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。     

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。     

神经网络模型与车险索赔频率预测

汽车保险广受社会关注,且在财产保险公司具有举足轻重的地位,因此汽车保险的索赔频率预测模型一直是非寿险精算理论和应用研究的重点之一。目前最为流行的索赔频率预测模型是广义线性模型,其中包括泊松回归、负二项回归和泊松-逆高斯回归等。本文基于一组实际的车险损失数据,对索赔频率的各种广义线性模型与神经网络模型和回归树模型进行了比较,得出了一些新的结论,即神经网络模型的拟合效果优于广义线性模型,在广义线性模型中,泊松回归的拟合效果优于负二项回归和泊松-逆高斯回归。线性回归模型的拟合效果最差,回归树模型的拟合效果略好于线性回归模型。     

拟合索赔数据的一种新方法：叠加分布模型

一、引言保险公司在收取续保费时,要充分利用每一个投保人的索赔历史记录,这些历史记录包括各投保期的索赔次数以及每一次索赔的大小等等。根据这些信息,保险公司利用损失分布模型将这些信息数据拟合出来,然后预测在续保期内投保人将给保险公司带来的损失。在对数据进行拟合以前,保险公司要选择合适的损失模型。就目前而言,拟合索赔大小的模型包括指数分布模型、伽马分布模型、对数正态分布模型、帕累托分布模型等;拟合索赔次数的损失模型有很多,包括泊松分布模型、负二项分布模型、泊松—逆高斯分布模型等。这些模型在拟合数据时都有比较良…     

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.     

大额索赔条件下的车险费率厘定

车险费率厘定是财险公司设计产品的核心内容之一。在传统的纯保费预测模型中,通常建立复合泊松-伽玛模型,该方法没有考虑到大额索赔出现的情况。为此,提出了一种处理大额索赔的频率-强度方法。基于一组机动车损失数据,对索赔频率和索赔强度分别建模。比较不同分布的索赔频率模型,得到零膨胀负二项模型效果较好;在索赔强度建模中,得到大额索赔伽玛模型比伽玛模型效果好。实证检验了带有大额索赔的频率-强度模型在车险费率厘定中的优越性。     

索赔次数的开放式混合泊松分布研究

本文建立了索赔次数的多风险类别混合泊松模型。首先，考虑索赔次数的零膨胀、厚尾性和异质性等特征，建立风险类别待定的开放式混合泊松模型，开放式结构使该模型对实际数据的多样特征和风险类别具有良好的自适应性；其次，定义混合权重参数的iSCAD惩罚函数，实现对权重参数的筛选；最后，借助EM算法求得模型参数，实现对各风险类别下索赔次数的估计。借助iSCAD惩罚函数，给出最优混合数，避免传统混合模型中主观选择的弊端，克服传统混合模型中结构复杂、参数估计没有显式表达式、估计结果不便于解释等问题。基于三组风险特征多样数据的实证分析，本文发现该模型可以显著改进现有模型的拟合效果。     

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。     

零膨胀模型在非寿险中应用

分类费率厘定中最常使用的模型之一是泊松回归模型，但当损失次数数据存在零膨胀特征时，通常会采用零膨胀模型来解决。本文讨论一些零膨胀模型在非寿险中的应用，并通过对一组汽车保险损失数据的拟合，发现零膨胀模型可以有效改善对实际损失数据的拟合效果。     

零膨胀泊松模型的改进在零次索赔建模中的应用

零膨胀是非寿险精算中的一种常见现象,国内外许多学者对此进行了研究分析,而最具影响力的方法是零膨胀泊松模型与Hurdle模型,但这两个方法在区分零之间的差别时存在不足。实际中,产生零次索赔的保单持有人并非全部同质,如何提取零中所包含的信息对保险公司来说是重要的。鉴此,基于零膨胀泊松模型与Hurdle模型的思想,提出修正的零膨胀泊松模型,并利用非寿险精算中的实际数据,对新模型进行了拟合分析。与零膨胀泊松模型拟合结果的比较说明,修正的零膨胀模型在零的处理上更符合实际情况,更能体现零中所包含的信息。     

