Computing Maximum-Likelihood Parameter Estimates of the Generalized Gamma Distribution by Numerical Root Isolation

Abstract

The generalized gamma distribution (GGD) is important in reliability modeling and includes as special cases the Weibull, Gamma, and exponential distributions. Unfortunately, the numerical difficulties associated with fitting the GGD to data have been a major impediment to its wider use. The parameter likelihood equations (LEs) can possess no solution or several solutions, and the Newton-Raphson method is, in general, poorly behaved when applied to the LEs. This paper describes an algorithm which uses a heuristic root isolation method to circumvent these difficulties. The algorithm can be used to determine the maximum likelihood estimates of the parameters of the GGD or to indicate that none exists.