18.6 The Bühlmann–Straub model
The Bühlmann model is the simplest of the credibility models because it effectively requires that the past claims experience of a policyholder comprise i.i.d. components with respect to each past year. An important practical difficulty with this assumption is that it does not allow for variations in exposure or size.
For example, what if the first year’s claims experience of a policyholder reflected only a portion of a year due to an unusual policyholder anniversary? What if a benefit change occurred part way through a policy year? For group insurance, what if the size of the group changed over time?
To handle these variations, we consider the following generalization of the Bühlmann model. Assume that X1,…, Xn are independent, conditional on , with common mean (as before)
but with conditional variances
where mj is a known constant measuring exposure. Note that mj need only be proportional to the size of the risk. This model would be appropriate if each Xj were the average of mj independent (conditional on ) random variables each with mean μ(θ) and variance ...