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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. Write
- the parameter for the distribution of the i-th risk
- premium for the i-th risk
Note: and are functions of random parameter
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 (1-Z) 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.[2]
Lemma. The problem can be stated alternatively as:
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
^Proof can be found on this site: Schmidli, Hanspeter. "Lecture notes on Risk Theory"(PDF). Institute of Mathematics, University of Cologne. Archived from the original(PDF) on August 11, 2013.
Frees, E.W.; Young, V.R.; Luo, Y. (1999). "A longitudinal data analysis interpretation of credibility models". Insurance: Mathematics and Economics. 24 (3): 229–247. doi:10.1016/S0167-6687(98)00055-9.