相依风险及平衡损失函数下的信度理论

发布时间：2018-09-18 06:49

【摘要】：在保险实践中,一项重要的工作是确定足够的保费应对风险。到目前为止,已出现许多保险价格确定的方法。其中信度厘定方法是一种被广泛认可的重要的技术,它通过风险集的过去索赔数据,来预测风险集未来的索赔。信度理论最早出现在二十世纪初,到二十世纪六十年代,Buhlmann模型则奠定了现代信度理论的基础,它通过最小二乘法确定的信度保费解决了无需分布的问题,并把表示成个体平均和集合平均二者的加权平均。以后又出现了Buhlmann-Straub模型、回归信度模型、分层模型等重要信度模型。但在实践中,Biihlmann模型的风险间独立的假设不一定满足,风险相关情形常常发生,具有风险相关结构的Biihlmann模型逐步受到重视。在经典的决策理论,损失函数往往只关注估计的精度,但拟合度也是一个重要的标准。Zellner首先在一般线性模型中引入了反映精度与拟合度的平衡损失函数,而在平衡损失函数下的信度厘定方法在最近几年也被开始研究。另外加权保费原理下的信度模型也不断被研究,这主要是由于加权损失函数可以减少纯保费原理中的负安全负载,如出现了如Esscher保费原理,调整保费原理等。本文主要研究二部分内容,其一是不同于经典信度模型中风险结构的信度模型研究；其二是不同于经典信度模型中损失函数的信度模型研究。本文的第一部分内容在第二章中研究。主要把风险相依、时间上条件独立的信度模型拓展为风险相依和时间相关的信度模型。首先给出了正交投影和信度估计的关系和由正交投影表示的信度估计公式：通过正交投影的方法,得到具有风险相依、时间相关Buhlmann和Buhlmann-Straub模型的风险保费的齐次与非齐次的信度估计；最后推导了具有共同效应风险结构的信度保费和风险等相关结构下的信度保费。在第二部分,研究把经典信度模型中二次损失函数修改为平衡损失函数和加权平衡损失函数时信度保费问题。在第三章中,首先给出在二次损失函数和平衡损失函数下信度保费的关系；然后推导了平衡损失函数下风险相依、时间相关一般信度保费表达式,具有共同效应风险结构的齐次与非齐次信度保费被重点研究,同时还给出信度公式中相关参数估计；最后平衡损失函数下指数保费原理的信度保费得到讨论。在第四章中,首先给出了在平衡损失函数下具有风险相依结构的线性回归信度保费的表达式；然后得到当平衡损失函数中目标估计取特殊估计时的信度保费；最后讨论了二种特殊风险结构即风险等相关和具有共同风险下的回归信度模型。在第五章中,考虑到安全负载缺失往往是由于损失函数造成,将经典信度模型中的二次损失函数用加权平衡二次损失函数代替。首先定义一种关于新测度的新期望；然后讨论了几类常见加权平衡损失函数下的简单Biihlmann模型信度保费,讨论了信度估计的相合性。研究一般平衡损失函数Lρ,ω,δ0(θ,δ)=ωρ(δ0,δ)+(1-ω)ρ(θ,δ)下的保费问题,最后通过泰勒展开式得到了在平衡熵损失函数和平衡Linex损失函数下的信度保费。
[Abstract]:In insurance practice, an important task is to determine sufficient premiums to cope with risks. So far, there have been many methods to determine the insurance price. Among them, the reliability determination method is a widely accepted and important technology. It predicts the future claims of the risk set through the past claim data of the risk set. Now in the early twentieth century, until the 1960s, Buhlmann model has laid the foundation of modern reliability theory. It solves the problem of no distribution by using the least square method to determine the premium of reliability, and expresses it as the weighted average of individual average and collective average. However, in practice, the assumption of risk independence in Biihlmann model is not necessarily satisfied. Risk-related situations often occur. Biihlmann model with risk-related structure is gradually taken seriously.
In classical decision theory, the loss function only focuses on the accuracy of estimation, but the fitness is also an important criterion. Zellner first introduced the balance loss function which reflects the accuracy and fitness in the general linear model, and the reliability determination method under the balance loss function has also been studied in recent years. The reliability model under premium principle is also studied continuously, mainly because the weighted loss function can reduce the negative security load in pure premium principle, such as Esscher premium principle, adjusting premium principle and so on.
This paper mainly studies two parts: one is the study of the reliability model which is different from the classical reliability model in risk structure; the other is the study of the reliability model which is different from the loss function in the classical reliability model.
The first part of this paper is studied in the second chapter. The risk-dependent and time-dependent reliability model is extended to the risk-dependent and time-dependent reliability model. The relationship between orthogonal projection and reliability estimation and the reliability estimation formula expressed by orthogonal projection are given. Homogeneous and inhomogeneous reliability estimates of risk premiums in time-dependent Buhlmann and Buhlmann-Straub models are given. Finally, reliability premiums with common effect risk structures and risk-related structures are deduced.
In the second part, we study the reliability premium problem when the quadratic loss function in the classical reliability model is modified into the balanced loss function and the weighted balanced loss function. The general reliability premium expression, homogeneous and inhomogeneous reliability premiums with common effect risk structure are studied emphatically, and the estimation of relevant parameters in the reliability formula is given. Finally, the reliability premium of exponential premium principle under the equilibrium loss function is discussed.
In Chapter 4, firstly, the linear regression reliability premium with risk-dependent structure under the balanced loss function is given; secondly, the reliability premium when the target estimator in the balanced loss function takes a special estimate is obtained; finally, the regression confidence under two special risk structures, i.e. risk equivalence correlation and common risk, is discussed. Degree model.
In Chapter 5, considering that the loss function is often responsible for the loss of security load, the quadratic loss function in the classical reliability model is replaced by the weighted balanced quadratic loss function. In this paper, we discuss the consistency of the reliability estimates. We study the premium problem under the general equilibrium loss function L_, _, _0 (theta, delta) = __ (delta 0, delta) + (1-_) Rho (theta, delta). Finally, we obtain the reliability premium under the equilibrium entropy loss function and the equilibrium Linex loss function by Taylor expansion.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：F840.3;O212.1

【共引文献】