# Bühlmann model

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Bühlmann model

The Bühlmann model is a variance components model used in credibility theory in actuarial science to determine the appropriate premium for a group of insurance contracts.

A simplified form of the classical Bühlmann model where the components are assumed independent and the variances are equal for all observations, as well as the number of policies in each cell, is called the balanced Bühlmann model. In this model the claim statistics are determined by

:$X_\left\{ij\right\}=m+Xi_i+Xi_\left\{ij\right\}$

where $Xi_i$ and $Xi_\left\{ij\right\}$ are independent random variables for which

:$operatorname\left\{E\right\} \left[Xi_i\right] =operatorname\left\{E\right\} \left[Xi_\left\{ij\right\}\right] =0, operatorname\left\{var\right\} \left[Xi_i\right] =a, operatorname\left\{var\right\} \left[Xi_\left\{ij\right\}\right] =s^2$

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