About divergence-based goodness-of-fit tests in the Dirichlet-multinomial model

The Pearson X(2)-statistic, the loglikelihood ratio statistic, the Freeman-Tukey statistic, and others well known statistics are considered as particular cases of appropriately normed phi-divergences D-phi((F) over cap(N), F) between empirical and hy pothetical distributions. All these statistics are asymptotically chi(2)-distributed. For the Dirichlet-multinomial F, conditions on phi are found under which the first two moments of the test statistics match the moments of chi(2), up to terms of order N-1. By calculating the exact sizes and powers of goodness-of-fit tests for 14 selected phi-divergence statistics it is shown that these conditions are of practical importance even for small sample sizes N.