Bounds for the chi-square approximation of the power divergence family of statistics

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.