ASYMPTOTIC AND NUMERICAL ASPECTS OF THE NONCENTRAL CHI-SQUARE DISTRIBUTION

The noncentral chi2-distribution is related with the series e(-x)SIGMA(n=0)infinity x(n)/n! P(mu + n, y) = 1 - e(-x) SIGMA(n=0)infinity x(n)/n! Q(mu + n, y), where P(alpha, z) and Q(alpha, z) are incomplete gamma functions (central chi2-distributions). Another representation is in terms of Q(mu)(x, y) := integral-infinity/y (z/x)1/2(mu-1) e(-z-x)I(mu-1)(2 square-root xz) dz, which is also known as the generalized Marcum Q-function; I(mu)(z) is the modified Bessel function. Q(mu)(x, y) plays a role in communication studies. From the integral representation recurrence relations for Q(mu)(x, y) are derived. Next, it is shown that Q(mu)(x, y) can be expressed in terms of the simpler integral F(mu)(xi, sigma) := integral-infinity/xi e-(sigma+1)(t)I(mu)(t) dt, where xi = 2 square-root xy and sigma = -1 + 1/2 (square-root y/x + square-root x/y). Two asymptotic expansions of Q(mu)(x, y) are derived. In one form, the function F(mu)(xi, sigma) is used with mu fixed and large xi, giving an expansion which holds uniformly with respect to or is-an-element-of (0, infinity). In a second expansion, both parameters xi and mu may be large. ln both asymptotic forms, an error function (the normal distribution function) is used to describe the behavior of Q(mu)(x, y) as y crosses the value x + mu. Series expansions in terms of incomplete gamma functions are discussed in connection with numerical evaluation of Q(mu)(x, y) or 1 - Q(mu)(x, y). It is also indicated when the asymptotic expansion can be used in order to obtain a certain relative accuracy.