Poisson and Gaussian approximations of the power divergence family of statistics

Consider the family of power divergence statistics based on n trials, each leading to one of r possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain regimes (e.g., when r is of order n² and the allocation is asymptotically uniform as n→∞) the power divergence statistic converges in distribution to a linear transformation of a Poisson random variable. We establish explicit error bounds in the Kolmogorov (or uniform) metric to complement this convergence result, which may be applied for any values of n, r and the index parameter λ for which such a finite-sample bound is meaningful. We further use this Poisson approximation result to derive error bounds in Gaussian approximation of the power divergence statistics.