Fast Computation of the Non-Central Chi Square PDF Outside the HDR Under a Requisite Precision Constraint

Computation of the non-central chi square probability density function is encountered in diverse fields of applied statistics and engineering. The distribution is commonly computed as a Poisson mixture of central chi square densities, where the terms of the sum are computed starting with the integer nearest the non-centrality parameter. However, for computation of the values in either tail region these terms are not the most significant and starting with them results in an increased computational load without a corresponding increase in accuracy. The most significant terms are shown to be a function of both the non-centrality parameter, the degree of freedom and the point of evaluation. A computationally simple approximate solution to the location of the most significant terms as well as the exact solution based on a Newton–Raphson iteration is presented. A quadratic approximation of the interval of summation is also developed in order to meet a requisite number of significant digits of accuracy. Computationally efficient recursions are used over these improved intervals. The method provides a means of computing the non-central chi square probability density function to a requisite accuracy as a Poisson mixture over all domains of interest.