聚合风险模型下的保费估计及信度估计的推广

发布时间：2019-01-12 18:23

【摘要】：在非寿险精算中,聚合风险模型是一种非常重要的数理模型,它描述了一段时间内某种保单的总索赔额。由于聚合风险模型的总索赔额的涉及到索赔次数和索赔额等多个随机变量的联合分布。求解聚合风险模型总索赔额分布是非寿险精算中的一大难点问题。因此,建立合适的聚合风险模型并对相关的保费进行估计是非常有意义的。本文重点讨论了聚合风险模型下的保费估计问题。一方面从统计上得到聚合风险模型在各种保费原理下的保费估计,得到相应的统计推断；另一方面在贝叶斯框架下讨论了聚合风险模型下的贝叶斯保费和信度保费估计,并对保费估计中结构参数提出合适的估计,使得保费估计可以直接运用于保险实际。第二章重点讨论聚合风险在期望值保费原理、方差保费原理、指数保费原理和Esscher保费原理下的经验估计,并在大样本下证明了估计值的强相合性和渐近正态性。并通过数值模拟验证了估计值的强相合性和渐近正态性。第三章利用信度理论的方法,在Bayes框架下,建立了聚合风险的信度模型,得到未来年总索赔的信度保费。结果表明,聚合风险模型下的信度保费仍然是历史索赔数据和聚合保费的加权平均。另外在多合同模型下,提出了结构参数的几个无偏估计,并证明了这些估计的统计性质。第四章对信度模型进行了进一步的推广,建立了分层随机效应线性模型,并利用信度理论估计了分层随机效应线性中随机参数。得到了参数的信度估计,并讨论了估计的统计性质。第五章对全文进行了总结并提出进一步研究的方向。
[Abstract]:In non-life insurance actuarial, aggregate risk model is a very important mathematical model, which describes the total claim amount of a certain policy for a period of time. Because of the joint distribution of multiple random variables such as the number of claims and the amount claimed, the aggregate risk model claims the total amount of claims. Solving aggregate risk model total claim distribution is a difficult problem in non-life insurance actuarial. Therefore, it is very meaningful to establish an appropriate aggregation risk model and estimate the relevant premiums. In this paper, we focus on the problem of premium estimation under the aggregation risk model. On the one hand, from the statistical point of view, we get the premium estimation of the aggregation risk model under various premium principles, and get the corresponding statistical inference; On the other hand, under Bayesian framework, Bayesian premium and reliability premium estimation under aggregation risk model are discussed, and the structural parameters in premium estimation are estimated appropriately, so that premium estimation can be directly applied to insurance practice. In chapter 2, we focus on the empirical estimation of polymeric risk under the expected premium principle, variance premium principle, exponential premium principle and Esscher premium principle, and prove the strong consistency and asymptotic normality of the estimated values under large samples. The strong consistency and asymptotic normality of the estimated values are verified by numerical simulation. In chapter 3, the reliability model of aggregation risk is established under the framework of Bayes by using the method of reliability theory, and the reliability premium of the total claim in the future is obtained. The results show that the reliability premium under aggregation risk model is still the weighted average of historical claim data and aggregate premium. In addition, several unbiased estimates of structural parameters are proposed under the multi-contract model, and their statistical properties are proved. In chapter 4, the reliability model is further extended, and the hierarchical random effect linear model is established, and the random parameters in the hierarchical random effect linear are estimated by using the reliability theory. The reliability estimation of the parameters is obtained and the statistical properties of the estimation are discussed. The fifth chapter summarizes the full text and puts forward the direction of further research.
【学位授予单位】：江西师范大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F840.4;F224

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