Sample Questions
Question 1
Easy
In the context of Bayesian credibility, the prior distribution represents:
Solution
B is correct.
In the Bayesian credibility (greatest accuracy) framework, the prior represents the distribution of the risk parameter across the heterogeneous population of risks. It captures how different risks are before any data on a specific risk is observed. When we select a risk at random, is drawn from this prior distribution.
In the Bayesian credibility (greatest accuracy) framework, the prior represents the distribution of the risk parameter across the heterogeneous population of risks. It captures how different risks are before any data on a specific risk is observed. When we select a risk at random, is drawn from this prior distribution.
Question 2
Medium
The semiparametric empirical Bayes approach to credibility differs from the nonparametric approach in which key way?
Solution
C is correct.
In the semiparametric empirical Bayes approach, the conditional distribution of losses given the risk parameter, , is assumed to belong to a parametric family (e.g., Poisson, normal). This allows the structural parameters and to be estimated using the known functional form of and . The prior remains completely unspecified (nonparametric component).
In the semiparametric empirical Bayes approach, the conditional distribution of losses given the risk parameter, , is assumed to belong to a parametric family (e.g., Poisson, normal). This allows the structural parameters and to be estimated using the known functional form of and . The prior remains completely unspecified (nonparametric component).
Question 3
Hard
A portfolio of 5 risks is observed for one year each. The claim counts are 3, 7, 2, 5, and 8. Using the Buhlmann credibility model, estimate the structural parameters and from the data, assuming each risk has a Poisson process with its own unknown rate.
Solution
B is correct.
With risks each observed for year, the overall mean is . For Poisson data with one observation per risk, the within-risk process variance estimate is (using Poisson mean-variance equality). The between-risk variance is estimated by removing sampling noise from total variance:
where .
With per risk: . .
With risks each observed for year, the overall mean is . For Poisson data with one observation per risk, the within-risk process variance estimate is (using Poisson mean-variance equality). The between-risk variance is estimated by removing sampling noise from total variance:
where .
With per risk: . .
