Loss Models: From Data to Decisions, 4th Edition

19.5 Exercises

19.1 Past claims data on a portfolio of policyholders are given in Table 19.4. Estimate the Bühlmann credibility premium for each of the three policyholders for year 4.

Table 19.4 Data for Exercise 19.1.

19.2 Past data on a portfolio of group policyholders are given in Table 19.5. Estimate the Bühlmann–Straub credibility premiums to be charged to each group in year 4.

Table 19.5 Data for Exercise 19.2.

19.3 For the situation in Exercise 17.3, estimate the Bühlmann credibility premium for the next year for the policyholder.

19.4 Consider the Bühlmann model in Example 19.1.

(a) Prove that Var(X_ij) = a + v.

(b) If {X_ij : i = 1, …, r and j = 1, …, n} are unconditionally independent for all i and j, argue that an unbiased estimator of a + v is

(c) Prove the algebraic identity

(d) Show that, conditionally,

(e) Comment on the implications of (b) and (d).

19.5 Suppose that the random variables Y₁, …, Y_n are independent with

Define b = b₁ + b₂ + ··· + b_n and = ∑ⁿ_j