A general optimal approach to Bühlmann credibility theory

Arguably almost all developments in modern credibility theory have been based on Bühlmann's fundamental Bayes approach to credibility. Despite its simple and widespread applicability, Bühlmann's approach leads to a linear Bayesian credibility estimator that is not robust and sensitive to heavy-tailed excess claims and may not accurately approximate a non-linear Bayesian credibility estimator. Since it is based on the sample mean, the linear credibility estimator cannot even be calculated when neither the sample mean nor the individual-level claim data are available. We present a mathematically rigorous extension of Bühlmann credibility theory and propose a general method based on an optimally weighted linear combination of multiple credibility estimators. Our approach allows various linear and nonlinear estimators with potentially different desirable properties such as robustness and efficiency to be incorporated in a dependence framework. We show that the best weights are optimal not only for finite samples but also converge to the asymptotic optimal weights. Furthermore, we introduce some finite-sample weights based on the leading terms of our asymptotic solution. These weights show remarkable performance compared with the optimal finite-sample weights while they are still relatively easy to compute for certain estimators. We perform Monte Carlo simulations to demonstrate the optimal performance in finite samples. We analyze a real-world insurance claims dataset to further illustrate the usefulness and the prediction accuracy of our proposed method.