Bivariate zero-inflated negative binomial regression model with applications

Count data often display excessive number of zero outcomes than are expected in the Poisson regression model. The zero-inflated Poisson regression model has been suggested to handle zero-inflated data, whereas the zero-inflated negative binomial (ZINB) regression model has been fitted for zero-inflated data with additional overdispersion. For bivariate and zero-inflated cases, several regression models such as the bivariate zero-inflated Poisson (BZIP) and bivariate zero-inflated negative binomial (BZINB) have been considered. This paper introduces several forms of nested BZINB regression model which can be fitted to bivariate and zero-inflated count data. The mean–variance approach is used for comparing the BZIP and our forms of BZINB regression model in this study. A similar approach was also used by past researchers for defining several negative binomial and zero-inflated negative binomial regression models based on the appearance of linear and quadratic terms of the variance function. The nested BZINB regression models proposed in this study have several advantages; the likelihood ratio tests can be performed for choosing the best model, the models have flexible forms of marginal mean–variance relationship, the models can be fitted to bivariate zero-inflated count data with positive or negative correlations, and the models allow additional overdispersion of the two dependent variables.     

A Bayesian Approach for Zero-Inflated Count Regression Models by Using the Reversible Jump Markov Chain Monte Carlo Method and an Application

In this study, estimation of the parameters of the zero-inflated count regression models and computations of posterior model probabilities of the log-linear models defined for each zero-inflated count regression models are investigated from the Bayesian point of view. In addition, determinations of the most suitable log-linear and regression models are investigated. It is known that zero-inflated count regression models cover zero-inflated Poisson, zero-inflated negative binomial, and zero-inflated generalized Poisson regression models. The classical approach has some problematic points but the Bayesian approach does not have similar flaws. This work points out the reasons for using the Bayesian approach. It also lists advantages and disadvantages of the classical and Bayesian approaches. As an application, a zoological data set, including structural and sampling zeros, is used in the presence of extra zeros. In this work, it is observed that fitting a zero-inflated negative binomial regression model creates no problems at all, even though it is known that fitting a zero-inflated negative binomial regression model is the most problematic procedure in the classical approach. Additionally, it is found that the best fitting model is the log-linear model under the negative binomial regression model, which does not include three-way interactions of factors.     

Bayesian estimation and case influence diagnostics for the zero-inflated negative binomial regression model

In recent years, there has been considerable interest in regression models based on zero-inflated distributions. These models are commonly encountered in many disciplines, such as medicine, public health, and environmental sciences, among others. The zero-inflated Poisson (ZIP) model has been typically considered for these types of problems. However, the ZIP model can fail if the non-zero counts are overdispersed in relation to the Poisson distribution, hence the zero-inflated negative binomial (ZINB) model may be more appropriate. In this paper, we present a Bayesian approach for fitting the ZINB regression model. This model considers that an observed zero may come from a point mass distribution at zero or from the negative binomial model. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. The approach can be easily implemented using standard Bayesian software, such as WinBUGS. The performance of the proposed method is evaluated with a simulation study. Further, a real data set is analyzed, where we show that ZINB regression models seems to fit the data better than the Poisson counterpart.     

Functional Form for the Zero-Inflated Generalized Poisson Regression Model

The generalized Poisson (GP) regression is an increasingly popular approach for modeling overdispersed as well as underdispersed count data. Several parameterizations have been performed for the GP regression, and the two well known models, the GP-1 and the GP-2, have been applied. The GP-P regression, which has been recently proposed, has the advantage of nesting the GP-1 and the GP-2 parametrically, besides allowing the statistical tests of the GP-1 and the GP-2 against a more general alternative. In several cases, count data often have excessive number of zero outcomes than are expected in the Poisson. This zero-inflation phenomenon is a specific cause of overdispersion, and the zero-inflated Poisson (ZIP) regression model has been proposed. However, if the data continue to suggest additional overdispersion, the zero-inflated negative binomial (ZINB-1 and ZINB-2) and the zero-inflated generalized Poisson (ZIGP-1 and ZIGP-2) regression models have been considered as alternatives. This article proposes a functional form of the ZIGP which mixes a distribution degenerate at zero with a GP-P distribution. The suggested model has the advantage of nesting the ZIP and the two well known ZIGP (ZIGP-1 and ZIGP-2) regression models, besides allowing the statistical tests of the ZIGP-1 and the ZIGP-2 against a more general alternative. The ZIP and the functional form of the ZIGP regression models are fitted, compared and tested on two sets of count data; the Malaysian insurance claim data and the German healthcare data.     

