Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

Abstract

The noncentral gamma distribution can be viewed as a generalization of the noncentral chi-squared distribution and it can be expressed as a mixture of a Poisson density function with a incomplete gamma function. The noncentral gamma distribution is not available in free conventional statistical programs. This paper aimed to propose an algorithm for the noncentral gamma by combining the method originally proposed by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) for the noncentral distributions with the method of inversion of the distribution function with respect to the noncentrality parameter using Newton–Raphson. The algorithms are available in pseudocode and implemented as R functions. To evaluate the accuracy and speed of computation of the algorithms implemented in R, results of the distribution function, density function, quantiles and noncentrality parameter of the noncentral incomplete gamma and its particular case, the noncentral chi-squared, were obtained for the arguments settings used by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) and Chen (J Stat Comput Simul 75(10):813–829, 2005). The implemented routines performed well and, in general, were as accurate than other approximations. The R package denoted ncg is available to download on the CRAN-R package repository http://cran.r-project.org/.