Bayesian Predictive Modeling for Exponential-Pareto Composite Distribution

Composite distributions have well-known applications in the insurance industry. In this paper, a composite exponential-Pareto distribution is considered, and the Bayes estimator under the squared error loss function is derived for the parameter q, which is the boundary point for the supports of the two distributions. A predictive density is constructed under an inverse gamma prior distribution for the parameter q, and the density is used to estimate the value at risk (VaR). Goodness of fit of the composite model is verified for a generated data set. The accuracy of the Bayes and VaR estimates is assessed via simulation studies. The “best” value for hyperparameters of the inverse gamma prior distribution are found via an upper bound on the variance of the prior distribution. Simulation studies indicate that when the “best” values of hyperparameters are used in the Bayes estimator, the estimator is consistently more accurate than maximum likelihood estimation.