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 & 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).

20 pages, 2 figures

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Cited by in corpus (1)

• Normal approximations for the multivariate inverse Gaussian distribution and asymmetric kernel smoothing on $d$-dimensional half-spaces