Computing discrete mixtures of continuous distributions: Noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient

In this article, we address the problem of computing the distribution functions that can be expressed as discrete mixtures of continuous distributions. Examples include noncentral chisquare, noncentral beta, noncentral F, noncentral t, and the distribution of squared sample multiple correlation. We illustrate the need for improved algorithms by pointing out situations where existing algorithms fail to compute meaningful values of the cumulative distribution functions (cdf) under study. To address this problem we recommend an approach that can be easily incorporated to improve the existing algorithms. For the distributions of the squared sample multiple correlation coefficient, noncentral t, and noncentral chisquare, we apply the approach and give a detailed explanation of computing the cdf values. We present results of comparison studies carried out to validate the calculated values and computational times of our suggested approach. Finally, we give the algorithms for computing the distributions of the squared sample multiple correlation coefficient, noncentral t, and noncentral chisquare so that they can be coded in any desired computer language. © 2002 Elsevier Science B.V. All rights reserved.