On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data

We study the rate of weak convergence of the distributions of the statistics $\{t_\lambda(\boldsymbol Y),\lambda\in\mathbb R\}$ from the power divergence family of statistics to the $\chi^2$ distribution. The statistics are constructed from $n$ observations of a random variable with three possible values. We show that
$$ \operatorname{Pr}(t_\lambda(\boldsymbol Y)<c)=G_2(c)+O(n^{-50/73}(\log n)^{315/146}), $$
where $G_2(c)$ is the $\chi^2$ distribution function of a random variable with two degrees of freedom. In the proof we use Huxley's theorem of 1993 on approximating the number of integer points in a plane convex set with smooth boundary by the area of the set.