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.

翻译：考虑基于$n$次试验的功率散度统计族，每次试验可能导致$r$种可能结果之一。这包括对数似然比统计量和皮尔逊统计量作为重要的特例。已知在某些条件下（例如，当$r$的阶为$n^2$且当$n\to\infty$时分配渐近均匀），功率散度统计量的分布收敛于泊松随机变量的线性变换。我们建立了Kolmogorov（或一致）度量下的显式误差界以补充该收敛结果，该误差界适用于任何$n$、$r$及指数参数$\lambda$，只要有限样本界有意义。我们进一步利用该泊松近似结果推导功率散度统计量高斯近似中的误差界。