零膨胀广义泊松回归模型与保险费率厘定

在保险产品的分类费率厘定中,最常使用的模型之一是泊松回归模型.当损失数据存在零膨胀(zero-in flated)特征时,通常会采用零膨胀泊松回归模型.在零膨胀泊松回归模型中,一般假设结构零的比例参数φ为常数,不受费率因子的影响,这有可能背离实际情况.假设参数φ与费率因子之间存在一定关系,并在此基础上建立了零膨胀广义泊松回归模型,即Z IGP(τ)回归模型.通过对一组汽车保险损失数据的拟合表明,Z IGP(τ)回归模型可以有效地改善对实际数据的拟合效果,从而提高费率厘定结果的合理性.     

Bayesian multivariate Poisson models for insurance ratemaking

When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or a homeowner’s insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to resolve this problem (and also allows us to derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claim. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models together with their zero-inflated versions.     

Capture-Recapture模型与车险事故总体预测

车险事故总体预测问题一直是车辆保险公司研究的重点内容之一,目前最为常用的方法是与泊松分布相关的模型.基于车辆保险中索赔数据的结构特征,构建了Capture-Recapture模型,并使用一组车辆保险数据,利用Capture-Recapture及常用的零膨胀泊松等模型分别建模分析,得出了一些新的结论,即Capture-Recapture模型拟合效果整体较优,从而为车辆保险公司更好预测事故总体提供一定的理论依据.     

含右删失数据的ZIP模型及其应用

零膨胀Poisson回归(ZIP)是处理零频数过多计数资料的有效模型,而计数数据一般含有删失或不精密的特点.本文将删失数据引入到ZIP模型中来,分别建立含右删失数据的固定效应ZIP模型,随机效应ZIP模型,通过极大边际似然函数估计法对模型进行参数估计.最后,利用实例分析验证了上述模型的可行性.     

A Bayesian model for repeated measures zero-inflated count data with application to outpatient psychiatric service use

In applications involving count data, it is common to encounter an excess number of zeros. In the study of outpatient service utilization, for example, the number of utilization days will take on integer values, with many subjects having no utilization (zero values). Mixed-distribution models, such as the zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB), are often used to fit such data. A more general class of mixture models, called hurdle models, can be used to model zero-deflation as well as zero-inflation. Several authors have proposed frequentist approaches to fitting zero-inflated models for repeated measures. We describe a practical Bayesian approach which incorporates prior information, has optimal small-sample properties, and allows for tractable inference. The approach can be easily implemented using standard Bayesian software. A study of psychiatric outpatient service use illustrates the methods.     

ZI数据的统计分析综述

ZI (zero-inflated)数据就是含零过多的数据.从上世纪90年代以来, ZI数据在各个研究领域受到越来越广泛的重视,现在仍然是数据分析的热点问题之一.本文首先通过2个实例说明ZI数据的实际意义,然后介绍ZI数据分析的研究概况和最新进展.另外文章还系统介绍了各种ZI数据模型、ZI纵向数据模型及其参数估计方法,同时也介绍了ZI数据的统计诊断等问题, 其中包括作者近年来的一些工作.最后, 本文列出了若干有待进一步研究的问题.     

Nonlife ratemaking and risk management with Bayesian generalized additive models for location,scale, and shape

Generalized additive models for location, scale and, shape define a flexible, semi-parametric class of regression models for analyzing insurance data in which the exponential family assumption for the response is relaxed. This approach allows the actuary to include risk factors not only in the mean but also in other key parameters governing the claiming behavior, like the degree of residual heterogeneity or the no-claim probability. In this broader setting, the Negative Binomial regression with cell-specific heterogeneity and the zero-inflated Poisson regression with cell-specific additional probability mass at zero are applied to model claim frequencies. New models for claim severities that can be applied either per claim or aggregated per year are also presented. Bayesian inference is based on efficient Markov chain Monte Carlo simulation techniques and allows for the simultaneous estimation of linear effects as well as of possible nonlinear effects, spatial variations and interactions between risk factors within the data set. To illustrate the relevance of this approach, a detailed case study is proposed based on the Belgian motor insurance portfolio studied in Denuit and Lang (2004).     

