both the null and the alternative hypothesis. Moreover we prove the asymptotic normality for the whole family. The results allow to compute analytically the power function for non local alternatives so that theoretical comparisons of the performance of the different statistics are possible.