Refinement on the convergence of one family of goodness-of-fit statistics to chi-squared distribution

Abstract

We consider a weak convergence of the power divergence family of statistics $\{T_{\lambda}(\boldsymbol{Y}),\lambda\in\mathbb{R}\}$ constructed from the multinomial distribution of degree $k$, to chi-squared distribution with $k-1$ degrees of freedom. We show that

\Pr(T_{\lambda}(\boldsymbol{Y})<c)=G_{k-1}(c)+ O(n^{-1+ 1/k})

where $G_r(c)$ is the distribution function of a chi-squared variable with $r$ degrees of freedom. In the proof we use E. Hlawka's theorem (1950) on the approximation of a number of integer points in a convex set with a closed smooth boundary by a volume of the set.