Statistical Inference for Point Process Models of Rainfall

Abstract

In this paper we develop maximum likelihood procedures for parameter estimation and model selection that apply to a large class of point process models that have been used to model rainfall occurrences, including Cox processes, Neyman‐Scott processes, and renewal processes. The statistical inference procedures are based on the stochastic intensity λ(t) = lim_s→0,s>0 (1/s)E[N(t + s) − N(t)|N(u), u < t]. The likelihood function of a point process is shown to have a simple expression in terms of the stochastic intensity. The main result of this paper is a recursive procedure for computing stochastic intensities; the procedure is applicable to a broad class of point process models, including renewal Cox process with Markovian intensity processes and an important class of Neyman‐Scott processes. The model selection procedure we propose, which is based on likelihood ratios, allows direct comparison of two classes of point processes to determine which provides a better model for a given data set. The estimation and model selection procedures are applied to two data sets of simulated Cox process arrivals and a data set of daily rainfall occurrences in the Potomac River basin.