Credibility in Favor of Unlucky Insureds

The classical Buhlmann credibility formula estimates the hypothetical mean of a particular insured, or risk, by a weighted average of the grand mean of the collection of risks with the sample mean of the given insured. If the insured is unfortunate enough to have had large claims in the previous policy period(s), then the estimate of future claims for that risk will also be large. In this paper we provide actuaries with a method for not overly penalizing an unlucky insured while still targeting the goal of accuracy in the estimate. We propose a credibility estimator that minimizes the expectation of a linear combination of a squared-error term and a first-derivative term. The squared-error term measures the accuracy of the estimator, while the first-derivative term constrains the estimator to be close to constant.