In credibility theory, a branch of study in actuarial science, the Bühlmann model is a random effects model (or "variance components model" or hierarchical linear model) used to determine the appropriate premium for a group of insurance contracts.
The model is named after Hans Bühlmann who first published a description in 1967.
[1] Consider i risks which generate random losses for which historical data of m recent claims are available (indexed by j).
A premium for the ith risk is to be determined based on the expected value of claims.
A linear estimator which minimizes the mean square error is sought.
The Bühlmann model is the solution for the problem: where
and arg min represents the parameter values which minimize the expression.
The solution for the problem is: where: We can give this result the interpretation, that Z part of the premium is based on the information that we have about the specific risk, and (1-Z) part is based on the information that we have about the whole population.
It is also more general, because it considers all linear estimators, while original proof considers only estimators based on average claim.
In our previous equation, we decompose minimized function in the sum of two expressions.
we have: We can simplify derivative, noting that: Taking above equations and inserting into derivative, we have: Right side doesn't depend on k. Therefore, all