GEE-based zero-inflated generalized Poisson model for clustered over or under-dispersed count data

The zero-inflated regression models such as zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB) or zero-inflated generalized Poisson (ZIGP) regression models can model the count data with excess zeros. The ZINB model can handle over-dispersed and the ZIGP model can handle the over or under-dispersed count data with excess zeros as well. Moreover, the count data may be correlated because of data collection procedure or special study design. The clustered sampling approach is one of the examples in which the correlation among subjects could be defined. In such situations, a marginal model using generalized estimating equation (GEE) approach can incorporate these correlations and lead up to the relationships at the population level. In this study, the GEE-based zero-inflated generalized Poisson regression model was proposed to fit over and under-dispersed clustered count data with excess zeros.     

Quasi-binomial zero-inflated regression model suitable for variables with bounded support

In recent years, a variety of regression models, including zero-inflated and hurdle versions, have been proposed to explain the case of a dependent variable with respect to exogenous covariates. Apart from the classical Poisson, negative binomial and generalised Poisson distributions, many proposals have appeared in the statistical literature, perhaps in response to the new possibilities offered by advanced software that now enables researchers to implement numerous special functions in a relatively simple way. However, we believe that a significant research gap remains, since very little attention has been paid to the quasi-binomial distribution, which was first proposed over fifty years ago. We believe this distribution might constitute a valid alternative to existing regression models, in situations in which the variable has bounded support. Therefore, in this paper we present a zero-inflated regression model based on the quasi-binomial distribution, taking into account the moments and maximum likelihood estimators, and perform a score test to compare the zero-inflated quasi-binomial distribution with the zero-inflated binomial distribution, and the zero-inflated model with the homogeneous model (the model in which covariates are not considered). This analysis is illustrated with two data sets that are well known in the statistical literature and which contain a large number of zeros.     

Analyzing clustered count data with a cluster-specific random effect zero-inflated Conway–Maxwell–Poisson distribution

Count data analysis techniques have been developed in biological and medical research areas. In particular, zero-inflated versions of parametric count distributions have been used to model excessive zeros that are often present in these assays. The most common count distributions for analyzing such data are Poisson and negative binomial. However, a Poisson distribution can only handle equidispersed data and a negative binomial distribution can only cope with overdispersion. However, a Conway–Maxwell–Poisson (CMP) distribution [4] can handle a wide range of dispersion. We show, with an illustrative data set on next-generation sequencing of maize hybrids, that both underdispersion and overdispersion can be present in genomic data. Furthermore, the maize data set consists of clustered observations and, therefore, we develop inference procedures for a zero-inflated CMP regression that incorporates a cluster-specific random effect term. Unlike the Gaussian models, the underlying likelihood is computationally challenging. We use a numerical approximation via a Gaussian quadrature to circumvent this issue. A test for checking zero-inflation has also been developed in our setting. Finite sample properties of our estimators and test have been investigated by extensive simulations. Finally, the statistical methodology has been applied to analyze the maize data mentioned before.     

Marginal zero-inflated regression models for count data

Data sets with excess zeroes are frequently analyzed in many disciplines. A common framework used to analyze such data is the zero-inflated (ZI) regression model. It mixes a degenerate distribution with point mass at zero with a non-degenerate distribution. The estimates from ZI models quantify the effects of covariates on the means of latent random variables, which are often not the quantities of primary interest. Recently, marginal zero-inflated Poisson (MZIP; Long et al. [A marginalized zero-inflated Poisson regression model with overall exposure effects. Stat. Med. 33 (2014), pp. 5151–5165]) and negative binomial (MZINB; Preisser et al., 2016) models have been introduced that model the mean response directly. These models yield covariate effects that have simple interpretations that are, for many applications, more appealing than those available from ZI regression. This paper outlines a general framework for marginal zero-inflated models where the latent distribution is a member of the exponential dispersion family, focusing on common distributions for count data. In particular, our discussion includes the marginal zero-inflated binomial (MZIB) model, which has not been discussed previously. The details of maximum likelihood estimation via the EM algorithm are presented and the properties of the estimators as well as Wald and likelihood ratio-based inference are examined via simulation. Two examples presented illustrate the advantages of MZIP, MZINB, and MZIB models for practical data analysis.