Credibility Using a Loss Function from Spline Theory: Parametric Models with a One-Dimensional Sufficient Statistic

Current formulas in credibility theory often estimate expected claims as a function of the sample mean of the experience claims of a policyholder. An actuary may wish to estimate future claims as a function of some statistic other than the sample arithmetic mean of claims, such as the sample geometric mean. This can be suggested to the actuary through the exercise of regressing claims on the geometric mean of prior claims. It can also be suggested through a particular probabilistic model of claims, such as a model that assumes a lognormal conditional distribution. In the first case, the actuary may lean towards using a linear function of the geometric mean, depending on the results of the data analysis. On the other hand, through a probabilistic model, the actuary may want to use the most accurate estimator of future claims, as measured by squared-error loss. However, this estimator might not be linear. In this paper, I provide a method for balancing the conflicting goals of linearity and accuracy. The credibility estimator proposed minimizes the expectation of a linear combination of a squared-error term and a second-derivative term. The squared-error term measures the accuracy of the estimator, while the second-derivative term constrains the estimator to be close to linear. I consider only those families of distributions with a one-dimensional sufficient statistic and estimators that are functions of that sufficient statistic or of the sample mean. Claim estimators are evaluated by comparing their conditional mean squared errors. In general, functions of the sufficient statistics prove to be better credibility estimators than functions of the sample mean.