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We build on recent works on Stein's method for functions of multivariatenormal random variables to derive bounds for the rate of convergence of someasymptotically chi-square distributed statistics. We obtain some general boundsand establish some simple sufficient conditions for convergence rates of order$n^{-1}$ for smooth test functions. These general bounds are applied toFriedman's statistic for comparing $r$ treatments across $n$ trials and thefamily of power divergence statistics for goodness-of-fit across $n$ trials and$r$ classifications, with index parameter $\lambda\in\mathbb{R}$ (Pearson'sstatistic corresponds to $\lambda=1$). We obtain a $O(n^{-1})$ bound for therate of convergence of Friedman's statistic for any number of treatments$r\geq2$. We also obtain a $O(n^{-1})$ bound on the rate of convergence of thepower divergence statistics for any $r\geq2$ when $\lambda$ is a positiveinteger or any real number greater than 5. We conjecture that the $O(n^{-1})$rate holds for any $\lambda\in\mathbb{R}$.

The rate of convergence of some asymptotically chi-square distributed statistics by Stein's method