Power-Divergence Statistics

In this chapter, we focus on approximation problems motivated by studies on the asymptotic behavior of power-divergence family of statistics. These statistics are the goodness-of-fit test statistics and include, in particular, the Pearson chi-squared statistic, the Freeman–Tukey statistic, and the log-likelihood ratio statistic. The distributions of the statistics converge to the chi-squared distribution as sample size \(n\) tends to \(\infty \). We show that the rate of convergence is of order \(n^{-\alpha } \) with \(\alpha : 1/2< \alpha < 1 \). Under some conditions \(\alpha \) is close to 1. The proofs are based on the fundamental number theory results about approximating the number of integer points in convex sets by the Lebesgue measure of the set.