The Premium Estimation Under Collective Risk Models And Its Generalization Of Credibility Estimate

In the field of insurance actuary, the collective model is a crucial mathematical model. The model describes the aggregate amount of the claims over a period of time. Because the collective model based on the aggregate amount of claims relates to the joint distribution of such random variables as the number of claims and the size of a single claim, it is difficult to solve the aggregate amount of claims under the collective model in the insurance actuary. Thus, it is meaningful to establish appropriate collective model and estimate the relevant premium under the construction.The thesis mainly discussed how to estimate the premium under the construction of the collective model. On the one hand, we obtained some premium estimations under various premium principles in the statistics and got some corresponding statistical inferences; on the other hand, we discussed the bayes premium and the credibility premium of the collective model under the bayes frame, and gave some appropriate estimators of structure parameters in the premium estimator. It makes the premium estimators directly apply to the insurance actual.Chapter2mainly discussed the empirical estimation of the collective model from the aspects of expected value premium principle, variance premium principle, exponential premium principle and Esscher premium principle. This paper also proved the strong consistency and asymptotic normality on the base of large samples. And the properties have been verified by the numerical simulation.Chapter3built up credibility models on the base of Bayes frames through making use of the credibility theory and obtained the credibility premium of total claims in the next year. The results showed that credibility estimator of the collective model can still be expressed as weighted sums of claim data and collective premium. Moreover, the thesis suggested the corresponding unbiased estimators of the unknown structure parameters under the models of multitude contract data. And some statistical properties of those estimators were proved.Chapter4extended the credibility model and built up the hierarchical linear model. We made use of the credibility theory to estimate the random parameters in the hierarchical model. We obtained the credibility estimators of these parameters and discussed the statistical properties of these credibility estimators.Chapter5made a summary of the thesis and put forward some suggestions for the future study.