Event count distributions from renewal processes: fast computation of probabilities

Discrete distributions derived from renewal processes, i.e. distributions of the number of events by some time t, are beginning to be used in management science, econometrics and health sciences. A new fast method is presented for computation of the probabilities for these distributions. This will enable practitioners in management science to exploit this rich class of models. We calculate the count probabilities by repeatedly convolving the discretized distribution, and then correct them using Richardson extrapolation. When just one probability is required, a second algorithm is described, an adaptation of De Pril's method, in which the computation time does not depend on the ordinality, so that even high-order probabilities can be rapidly found. Any survival distribution can be used to model the inter-arrival times, which gives models with great flexibility for modelling both underdispersed and overdispersed data. This work could pave the way for the routine use of these distributions as an additional tool for modelling event count data. An empirical example using fertility data illustrates the use of the method and has been fully implemented using an R package Countr developed by the authors and available from the Comprehensive R Archive Network (CRAN).