Poisson approximation in $\chi^2$ distance by Chen-Stein approach

TL;DR

This paper explores using the Chen-Stein method to efficiently estimate the $$ distance between a Poisson distribution and a sum of independent indicators, offering a simpler proof with comparable bounds.

Contribution

It introduces a novel application of the Chen-Stein approach to $$ distance estimation, simplifying previous complex analytical methods.

Findings

Chen-Stein approach provides quick upper bounds for $$ distance.

Bounds are comparable to earlier, more complex estimates.

Method simplifies the process of $$ distance estimation.

Abstract

The main purpose of the paper is to investigate the possibility of applying Chen-Stein approach to estimate the $\chi^2$ distance between Poisson distribution and a sum of independent indicators. Earlier results concerning $\chi^2$ distance between above mentioned distributions either used analytical approach heavily based on the analysis of the generating functions or on rather lengthy and complicated elementary calculations. Applying Chen-Stein approach we succeed in providing a very quick proof of upper bounds for $\chi^2$ distance that are of comparable strength to the earlier estimates obtained by other approaches.