Practical Credibility Theory with Emphasis on Optimal Parameter Estimation

Abstract

We develop Hachemeister's regression model in credibility theory (without proofs) and indicate how the involved structural parameters can be estimated from the observable variables (with proofs for the simple results and those not yet published). Large families of unbiased estimators are available. From the practical viewpoint this is rather a handicap because it creates the problem to decide what estimators actually to use. In order to fix optimal estimators, we adopt the small-sample criterion of minimum-variance. But in the research for general solutions three kinds of difficulties arise. (i) The calculations become too lengthy. (ii) The optimal estimators depend on some of the parameters to be estimated. (Then we call them pseudo-estimators). (iii) The optimal estimators depend on new structural parameters defined in terms of fourth-order moments. Only a compromise allows to cope with this reality. Situation (iii) creates new estimation problems. They can only be avoided at the cost of the introduction of special assumptions or approximations. Then problem (i) is more or less automatically solved. By an obvious method of successive approximations pseudo-estimators can serve as true estimators. Thus (ii) is no real problem.