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量族，每次试验可能产生 $r$ 种结果之一。该族包含对数似然比统计量和皮尔逊统计量作为重要特例。已知在某些情形下（例如当 $r$ 的量级为 $n^2$ 且当 $n\to\infty$ 时各结果渐近均匀分布），功率散度统计量依分布收敛为泊松随机变量的线性变换。我们建立了Kolmogorov（或均匀）度量下的显式误差界以补充该收敛结果，该误差界适用于任意使得此类有限样本界有意义的 $n$、$r$ 及指标参数 $\lambda$ 取值。进一步地，我们利用此泊松近似结果导出了功率散度统计量高斯近似的误差界。