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

Abstract

In this paper, we prove a local limit theorem for the chi-square distribution with r>0 degrees of freedom and noncentrality parameter $\lambda \ge 0$. 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 and Murphy [On the convergence of the chi square and noncentral chi square distributions to the normal distribution. IEEE Commun Lett. 2013;17(12):2233–2236. DOI:10.1109/LCOMM.2013.111113.131879] and Seri [A tight bound on the distance between a noncentral chi square and a normal distribution. IEEE Commun Lett. 2015;19(11):1877–1880. DOI:10.1109/LCOMM.2015.2461681]. 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 [Energy detection of unknown deterministic signals. Proc IEEE. 1967;55(4):523–531. DOI:10.1109/PROC.1967.5573], 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 [On some accurate bounds for the quantiles of a noncentral chi squared distribution. Stat Probab Lett. 1990;10(2):101–106. Available from: https://www.ams.org/mathscinet-getitem?mr=MR1072495].

Keywords:

• Asymptotic statistics
• local limit theorem
• Gaussian approximation
• normal approximation
• chi-square distribution
• noncentrality
• noncentral chi-square
• error bound
• survival function
• percentage point
• median
• quantiles
• detection theory

2020 Mathematics Subject Classifications:

• Primary: 62E20
• Secondary: 60F99

Acknowledgements

We thank Robert Ferydouni (University of California – Santa Cruz) for collecting some of the references in Section 2 and helping us use the latex2exp package in R.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

F. Ouimet is supported by postdoctoral fellowships from the Natural Sciences and Engineering Research Council of Canada and the Fond québécois de la recherche – Nature et technologies (B3X supplement and B3XR) .