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 λ (Y), λ ∈ ℝ} from the power divergence family of statistics to the χ 2 distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that (t_λ (Y) < c) = G₂ (c) + O(n- 50/73 (log n)315/146 ), where G 2(c) is the χ 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.