A Bayesian Gamma Frailty Model Using the Sum of Independent Random Variables: Application of the Estimation of an Interpurchase Timing Model

In statistics, researchers have rigorously investigated the reproductive property, which maintains that the sum of independent random variables with the same distribution follows the same family of distributions. However, even if a distribution of the sum of random variables demonstrates the reproductive property, estimating parameters appropriately from only summed observations is difficult. This is because of identification problems when component random variables have different parameters. In this study, we develop a method to effectively estimate parameters from the sum of independent random variables with different parameters. In particular, we focus on the sum of Gamma random variables composed of two types of distributions. We generalize the result according to Moschopoulos(1985) to a proportional hazard model with covariates and a frailty model to capture individual heterogeneities. Additionally, to estimate each parameter from the sum of random variables, we incorporate auxiliary information using quasi-Bayesian methods, and we propose the estimation procedure by Markov chain Monte Carlo. We confirm the effectiveness of the proposed method through a simulation study and apply it to the interpurchase timing model in marketing.