Nonparametric Bayesian Modeling and Estimation for Renewal Processes

Abstract

ABSTRACT We propose a flexible approach to modeling for renewal processes. The model is built from a structured mixture of Erlang densities for the renewal process inter-arrival density. The Erlang mixture components have a common scale parameter, and the mixture weights are defined through an underlying distribution function modeled nonparametrically with a Dirichlet process (DP) prior. This model specification enables nonstandard shapes for the inter-arrival time density, including heavy tailed and multimodal densities. Moreover, the choice of the DP centering distribution controls clustering or declustering patterns for the point process, which can therefore be encouraged in the prior specification. Using the analytically available Laplace transforms of the relevant functions, we study the renewal function and the directly related K function, which can be used to infer about clustering or declustering patterns. From a computational point of view, the model structure is attractive as it enables efficient posterior simulation while properly accounting for the likelihood normalizing constant implied by the renewal process. A hierarchical extension of the model allows for the quantification of the impact of different levels of a factor. The modeling approach is illustrated with several synthetic datasets, earthquake occurrences data, and coal-mining disaster data.