In
credibility theory
Credibility theory is a form of statistical inference used to forecast an uncertain future event developed by Thomas Bayes. It is employed to combine multiple estimates into a summary estimate that takes into account information on the accuracy o ...
, a branch of study in
actuarial science, the Bühlmann model is a
random effects model
In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are dra ...
(or "variance components model" or
hierarchical linear model
Multilevel models (also known as hierarchical linear models, linear mixed-effect model, mixed models, nested data models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parame ...
) used to determine the appropriate
premium
Premium may refer to:
Marketing
* Premium (marketing), a promotional item that can be received for a small fee when redeeming proofs of purchase that come with or on retail products
* Premium segment, high-price brands or services in marketing, ...
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 (indexed by ''j''). 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
* ''X''
ij for the ''j''-th claim on the ''i''-th risk (we assume that all claims for ''i''-th risk are
independent and identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
)
*
for the average value.
*
- the parameter for the distribution of the i-th risk
*