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

Abstract

We study the rate of weak convergence of the distributions of the statistics {tλ(Y), λ ∈ ℝ} from the power divergence family of statistics to the χ2 distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that $$\Pr (t_\lambda (Y) < c) = G_2 (c) + O(n^{ - 50/73} (\log n)^{315/146} ),$$ where G2(c) is the χ2 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.