Credibility Theory Based on Winsorizing

Abstract: The classical B\"{u}hlmann credibility model has been widely applied to premium estimation for group insurance contracts and other insurance types. In this paper, we develop a robust B\"{u}hlmann credibility model using the winsorized version of loss data, also known as the winsorized mean (a robust alternative to the traditional individual mean). This approach assumes that the observed sample data come from a contaminated underlying model with a small percentage of contaminated sample data. This framework provides explicit formulas for the structural parameters in credibility estimation for scale-shape distribution families, location-scale distribution families, and their variants, commonly used in insurance risk modeling. Using the theory of (L)-estimators (different from the influence function approach), we derive the asymptotic properties of the proposed method and validate them through a comprehensive simulation study, comparing their performance to credibility based on the trimmed mean. By varying the winsorizing/trimming thresholds in several parametric models, we find that all structural parameters derived from the winsorized approach are less volatile than those from the trimmed approach. Using the winsorized mean as a robust risk measure can reduce the influence of parametric loss assumptions on credibility estimation. Additionally, we discuss non-parametric estimations in credibility. Finally, a numerical illustration from the Wisconsin Local Government Property Insurance Fund indicates that the proposed robust credibility approach mitigates the impact of model mis-specification and captures the risk behavior of loss data from a broader perspective.