A gamma-minimax result in credibility theory

The statistical decision problem of experience ratemaking has been described by Bühlmann (1975) as a two-person game of the actuary (player 2) against (malevolent) nature (player 1). Within this framework premium calculation procedures are strategies of player 2 and some types of credibility formulae can be regarded as Bayesian strategies or in terms of statistical decision theory as (linear) Bayes estimators. Since the application of the Bayes principle of game theory to insurance ratemaking is not appropriate — usually the actuary has not enough information for identifying one single prior — it is quite natural to select a premium calculation procedure which is optimal according to the minimax principle of game theory or according to the so-called gamma-minimax principle. This principle is more adaptable than the minimax principle since it allows to take into account vague prior information. In this paper credibility formulae are derived which are gamma-minimax for special types of prior information similar to those in Bühlmann's paper.