基于联合定价模型的奖惩因子的扩展与比较

信度模型是经验费率厘定的主要方法,其缺陷在于隐含的正态分布假设并不适用于索赔次数,同时也无法分析费率因子对预期保费的影响。若将信度模型与广义线性混合模型相结合,同时考虑保单已知的风险特征信息和潜在的个体风险特征信息,将正态分布假设推广到泊松分布,放宽随机效应假设,即可构建一种扩展的联合定价模型。扩展的联合定价模型不仅能解决定价过程中风险信息重叠的问题,其预测值还具有类似信度模型"收缩估计"的性质。对一组保单索赔次数数据的研究发现,扩展的联合定价模型(泊松-伽马模型)对索赔次数的拟合更加合理,解决了奖惩因子的"过度奖惩"的问题,有效改进了预测结果。     

Hurdle模型在非寿险分类费率厘定中的应用

泊松回归模型是常用的索赔次数预测模型。但在实务中,索赔次数往往具有零膨胀特征,如果继续使用泊松模型会低估参数的标准误差,高估其显著性水平,从而在模型中保留多余的解释变量,产生不准确费率厘定结果。Hurdel模型是一个二阶段模型,可以将索赔次数分为两个部分来处理。因此,利用该模型的这一性质来处理费率厘定中具有零膨胀特征的索赔数据,可以有效地改善拟合效果。     

多项式风险的期望贴现惩罚函数

文章提出了马氏环境下相关多险种的多项式风险过程,总索赔及各类险种间索赔次数服从泊松-多项式分布,同时引入环境过程刻画随机因素对索赔大小及索赔次数的影响,利用Lund-berg基本方程,文章得到了期望贴现惩罚函数Laplace变换的表达式,在初始环境状态给定,初始资金为0时,得到了的期望贴现惩罚函数的确切表达式,推出了破产瞬间盈余及破产赤字贴现的联合密度函数及相应的边缘密度.     

随机效应零膨胀索赔次数回归模型

索赔频率预测是非寿险费率厘定的重要组成部分。最常使用的索赔频率预测模型是泊松回归和负二项回归,以及与它们相对应的零膨胀回归模型。但是,当索赔次数观察值既具有零膨胀特征,又存在组内相依结构时,上述模型都不能很好地拟合实际数据。为此,本文在泊松分布、负二项分布、广义泊松分布、P型负二项分布等条件下分别建立了随机效应零膨胀损失次数回归模型。为了改进模型的预测效果,对于连续型的解释变量,还引入了二次平滑项,并建立了结构性零比例与解释变量之间的回归关系。基于一组实际索赔次数数据的实证分析结果表明,该模型可以显著改进现有模型的拟合效果。     

负二项回归模型在过离散型索赔次数中的应用研究

索赔次数预测模型中通常考虑泊松回归模型，但当索赔次数中出现过离散问题时，泊松回归模型就不再适合。本文讨论了两种分布形式的负二项回归模型，并利用它们对一组车险数据进行了拟合，效果得到了明显改善。     

非寿险费率厘定的索赔频率预测模型及其应用

在非寿险分类费率厘定中,泊松回归模型是最常使用的索赔频率预测模型,但实际的索赔频率数据往往存在过离散特征,使泊松回归模型的结果缺乏可靠性.因此,讨论处理过离散问题的各种回归模型,包括负二项回归模型、泊松-逆高斯回归模型、泊松-对数正态回归模型、广义泊松回归模型、双泊松回归模型、混合负二项回归模型、混合二项回归模型、Delaporte回归模型和Sichel回归模型,并对其进行系统比较研究认为:这些模型都可以看做是对泊松回归模型的推广,可以用于处理各种不同过离散程度的索赔频率数据,从而改善费率厘定的效果;同时应用一组实际的汽车保险数据,讨论这些模型的具体应用.     

指数族分布中参数的经验贝叶斯估计

指数族分布是一类应用广泛的分布类,包括了泊松分布、Gamma分布、Beta分布、二项分布等常见分布.在非寿险中,索赔额或索赔次数过程常常被假定服从指数族分布,由于风险的非齐次性,指数族分布中的参数θ也为随机变量,假定服从指数族共轭先验分布.此时风险参数的估计落入了Bayes框架,风险参数θ的Bayes估计被表达“信度”形式.然而,在实际运用中,由于先验分布与样本分布中仍然含有结构参数,根据样本的边际分布的似然函数估计结构参数,从而获得风险参数的经验Bayes估计,最后证明了该经验Bayes估计是渐近最优的.     

零膨胀计数数据的联合建模及变量选择

零膨胀计数数据破坏了泊松分布的方差-均值关系，可由取值服从泊松分布的数据和取值为零(退化分布)的数据各占一定比例所构成的混合分布所解释。本文基于自适应弹性网技术， 研究了零膨胀计数数据的联合建模及变量选择问题.对于零膨胀泊松分布，引入潜变量，构造出零膨胀泊松模型的完全似然, 其中由零膨胀部分和泊松部分两项组成.考虑到协变量可能存在共线性和稀疏性，通过对似然函数加自适应弹性网惩罚得到目标函数,然后利用EM算法得到回归系数的稀疏估计量，并用贝叶斯信息准则BIC来确定最优调节参数.本文也给出了估计量的大样本性质的理论证明和模拟研究，最后把所提出的方法应用到实际问题中。     

同质性保单索赔次数的一种分布类讨论

受免赔额和无赔款优待等因素的影响,使得保单组合中索赔次数为零保单数相对较多,文章根据这个特点引出了同质性保单索赔次数的一种分布类,即调零的复合泊松分布类.然后讨论了这类分布中两种特殊的索赔次数分布模型,讨论了模型中相应参数的极大似然估计.最后给出数值算例,并对拟合效果进行了分析.     

