An extension of ruben’s result on the probability intergral of noncentral chi-square

Abstract

Ruben (1974a) has obtained infinite series expressions for the distribution function (and its complement) of noncentral chi-square with even degrees of freedom. The tarms of these series are products of Poisson probabilities and the distribution function (or its complement) of central chi-square with even degrees of freedom and with the noncentrality parameter as argument. An alternative proof of Ruben’s results is provided here. At the same time the results are generalized to include odd as well as even degrees of freedom by replacing the Poisson probabilities by chi-squara densities. A recursion is given for the terms of the series.

Keywords:

• noncentral chi-square
• series represetation of noncentral chi-square distribution function