Efficient empirical Bayes estimates for risk parameters of Pareto distributions

Abstract

Pareto distributions are useful for modeling the loss data in many fields such as actuarial science, economics, insurance, hydrology and reliability theory. In this paper, we consider the simultaneous estimation of the risk parameters of Pareto distributions from the perspective of empirical Bayes, novel SURE-type shrinkage estimators are developed by employing the Stein’s unbiased estimate of risk (SURE). Specifically, due to the lacking of the analytic form for the risk function, we propose to estimate the hyperparameters by minimizing an unbiased estimate of an approximation of the risk function. Under mild conditions, we prove the optimality of the new shrinkage estimators. The performance of our estimators is illustrated with simulation studies and an analysis of a real auto insurance claim dataset.

Keywords:

• Asymptotic optimality
• empirical Bayes
• Pareto distribution
• shrinkage estimator
• Stein’s unbiased risk estimate (SURE)

Acknowledgements

We are very grateful to the editor and the reviewers for helpful comments and suggestions which have improved the presentation of the paper. This work is supported by the National Natural Science Foundation of China (11571154) and the Fundamental Research Funds for the Central Universities (lzujbky-2018-110).