Bühlmann model
In credibility theory, a branch of study in actuarial science, the Bühlmann model is a random effects 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.
Model description
Consider i risks which generate random losses for which historical data of m recent claims are available. A premium for the i-th risk is to be determined based on the expected value of claims. A linear estimator which minimizes the mean square error is sought. Write- Xij for the j-th claim on the i-th risk
- for the average value.
- - the parameter for the distribution of the i-th risk
- - premium for the i-th risk
The Bühlmann model is the solution for the problem:
where is the estimator of premium and arg min represents the parameter values which minimize the expression.
Model solution
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 part is based on the information that we have about the whole population.
Proof
The following proof is slightly different from the one in the original paper. It is also more general, because it considers all linear estimators, while original proof considers only estimators based on average claim.Proof:
The last equation follows from the fact that
We are using here the law of total expectation and the fact, that
In our previous equation, we decompose minimized function in the sum of two expressions. The second expression does not depend on parameters used in minimization. Therefore, minimizing the function is the same as minimizing the first part of the sum.
Let us find critical points of the function
For 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 are constant
From the solution for we have
Finally, the best estimator is