Empirical Bayes Methods in Actuarial Credibility
Understand the Buhlmann and Buhlmann-Straub empirical Bayes credibility models for actuarial exams.
Buhlmann Credibility Model
The Buhlmann model provides a least-squares credibility framework. For a risk with parameter theta, the credibility premium is Z times the observed mean plus (1-Z) times the overall mean. The credibility factor Z = n/(n+k) where k = v/a. Here v is the expected value of the process variance (within-risk variance) and a is the variance of the hypothetical means (between-risk variance). The key insight is that as data grows, Z approaches 1 and the estimate converges to the individual risk's experience.
Buhlmann-Straub Extension
The Buhlmann-Straub model extends Buhlmann by allowing varying exposure weights across observations. Each observation has weight w_j, and the credibility factor becomes Z = W/(W+k) where W is total weight. Estimating v and a from data requires the within-risk and between-risk sum of squares formulas. Exam STAM problems frequently ask you to compute the credibility premium using sample estimates of v and a from a table of risk class data.