Copula-based probability for occurrence of the event due to one of dependent NHPPs before the other, past a truncated time

Abstract

The bivariate Gumbel-Hougaard copula is considered for two dependent non-homogenous Poisson processes, The purpose of this article is to provide a computational tool for estimation of the probability that the next event arrival after a truncated time τ, is due to a specific process. There are many situations in which non-homogenous Poisson processes could be dependent, such as failure times of a two-component repairable system in which the failure of the system at a given time is due to only one of the components. Power Law is used as the intensity function for both non-homogenous Poisson Processes. Maximum likelihood and Bayes estimators for the probability based on gamma priors are derived. It is verified through simulations that Bayes estimator outperforms maximum likelihood estimator with regard to their accuracy. Two examples for illustration of computation via Mathematica code (in the Appendix) are provided.