#### Thesis Title

An Estimation Problem of Large Claims on Actuarial Decisions

Summer 2002

#### Degree

Master of Science in Mathematics

Mathematics

George Mathew

Mathematics

#### Abstract

The problem of estimating large claims is important in order to help insurance companies set premiums at an appropriate level. The basic insurance risk model is based on the Cramer-Lundberg renewal model. This model outlines five conditions related to the claim size process, claim times, claim arrival process, inter-arrival times of the claims, and independence. When these conditions are met, the number of claims follows a Poisson process. The expected value of the total claim amount is determined by calculating the product of the expected value of a random sample of claims and the expected value of the number of claims. These expected values are calculated by assuming the claim amounts follow an exponential distribution and by applying the Cramer-Lundberg model so that the number of claims follows a Poisson process. Since these expected values are based on two parameters, which may be unknown, estimates for these parameters are also calculated. The estimates calculated are the maximum likelihood estimators for each of these parameters. These estimates satisfy the statistical properties of unbiasedness and consistency. We also show that the product of unbiased, consistent estimators is proved to be unbiased and consistent. In addition, an example of estimation of the net premium for excess claims is shown using sample data. Finally, we investigate the long time behavior of the large amount of claims under the assumption that these claim amounts follow an exponential distribution.