Subgroup analysis of zero-inflated Poisson regression model with applications to insurance data

Customized personal rate offering is of growing importance in the insurance industry. To achieve this, an important step is to identify subgroups of insureds from the corresponding heterogeneous claim frequency data. In this paper, a penalized Poisson regression approach for subgroup analysis in claim frequency data is proposed. Subjects are assumed to follow a zero-inflated Poisson regression model with group-specific intercepts, which capture group characteristics of claim frequency. A penalized likelihood function is derived and optimized to identify the group-specific intercepts and effects of individual covariates. To handle the challenges arising from the optimization of the penalized likelihood function, an alternating direction method of multipliers algorithm is developed and its convergence is established. Simulation studies and real applications are provided for illustrations.     

A non-negative matrix factorization model based on the zero-inflated Tweedie distribution

Non-negative matrix factorization (NMF) is a technique of multivariate analysis used to approximate a given matrix containing non-negative data using two non-negative factor matrices that has been applied to a number of fields. However, when a matrix containing non-negative data has many zeroes, NMF encounters an approximation difficulty. This zero-inflated situation occurs often when a data matrix is given as count data, and becomes more challenging with matrices of increasing size. To solve this problem, we propose a new NMF model for zero-inflated non-negative matrices. Our model is based on the zero-inflated Tweedie distribution. The Tweedie distribution is a generalization of the normal, the Poisson, and the gamma distributions, and differs from each of the other distributions in the degree of robustness of its estimated parameters. In this paper, we show through numerical examples that the proposed model is superior to the basic NMF model in terms of approximation of zero-inflated data. Furthermore, we show the differences between the estimated basis vectors found using the basic and the proposed NMF models for \(\beta \) divergence by applying it to real purchasing data.     

基于零膨胀模型的我国银企关系规模影响因素研究

银企关系是学术界和实务界关注的焦点之一,然而,国内学者鲜有探讨银企关系数量的影响因素。本文使用我国A股上市公司2006-2013年的银企关系计数资料,利用零膨胀模型对企业建立银企关系规模的影响因素进行了分析。研究发现:规模大、资产负债率高、获利能力强的公司倾向于建立更多的银企关系;企业的长期负债率、第一大股东持股比例,是否是国有产权属性和企业的经营风险与银企关系的规模(数量)显著负相关;信贷合约的期限和信贷金额与银企关系的数量显著正相关;进一步比较了零膨胀模型与Poisson回归、负二项分布回归模型等计数模型,统计检验显示,零膨胀模型比较适合零值过多和过度离散的数据结构资料。     

Score tests for zero-inflated double poisson regression models

Count data with excess zeros encountered in many applications often exhibit extra variation. Therefore, zero-inflated Poisson (ZIP) model may fail to fit such data. In this paper, a zero-inflated double Poisson model (ZIDP), which is generalization of the ZIP model, is studied and the score tests for the significance of dispersion and zero-inflation in ZIDP model are developed. Meanwhile, this work also develops homogeneous tests for dispersion and/or zero-inflation parameter, and corresponding score test statistics are obtained. One numerical example is given to illustrate our methodology and the properties of score test statistics are investigated through Monte Carlo simulations.     

On the Positive Definiteness of the Information Matrix Under the Binary and Poisson Mixed Models

Binary and Poisson generalized linear mixed models are used to analyse over/under-dispersed proportion and count data, respectively. As the positive definiteness of the information matrix is a required property for valid inference about the fixed regression vector and the variance components of the random effects, this paper derives the relevant necessary and sufficient conditions under both these models. It is found that the conditions for the positive definiteness are not identical for these two nonlinear mixed models and that a mere analogy with the usual linear mixed model does not dictate these conditions.     

Sieve maximum likelihood estimation for doubly semiparametric zero-inflated Poisson models

For nonnegative measurements such as income or sick days, zero counts often have special status. Furthermore, the incidence of zero counts is often greater than expected for the Poisson model. This article considers a doubly semiparametric zero-inflated Poisson model to fit data of this type, which assumes two partially linear link functions in both the mean of the Poisson component and the probability of zero. We study a sieve maximum likelihood estimator for both the regression parameters and the nonparametric functions. We show, under routine conditions, that the estimators are strongly consistent. Moreover, the parameter estimators are asymptotically normal and first order efficient, while the nonparametric components achieve the optimal convergence rates. Simulation studies suggest that the extra flexibility inherent from the doubly semiparametric model is gained with little loss in statistical efficiency. We also illustrate our approach with a dataset from a public health study.     

