Copula-based Bayesian estimation of probabilities for two dependent non homogenous compound Poisson processes

Bivariate copulas for two dependent non homogeneous compound Poisson processes (NCPP) are considered. In this scenario, the arrival times of events are assumed to follow two independent non homogeneous Poisson processes (NHPP), but the corresponding renewal variables are dependent through a bivariate copula. The article aims to provide a computational tool for the estimation of the probability that the random sum associated with the first NCPP is less than the other random sum by a truncated time. The proposed ML and Bayesian estimation methods in a general setting can be applied using bivariate copulas such as Gumbel (for upper tail dependency), Clayton (for lower tail dependency), and Frank (for symmetric dependency). Power Law (PL) intensity function for the NHPPs and Inverse-Gaussian as renewal distribution are considered as examples. Bayes estimates of NHPPs parameters use gamma priors, and for Bayes estimates of Inverse-Gaussian distribution, an MCMC algorithm is employed. Our method has a broad range of practical applications, especially in the insurance sector, where the renewal variable may represent the dollar amount associated with a claim. To illustrate our approach, we analyze injury claims in North Carolina and South Carolina over a specific period, using a dataset sourced from docs.databricks.com.