一种车险先验风险分布的参数估计方法

采用全体车险保单组合的风险损失数据(即先验信息)作为定价的信度补充,是车险精算定价的主流方法;而得到风险损失的先验分布或特征信息是经验费率定价的基础.文章引入过程和结构方差分析方法对车险索赔过程的先验分布参数进行估计;并提出了针对索赔频率和索赔额模型的参数估计方法.该方法能快速近似估计多参数分布模型,优于传统参数估计方法.     

Dividend barrier and ruin problems for a risk model with delayed claims

In this paper, a compound Poisson risk model in the presence of a constant dividend barrier is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and and the time of delay for the claim is assumed to be random. A system of integro-differential equations with certain boundary conditions for the expected discounted penalty function is derived. We show that its solution can be expressed as the solution to the expected discounted penalty function in the same risk model with the absence of a barrier plus a linear combination of two linearly independent solutions to the associated homogeneous integro-differential equation. Using systems of integro-differential equations for the moment-generating function as well as for the arbitrary moments of the sum of discounted dividend payments until ruin, a matrix version of the dividends–penalty type relationship is derived. We also prove that ruin is certain under constant dividend barrier strategy. The closed form expressions are given when the claim amounts from both classes are exponentially distributed. Finally, a numerical example is presented to illustrate the solution procedure.     

Doubly periodic non-homogeneous Poisson models for hurricane data

Non-homogeneous Poisson processes with periodic claim intensity rate have been proposed as claim counts in risk theory. Here a doubly periodic Poisson model with short- and long-term trends is studied. Beta-type intensity functions are presented as illustrations. The likelihood function and the maximum likelihood estimates of the model parameters are derived.Doubly periodic Poisson models are appropriate when the seasonality does not repeat exactly the same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. An application of the model to the data set of Atlantic hurricanes affecting the United States (1899–2000) is discussed in detail.     

On the Gerber–Shiu discounted penalty function in a risk model with delayed claims

In this paper, we consider an extension to the continuous time risk model for which the occurrence of the claim may be delayed and the time of delay for the claim is assumed to be random. Two types of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim. The time of occurrence of a by-claim is later than that of its associate main claim and the time of delay for the occurrence of a by-claim is random. An integro-differential equations system for the Gerber–Shiu discounted penalty function is established using the auxiliary risk models. Both the system of Laplace transforms of the Gerber–Shiu discounted penalty functions and the Gerber–Shiu discounted penalty functions with zero initial surplus are obtained. From Lagrange interpolating theorem, we prove that the Gerber–Shiu discounted penalty function satisfies a defective renewal equation. Exact representation for the solution of this equation is derived through an associated compound geometric distribution. Finally, examples are given with claim sizes that have exponential and a mixture of exponential distributions.     

The discrete Lindley distribution: properties and applications

Modelling count data is one of the most important issues in statistical research. In this paper, a new probability mass function is introduced by discretizing the continuous failure model of the Lindley distribution. The model obtained is over-dispersed and competitive with the Poisson distribution to fit automobile claim frequency data. After revising some of its properties a compound discrete Lindley distribution is obtained in closed form. This model is suitable to be applied in the collective risk model when both number of claims and size of a single claim are implemented into the model. The new compound distribution fades away to zero much more slowly than the classical compound Poisson distribution, being therefore suitable for modelling extreme data.     

The Gerber-Shiu function for the compound Poisson Omega model with a three-step premium rate

Abstract

The compound Poisson Omega model is considered in the presence of a three-step premium rate. Firstly, the integral equations and the integro-differential equations for the Gerber-Shiu expected discounted penalty function are derived. Secondly, the integro-differential equations for the Gerber-Shiu expected discounted penalty function are determined in three different initial conditions. The results are then used to find the bankruptcy probability. Finally, the special cases where the claim size distribution is exponential be discussed in some detail in order to illustrate the effect of the model with three-step premium rate.     

Omega model for a jump–diffusion process with a two-step premium rate

In this paper, a jump–diffusion Omega model with a two-step premium rate is studied. In this model, the surplus process is a perturbation of a compound Poisson process by a Brown motion. Firstly, using the strong Markov property, the integro-differential equations for the Gerber–Shiu expected discounted penalty function and the bankruptcy probability are derived. Secondly, for a constant bankruptcy rate function, the renewal equations satisfied by the Gerber–Shiu expected discounted penalty function are obtained, and by iteration, the closed-form solutions of the function are also given. Further, the explicit solutions of the Gerber–Shiu expected discounted penalty function are obtained when the individual claim size is subject to exponential distribution. Finally, a numerical example is presented to illustrate some properties of the model.     

The truncated generalized poisson distribution and its estimation

A discrete probability model always gets truncated during the sampling process and the point of truncation depends upon the sample size. Also, the generalized Poisson distribution cannot be used with full justification when the second parameter is negative. To avoid these problems a truncated generalized Poisson distribution is defined and studied. Estimation of its parameters by moments method, maximum likelihood method and a mixed method are considered. Some examples are given to illustrate the effect on the parameters’ estimates when a non-truncated GPD is used instead of a truncated GPD.     

Risk aggregation with dependence and overdispersion based on the compound Poisson INAR(1) process

Abstract

This paper considers an extension of the classical discrete time risk model for which the claim numbers are assumed to be temporal dependence and overdispersion. The risk model proposed is based on the first-order integer-valued autoregressive (INAR(1)) process with discrete compound Poisson distributed innovations. The explicit expression for the moment generating function of the discounted aggregate claim amount is derived. Some numerical examples are provided to illustrate the impacts of dependence and overdispersion on related quantities such as the stop-loss premium, the value at risk and the tail value at risk.