Chi-square approximation by Stein's method with application to Pearson's statistic

TL;DR

This paper develops Stein's method for chi-square approximation, providing explicit bounds on the distributional distance between Pearson's statistic and its limiting chi-square distribution, with applications in statistical analysis.

Contribution

It introduces new bounds for derivatives of the gamma Stein equation and applies Stein's method to derive explicit bounds for Pearson's statistic.

Findings

Explicit bounds of order n^{-1} for distributional distance

New bounds involving shape parameter and derivative order

Application to Pearson's statistic in statistical tests

Abstract

This paper concerns the development of Stein's method for chi-square approximation and its application to problems in statistics. New bounds for the derivatives of the solution of the gamma Stein equation are obtained. These bounds involve both the shape parameter and the order of the derivative. Subsequently Stein's method for chi-square approximation is applied to bound the distributional distance between Pearson's statistic and its limiting chi-square distribution, measured using smooth test functions. In combination with the use of symmetry arguments, Stein' method yields explicit bounds on this distributional distance of order $n^{-1}$.