Convergence Rate of Multinomial Goodness-of-fit Statistics to Chi-square Distribution

Abstract

Let $\boldsymbol{Y}=\left(Y_1, Y_2,\dots, Y_k\right)'$ be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics $\{I^\lambda(\boldsymbol{Y}),\lambda\in\mathbb{R}\}$ introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every $k\ge4$

$\Pr(2nI^\lambda(\boldsymbol{Y})<c)=G_{k-1}(c)+O(n^{-1+\mu(k-1)}),$

where $G_r(c)$ is the distribution function of chi-square random variable with $r$ degrees of freedom, $\mu(r)={6}/{(7r + 4)}$ for $3\le r\le 7$, $\mu(r)={5}/{(6r+2)}$ for $r\ge 8$. This refines Zubov and Ulyanov's result (2008). The proof uses Krätzel-Nowak's theorem (1991) on the number of integer points in a convex body with smooth boundary.