It is well-known that each statistic in the family of power divergence of statistics, across $n$ trials and $r$ classifications with index parameter $\lambda\in\mathbb{R}$ (the Pearson, likelihood ratio and Freeman-Tukey statistics correspond to $\lambda=1,0,-1/2$, respectively) is asymptotically chi-square distributed as the sample size tends to infinity. In this paper, we obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample $n$, and all index parameters ($\lambda>-1$) for which such finite sample bounds are meaningful. We obtain bounds that are of the optimal order $n^{-1}$. The dependence of our bounds on the index parameter $\lambda$ and the cell classification probabilities is also optimal, and the dependence on the number of cells is also respectable. Our bounds generalise, complement and improve on recent results from the literature.

翻译：众所周知,在以美元计的试验和以美元计的指数参数(Pearson, 概率比率和Freeman-Tukey统计数据分别对应$\lambda=1,0,-1/2美元)的分类中,统计中各权力系差异的每一项统计数据都是零星分布的,因为样本大小往往无穷无尽。在本文中,我们获得了这一分布式近似的清晰界限,通过光滑测试功能测量,该近似分布式近似保持了一定的样本($)和所有指数参数($\lambda>-1美元),对于这些有限样本界限是有意义的。我们获得了最优顺序的界限 $n ⁇ -1美元。我们的界限对指数参数 $\lambda$和细胞分类概率的依赖性也是最佳的,对细胞数量的依赖也是可尊重的。我们对文献的最新结果的界限、补充和改进。