Convergence rate of multinomial goodness-of-fit statistics to chi-square distribution

Let Y = (Y₁, Y₂,..., Y_k)' be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics {I^λ, λ ∈ R} introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every k ≥ 4 Pr(2nI ^λ (Y) < c) = G_k-1 (c) + O(n^-1+μ(k-1)), where G_r (c) is the distribution function of chi-square random variable with r degrees of freedom, μ(r) = 6/(7r + 4) for 3 ≤ r ≤ 7, μ(r) = 5/(6r + 2) for r ≥ 8. This refines Zubov and Ulyanov's result (2008). The proof uses Krä tzel-Nowak's theorem (1991) on the number of integer points in a convex body with smooth boundary.