  It is well-known that each statistic in the family of power divergence
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.
