A simple algorithm for computing the probabilities of count models based on pure birth processes

Recently, non-monotonic rate sequences of pure birth processes have been the focus of much attention in the analysis of count data due to their ability to provide a combination of over-, under-, and equidispersed distributions without the need to reuse covariates (traditional methods). They also permit the modeling of excess counts, a frequent issue arising when using count models based on monotonic rate sequences such as the Poisson, gamma, Weibull, Conway-Maxwell-Poisson (CMP), Faddy (1997), etc. Matrix-exponential approaches have always been used for computing the probabilities for count models based on pure birth processes, although none have been proposed for them as a specific algorithm. It is intractable to calculate these pure birth probabilities numerically in an analytic form because severe numerical cancellations may occur. However, we circumvent this difficulty by exploiting a Taylor series expansion, and then a new analytic form is derived. We developed a simple algorithm for efficiently implementing the new formula and conducted numerical experiments to study the efficiency and accuracy of the developed algorithm. The results indicate that this new approach is faster and more accurate than the matrix-exponential methods.