The rate of convergence of some asymptotically chi-square distributed statistics by Stein's method (1603.01889v5)

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