Refined normal approximations for the central and noncentral chi-square distributions and some applications

TL;DR

This paper develops refined normal approximations for chi-square distributions with improved error bounds, enabling more efficient energy detection and providing new bounds on distribution metrics and median estimates.

Contribution

It introduces a local limit theorem and refined approximations with errors of order r^{-2}, significantly improving previous bounds and reducing sample size requirements in applications.

Findings

Maximal errors reduced to order r^{-2}

Energy detection sample size reduced from 250 to 8

Provides bounds on distribution metrics and median approximations

Abstract

In this paper, we prove a local limit theorem for the chi-square distribution with $r > 0$ degrees of freedom and noncentrality parameter $\lambda \geq 0$. We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of $r^{-2}$, which is significantly smaller than the maximal error bounds of order $r^{-1/2}$ recently found by Horgan & Murphy (2013) and Seri (2015). Our results allow us to drastically reduce the number of observations required to obtain negligible errors in the energy detection problem, from $250$, as recommended in the seminal work of Urkowitz (1967), to only $8$ here with our new approximations. We also obtain an upper bound on several probability metrics between the central and noncentral chi-square distributions and the standard normal distribution, and we obtain an approximation for the median that improves the lower bound previously obtained by Robert (1990).