Robust E-Bayesian Estimation and Prediction of Esscher Premium for Car Insurance Claim Frequencies Under a Negative Binomial Model

Abstract

Car insurance claim frequencies often exhibit overdispersion, making the Poisson model too restrictive for premium estimation. This study develops a Negative Binomial-based Bayesian and E-Bayesian framework for estimating and predicting Esscher premiums using aggregated car insurance claim-frequency data. The novelty of this study lies in combining a closed-form Esscher premium under the Negative Binomial model with Bayesian conjugate updating and E-Bayesian hyperparameter averaging to reduce sensitivity to prior specification. A simulated aggregated claim-frequency table with 1,200 exposure units was used to illustrate the proposed framework. The data contained 642 total claims, with an empirical mean of 0.5350 and a sample variance of 0.8378. The variance-to-mean ratio of 1.5660 confirmed overdispersion and supported the use of the Negative Binomial model over the Poisson model. The method-of-moments estimates were  for the dispersion parameter and  for the success probability. At the representative tilt parameter , the Esscher premiums obtained from the method-of-moments, Bayesian, and E-Bayesian approaches ranged from 0.9004869 to 0.9059240, indicating stable premium estimates across alternative estimation methods. The sensitivity analysis showed that the Esscher premium increases nonlinearly with the tilt parameter, confirming its role as a risk-loading control. The prequential illustration further showed that the framework can support one-step-ahead prediction for aggregated claim totals. Overall, the proposed approach provides a robust and analytically tractable premium estimation framework for overdispersed car insurance claim-frequency data when prior information is uncertain and only aggregated data are available.