Non-asymptotic approximations for Pearson's chi-square statistic and its application to confidence intervals for strictly convex functions of the probability weights of discrete distributions

TL;DR

This paper introduces non-asymptotic approximations for Pearson's chi-square statistic, providing bounds and methods to construct confidence intervals for convex functions of discrete distribution weights, with broad applicability.

Contribution

It develops a non-asymptotic local normal approximation for multinomial probabilities and derives bounds and inequalities for Pearson's chi-square statistic compared to multivariate normals.

Findings

Established total variation bounds between multinomials and normals.

Derived quantile coupling inequalities for Pearson's chi-square statistic.

Applied results to construct confidence intervals for negative entropy.

Abstract

In this paper, we develop a non-asymptotic local normal approximation for multinomial probabilities. First, we use it to find non-asymptotic total variation bounds between the measures induced by uniformly jittered multinomials and the multivariate normals with the same means and covariances. From the total variation bounds, we also derive a comparison of the cumulative distribution functions and quantile coupling inequalities between Pearson's chi-square statistic (written as the normalized quadratic form of a multinomial vector) and its multivariate normal analogue. We apply our results to find confidence intervals for the negative entropy of discrete distributions. Our method can be applied more generally to find confidence intervals for strictly convex functions of the weights of discrete distributions.