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Computation 2015, 3(2), 326-335; doi:10.3390/computation3020326

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

College of Engineering, The University of Massachusetts Dartmouth, North Dartmouth, MA 02747-2300, USA
Academic Editor: Dominik Obrist
Received: 27 February 2015 / Revised: 11 June 2015 / Accepted: 15 June 2015 / Published: 19 June 2015
(This article belongs to the Section Computational Engineering)
View Full-Text   |   Download PDF [406 KB, uploaded 19 June 2015]   |  

Abstract

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. View Full-Text
Keywords: non-central chi-square probability density function; discrete mixture density representation; computational efficiency non-central chi-square probability density function; discrete mixture density representation; computational efficiency
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Gendron, P.J. Fast Computation of the Non-Central Chi Square PDF Outside the HDR Under a Requisite Precision Constraint. Computation 2015, 3, 326-335.

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