Buhlmann Credibility: Theory and Exam STAM Calculations

The Buhlmann Model

Buhlmann credibility finds the best linear approximation to the Bayesian estimate. Given n observations X1, ..., Xn from a risk with parameter Theta, the Buhlmann premium is: P_{B} = Z * X_bar + (1 - Z) * mu, where mu = E[mu(Theta)] is the overall expected claim, X_bar is the sample mean, and Z = n / (n + k) is the credibility factor with k = v/a. Here v = E[Var(X|Theta)] (expected process variance) and a = Var(E[X|Theta]) (variance of hypothetical means).

The Buhlmann premium minimizes E[(X_{n+1} - estimate)^2] over all linear functions of X1, ..., Xn.

Computing the Buhlmann Premium

Step 1: Compute mu(theta) = E[X|Theta = theta] and v(theta) = Var(X|Theta = theta) for each theta value. Step 2: Compute mu = E[mu(Theta)], v = E[v(Theta)], and a = Var(mu(Theta)) = E[mu(Theta)^2] - mu^2. Step 3: k = v/a. Step 4: Z = n/(n+k). Step 5: P_B = Z * X_bar + (1 - Z) * mu.

For discrete Theta with known distribution, expectations are sums. For continuous Theta, they are integrals.

Exam STAM Problem Types

Typical problems give the conditional distribution X|Theta (often Poisson, exponential, or normal) and the prior distribution of Theta, then ask for the Buhlmann credibility premium. Some problems give v and a directly. Others ask you to determine how many observations are needed for the credibility factor to reach a target value (e.g., Z = 0.95 requires n = 0.95*k/0.05 = 19k). The key skill is correctly computing v and a from the given model, as these determine everything else. Practice with Poisson-Gamma, Bernoulli-Beta, and Normal-Normal models.