关于（a,b）类分布的统计性质研究

发布时间：2018-02-13 12:34

本文关键词： (a b)类分布矩估计极大似然估计贝叶斯估计　出处：《吉林大学》2017年硕士论文　论文类型：学位论文

【摘要】：从改革开放后的第一家保险公司成立至今,保险业飞速发展.如今的保险业已覆盖了衣食住行各个方面.AIG(American International Group,Inc.)是一个全球领先的保险组织,AIG于今年提出了在保险行业立身之本的四大要素,其中包括:成本的控制以及承保的盈利;利润的模式;多元文化的包容性;投资结构的稳健.由此看出AIG把承保的盈利作为第一条突出了条款理赔和利润的重要性.对于不同险种的承保盈利,赔付率的估计是影响险种定价以及公司利润的重要因素.其中索赔次数和索赔金额在厘定保费中起最关键的作用,本文则是基于对索赔次数的分布进行了更深的探索.一般来说,描述损失分布的常用方法有:几何分布,二项分布,负二项分布,Poisson分布和正态分布.获得损失分布的方法则有:经典统计法,随机模拟法和Bayes法.在实际的应用中,保险行业常用一些特殊的取值非负的计数随机变量来刻画损失次数,这类特殊的损失次数是不属于一般的损失分布的,所以针对不同的数据样本确定不同的损失分布尤为关键.根据这一想法,本文考虑(a,b)类分布来刻画损失次数的问题.从整体的角度,对(a,b)类分布进行参数估计,同时研究了该分布的方差与期望的关系.并给出了a与b的矩估计与极大似然估计.在极大似然估计的基础上,利用Lindley逼近引理,进一步给出(a,b,0)类分布的贝叶斯估计,最后运用Matlab进行蒙特卡洛模拟.从模拟结果中可看出,对于(a,b,0)类分布的估计中,若样本数量足够大,我们优先选择使用贝叶斯估计,否则优先选用矩估计较为合理。
[Abstract]:Since the establishment of the first insurance company after the reform and opening up, The insurance industry is booming. Today's insurance industry has covered all aspects of clothing, food, housing and transportation. AIGAmerican International Group Inc.is a leading global insurance organization that this year proposed four key elements for a foothold in the insurance industry. These include: cost control and underwriting profit; profit model; multiculturalism; The soundness of the investment structure shows that AIG regards underwritten profits as the first item highlighting the importance of clause settlement and profit. The estimation of the compensation rate is an important factor affecting the pricing of insurance and the profits of the company, in which the number of claims and the amount of the claim play the most important role in determining the premium. This paper is based on the distribution of the number of claims. Generally speaking, the commonly used methods to describe the distribution of losses are: geometric distribution, binomial distribution, The negative binomial distribution is Poisson distribution and normal distribution. The methods to obtain the loss distribution are classical statistical method, stochastic simulation method and Bayes method. In the insurance industry, some special non-negative counting random variables are used to describe the number of losses. Therefore, it is very important to determine the different loss distribution for different data samples. According to this idea, this paper considers the problem of describing the number of loss by using the class distribution. At the same time, the relationship between the variance and expectation of the distribution is studied, and the moment and maximum likelihood estimates of a and b are given. On the basis of the maximum likelihood estimation, the Bayesian estimation of the distribution of a and b is further given by using the Lindley approximation Lemma. Finally, Monte Carlo simulation is carried out by using Matlab. It can be seen from the simulation results that if the number of samples is large enough, we prefer Bayesian estimation, otherwise the moment estimation is more reasonable.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.3

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