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

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 $\lambda\in\mathbb{R}$.Comment: 32 page