Loss Models: From Data to Decisions, 4th Edition

18.7 Exact credibility

In Examples 18.15-18.17, we found that the credibility premium and the Bayesian premium are equal. From (18.19), one may view the credibility premium as the best linear approximation to the Bayesian premium in the sense of squared error loss. In these examples, the approximation is exact because the two premiums are equal. The term exact credibility is used to describe the situation when the credibility premium equals the Bayesian premium.

At first glance it appears to be unnecessary to discuss the existence and finiteness of the credibility premium in this context, because exact credibility as defined is clearly not possible otherwise. However, in what follows, there are some technical issues to be considered, and their treatment is clearer if it is tacitly remembered that the credibility premium must be well defined, which requires that E(X_j) < ∞, Var(X_j) < ∞, and Cov(X_i, X_j) < ∞, as is obvious from the normal equations (18.15) and (18.17). Exact credibility typically occurs in Bühlmann (and Bühlmann–Straub) situations involving linear exponential family members and their conjugate priors. It is clear that the credibility premium’s existence requires that the structural parameters E[μ()], E[Var()], and Var[μ()] be finite.

Consider E[μ()] in this situation. ...