{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:24:12Z","timestamp":1760243052708,"version":"build-2065373602"},"reference-count":8,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T00:00:00Z","timestamp":1434672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>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\u2013Raphson 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.<\/jats:p>","DOI":"10.3390\/computation3020326","type":"journal-article","created":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T10:23:33Z","timestamp":1434709413000},"page":"326-335","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fast Computation of the Non-Central Chi Square PDF Outside the HDR Under a Requisite Precision Constraint"],"prefix":"10.3390","volume":"3","author":[{"given":"Paul","family":"Gendron","sequence":"first","affiliation":[{"name":"College of Engineering, The University of Massachusetts Dartmouth, North Dartmouth, MA 02747-2300, USA"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1016\/j.csda.2008.11.025","article-title":"A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables","volume":"53","author":"Liu","year":"2009","journal-title":"Comput. Stat. Data Anal"},{"key":"ref_2","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1970). Distributions in Statistics: Continuous Univariate Distributions-2, John Wiley & Sons."},{"key":"ref_3","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1970). Distributions in Statistics: Continuous Univariate Distributions-1, John Wiley & Sons."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (2005). Univariate Discrete Distributions, John Wiley & Sons. [3rd ed].","DOI":"10.1002\/0471715816"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1109\/TIT.1975.1055327","article-title":"Some integrals involving the Qm function","volume":"21","author":"Nuttall","year":"1975","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1109\/18.761294","article-title":"Algorithm for calculating the non-central chi-square distribution","volume":"45","author":"Ross","year":"1999","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1214\/ss\/1177013621","article-title":"Memoir on the probability of causes of events","volume":"1","author":"Laplace","year":"1986","journal-title":"Statist. Sci"},{"key":"ref_8","unstructured":"Abramowitz, M., and Stegun, I. (1964). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Dover Publications, Inc."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/3\/2\/326\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:48:13Z","timestamp":1760215693000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/3\/2\/326"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,19]]},"references-count":8,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,6]]}},"alternative-id":["computation3020326"],"URL":"https:\/\/doi.org\/10.3390\/computation3020326","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2015,6,19]]}}}