  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 $n^2$ and the allocation is asymptotically
uniform as $n\to\infty$) 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 $\lambda$ 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.
