Poisson and Gaussian approximations of the power divergence family of statistics

TL;DR

This paper provides explicit error bounds for Poisson and Gaussian approximations of the power divergence family of statistics, enhancing understanding of their distributional behavior in finite samples.

Contribution

It establishes finite-sample error bounds for Poisson and Gaussian approximations of power divergence statistics, extending previous asymptotic results.

Findings

Explicit Kolmogorov error bounds for Poisson approximation

Finite-sample Gaussian approximation error bounds

Applicable for various sample sizes, outcome counts, and divergence parameters

Abstract

Consider the family of power divergence statistics based on $n$ trials, each leading to one of $r$ possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain regimes (e.g., when $r$ is of order $n^2$ and the allocation is asymptotically uniform as $n\to\infty$) the power divergence statistic converges in distribution to a linear transformation of a Poisson random variable. We establish explicit error bounds in the Kolmogorov (or uniform) metric to complement this convergence result, which may be applied for any values of $n$, $r$ and the index parameter $\lambda$ for which such a finite-sample bound is meaningful. We further use this Poisson approximation result to derive error bounds in Gaussian approximation of the power divergence statistics.