Loss Models: From Data to Decisions, 4th Edition

18.5 The Bühlmann model

The simplest credibility model, the Bühlmann model specifies that, for each policyholder (conditional on ), past losses X₁,…, X_n have the same mean and variance and are i.i.d. conditional on .

Thus, define

and

As discussed previously, μ(θ) is referred to as the hypothetical mean whereas v(θ) is called the process variance. Define

(18.21)

(18.22)

and

(18.23)

The quantity μ in (18.21) is the expected value of the hypothetical means, v in (18.22) is the expected value of the process variance, and a in (18.23) is the variance of the hypothetical means. Note that μ, is the estimate to use if we have no information about θ (and thus no information about μ(θ)). It will also be referred to as the collective premium.

The mean, variance, and covariance of the X_js may now be ...