Studies of compound risk models with dependence and parameter uncertainty
Abstract
This thesis focuses on two areas in actuarial science: compound risk models with dependence, and parameter uncertainty.
In the first part of the thesis, we study a hierarchical risk model where an accident can cause a combination of different types of claims, whose sizes could be dependent. In addition, the frequencies of accidents that cause the different combinations of claims are dependent. We first derive formulas for computing risk measures, such as the tail conditional expectation and tail variance of the aggregate losses for a portfolio of businesses. Then, we present formulas for performing the associated capital allocation to different types of claims in the portfolio. The main tool we used is the moment (or size-biased) transform of the multivariate distributions.
In the second part of the thesis, we consider an extension of the classical compound Poisson risk model with a dependence structure between the inter-claim time and the subsequent claim size. We derive formulas for asymptotic tail moments of the aggregate claims when the claim amounts are heavy tail distributed. Our results extended those in the literature. Again, the primary technique we utilized is the moment transform.
In the third part of the thesis, we investigate actuarial credibility theory when the information about the loss model or the prior distribution of its parameters is imprecise or vague. Several approaches, such as robust Bayesian method and imprecise probability, have been proposed in the literature to study such problems. We propose to represent the imprecise/partial/vague information about model parameters as fuzzy numbers and derive formulas for “fuzzy credibility premiums”. The results extend those exist in the literature.