Inferential aspects of the zero-inflated Poisson INAR(1) process

A first-order INteger-valued AutoRegressive (INAR) process with zero-inflated Poisson distributed innovations was proposed by Jazi, Jones and Lai (2012) [First-order integer valued AR processes with zero inflated Poisson innovations. Journal of Time Series Analysis. 33, 954–963.], which is able for dealing with zero-inflated/deflated count time series data. The inferential aspects of this model were not well explored by the authors, only a conditional maximum likelihood approach was briefly discussed. In this paper, we explore the inferential aspects of this zero-inflated Poisson INAR(1) process. We propose parameter estimation through Two-Step Conditional Least Squares and Yule–Walker methods. The asymptotic properties of the estimators are provided. Simulation results about the finite-sample behavior of both estimation methods and comparisons with the conditional maximum likelihood approach are presented under correct model specification and misspecification. Two empirical applications to real data sets are considered in order to illustrate the usefulness of the proposed methodology in practical situations.     

On familial longitudinal Poisson mixed models with gamma random effects

Poisson mixed models are used to analyze a wide variety of cluster count data. These models are commonly developed based on the assumption that the random effects have either the log-normal or the gamma distribution. Obtaining consistent as well as efficient estimates for the parameters involved in such Poisson mixed models has, however, proven to be difficult. Further problem gets mounted when the data are collected repeatedly from the individuals of the same cluster or family. In this paper, we introduce a generalized quasilikelihood approach to analyze the repeated familial data based on the familial structure caused by gamma random effects. This approach provides estimates of the regression parameters and the variance component of the random effects after taking the longitudinal correlations of the data into account. The estimators are consistent as well as highly efficient.     

Poisson-Lindley回归模型基于EM算法的诊断分析(英文)

利用EM算法研究了来自于Lindley分布权重的混合Poisson模型,即Poisson-Lindley回归模型,从而利用基于完全数据似然函数的条件期望进行统计诊断和局部影响分析,得到了几个有用的诊断统计量,并用一个数值实例说明了所得统计量的有效性.     

广义泊松回归模型的推广及其在医疗保险中应用

计数数据往往存在过离散(over-dispersed)即方差大于均值特征,若利用传统的泊松回归模型拟合数据往往会导致其参数的标准误差被低估,显著性水平被高估的错误结论。负二项回归模型、广义泊松回归模型通常被用来处理过离散特征数据。本文以两类广义泊松回归模型GP-1和GP-2模型为基础,将其推广为更为一般的GP-P形式,其中P为参数。此时,P=1或P=2,GP-P模型就退化为GP-1和GP-2模型。文中最后利用此类推广的GP-P模型处理了一组医疗保险数据,并与泊松回归模型、负二项回归模型拟合结果进行了比较。结果表明,推广后的GP-P模型的拟合效果更优。     

Semiparametric analysis of longitudinal zero-inflated count data

In this article, we consider a semiparametric zero-inflated Poisson mixed model that postulates a possible nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate in the longitudinal studies. A penalized log-likelihood function is proposed and Monte Carlo expectation-maximization algorithm is used to derive the estimates. Under some mild conditions, we establish the consistency and asymptotic normality of the resulting estimators. Simulation studies are carried out to investigate the finite sample performance of the proposed method. For illustration purposes, the method is applied to a data set from a pharmaceutical company where the variable of interest is the number of episodes of side effects after the patient has taken the treatments.     

Estimating Frailty Models via Poisson Hierarchical Generalized Linear Models

Frailty models extend proportional hazards models to multivariate survival data. Hierarchical-likelihood provides a simple unified framework for various random effect models such as hierarchical generalized linear models, frailty models, and mixed linear models with censoring. Wereview the hierarchical-likelihood estimation methods for frailty models. Hierarchical-likelihood for frailty models can be expressed as that for Poisson hierarchical generalized linear models. Frailty models can thus be fitted using Poisson hierarchical generalized linear models. Properties of the new methodology are demonstrated by simulation. The new method reduces the bias of maximum likelihood and penalized likelihood estimates.     

基于重复测量数据的Poisson回归模型的偏大离差的Score检验

Poisson回归模型广泛应用于分析计数型数据 ,Dean&Lawless(1989)和Dean(1992 )讨论了非重复测量得到的计数型数据的偏大离差存在性的检验问题 .本文分别利用随机系数模型和对数非线性模型讨论了基于重复测量得到的计数型数据的偏大离差的检验问题 ,得到了检验的score统计量 .