Computation of the Noncentral Chi-Square Distribution

The Non-Central Chi-Square Distribution is encountered in manystatistical problems, one of the most important in communications studies being the detection of signals in noise using a square-law detector.1 Marcum discussed this application but concluded that a satisfactory algorithm for computing system performance could not be based on the formula he used.2 This article shows that a satisfactory algorithm can be based on the formula that Marcum derived if the expression is expanded in a power series and the terms are properly grouped before being evaluated. More recently Urkowitz3 discussed detection system performance in which the above distribution arose and recommended that approximations in terms of the chi-square distribution, given by Patnaik, 1 be used for computation. While these approximations are adequate for some purposes, it is desirable to have a reliable and accurate method of computing values, if only to check the approximations. * This work was supported by the U. S. Navy under contract N00039-68-C-3584. 201