Asymptotic and numerical aspects of the noncentral chi-square distribution

Abstract

The concentral χ 2-distribution is related with the series e −x ∑ n=0 ∞ x n n! P(μ+ n, y)=1− e −x ∑ n=0 ∞ ∞ n n! Q(μ+n, y) where P( α, z) and Q( α, z) are incomplete gamma functions (central χ 2-distributions). Another representation is in terms of Q μ(x,y)≔ ∫ y ∞ z x 1 2 (μ−1) e −z−xI μ−1(2 xz )dz which is also known as the generalized Marcum Q-function; I μ ( z) is the modified Bessel function. Q μ ( x, y) plays a role in communication studies. From the integral representation recurrence relations for Q μ ( x, y) are derived. Next, it is shown that Q μ ( x, y) can be expressed in terms of the simpler integral F μ(ξσ)≔ ∫ ξ ∞ e −(σ+1)tI μ(t)dt where ξ=2 xy andσ=−1+ 1 2 y x + x y Two asymptotic expansions of Q μ( x, y) are derived. In one form, the function F μ ( ξ, σ) is used with μ fixed and large ξ, giving an expansion which holds uniformly with respect to σ ϵ (0, ∞). In a second expansion, both parameters ξ and μ may be large. In both asymptotic forms, an error function (the normal distribution function) is used to describe the behavior of Q μ ( x, y) as y crosses the value x+ μ. Series expansions in terms of incomplete gamma functions are discussed in connection with numerical evaluation of Q μ ( x, y) or 1 − Q μ ( x, y). It is also indicated when the asymptotic expansion can be used in order to obtain a certain relative accuracy.