二元泊松回归模型及其推广模型的应用
|
摘要
单变量Poisson回归模型是对离散数据分析应用最多的回归模型之一,在对具有相关性的二维数据组的处理上,二元Poisson回归模型作为Poisson回归模型的自然推广,由于其考虑了数据间的相关性和过度离势,使模型的适用性与结果的可信性进一步得到提升,越来越受到国内外统计学家的研究与应用.而由于对特殊样本数据组,如(0,0)占较大比例的样本数据的处理上具备良好性质,二元Poisson回归模型的推广之一,二元Poisson零堆积回归模型也成为统计领域中热门研究课题之一
本文主要综述了二元Poisson、二元Poisson零堆积分布的性质及其回归模型的参数回归、参数估计与实际应用.本文第一章为引言部分,第二章综述了二元Poisson分布,二元Poisson零堆积分布的产生、性质与发展.第三章与第四章分别对二元Poisson、二元Poisson零堆积回归模型及相应的参数估计方法进行了详细的介绍与综述.第五章主要介绍了二元Poisson回归模型在保险、体育比赛结果分析等多个方面的应用.
There have been a great development for Bivariate continuous data analysis by using Bivariate continuous distributions. however, because of the complicated forms of joint probability function, Bivariate discrete distributions, especial the Bivariate Poisson distribution which is the most important distribution of Bivariate discrete distributions, have few applications.
In order to deal with Bivariate discrete data, there are two methods:first, we deal with discrete count data approximately by using continuous normal distribution. but, there often will be a great error; and the second one,on the independent assumption, we treat it as the double independent Poisson distribution data. If there exist dependence between in the count data, this method is also not appropriate.
Because of the development of computer performance, there have been a gradual increase in the applications of Bivariate Poisson models and its extensions. Bivariate Poisson distribution is provided by Holgate (1964), its joint distribution can be gener-ated in many ways(see Kocherlakota & Kocherlakota(1992), here, we use the trivari-ate reduction method which have been provided in Holgate (1964),Kocherlakota &Kocherlakota(1992). If we let Wi,i=1,2,3 are independent random variables and Wi～Poisson(λi), i=1,2,3,
Let N1=W1+W3 and N2=W2+W3, then the random variables N1,N2 follow jointly a Bivariate Poisson distribution: (N1,N2)～BP(λ1,λ2,λ3),
We can give the joint probability function
It have some useful properties:
(1). From marginal means we known the marginal random variables are all to be Poisson with parametersλi+λ3,i=1,2.
(2). Because soλ3 is a measure of the dependence for random count variables.
(3). Let N=N1+N2, we can get the variance is and the mean is E(N)=E(N1+N2)=λ1+λ2+2λ3, so Var(N)> EN, and it is overdispersion.
We can summarize the improvement of Bivariate Poisson distribution repalacing the independence assumption:
(1). Bivariate Poisson distribution considered the correlation of count random variables;
(2). Because most of the observed data is overdispersion, and from the properties of Bivariate Poisson distribution we know that Bivariate Poisson distribution is more suitable for data than double Poisson(it is equaldispersion);
(3). Bivariate Poisson distribution can improve model fit and predictions of out-comes.
Otherwise, this article also summarizes other properties, for instance, development of Bivariate Poisson, multivariate distributions, estimation of the parameters and so on.
As the extension of Bivariate Poisson distribution, Zero-Inflated Bivariate Poisson distribution considered the large proportion of (0,0), it can give a useful describing for this data.Taken from Wang et al(2003),we obtain its joint probability function is where n1, n2 can't both equal to zero and 0< p< 1.
From Kocherlakota & Kocherlakota(2001) we know, If we introduce the Bivari-ate Poisson and its extension to Model, we need to make regression for the param-eters. we follow the regressive method from Denuit(2007),like univariate Poisson regression, loglinear function of explanatory variables. we learn from Kocherlakota & Kocherlakota(2001),Morata(2009),Bivariate Poisson Regression Model is where i=1,2,3,..n,βj= (β1j,β2j,…βpj)′,j=1,2,3 are corresponding vectors of regression coefficients. and are vectors of explanatory variables.
Learn from Walhin(2001),Wang et al(2003),Zero-Inflated Bivariate Regres-sion is
(1). Take a logistic function of covariates Z with coefficients a as the parameter p and the Model is
(2). we suppose that the logistic component concerning p is fixed and the Poisson parameterλi are related to the set of covariates, so the Model is
this article summarize the methods of estimated parameters of Regression Mod-els. About the estimations of the parameters of Regression Models, there have many methods. Numerical methods for maximizing the likelihood, Newton-Raphson itera-tive, have been used in many articles(see Jung & Winkelmann(1993),Kocherlakota & Kocherlakota(2001)); Ho & Singer(2001)used a generalized least squares method; EM algorithm to obtain maximum likelihood estimations are also used to estimate the parameters(see Karlis& Ntzoufras(2003)). otherwise, From a Bayesian point of view, MCMC method are also used to estimated parameters(see Karlis & Meligkotsidou(20 05)).
In the last part, we summarized the applications of Regression Models in many fields,such as sports data analysis, ruin theory, labor mobility, especially in insurance.
本文主要综述了二元Poisson、二元Poisson零堆积分布的性质及其回归模型的参数回归、参数估计与实际应用.本文第一章为引言部分,第二章综述了二元Poisson分布,二元Poisson零堆积分布的产生、性质与发展.第三章与第四章分别对二元Poisson、二元Poisson零堆积回归模型及相应的参数估计方法进行了详细的介绍与综述.第五章主要介绍了二元Poisson回归模型在保险、体育比赛结果分析等多个方面的应用.
There have been a great development for Bivariate continuous data analysis by using Bivariate continuous distributions. however, because of the complicated forms of joint probability function, Bivariate discrete distributions, especial the Bivariate Poisson distribution which is the most important distribution of Bivariate discrete distributions, have few applications.
In order to deal with Bivariate discrete data, there are two methods:first, we deal with discrete count data approximately by using continuous normal distribution. but, there often will be a great error; and the second one,on the independent assumption, we treat it as the double independent Poisson distribution data. If there exist dependence between in the count data, this method is also not appropriate.
Because of the development of computer performance, there have been a gradual increase in the applications of Bivariate Poisson models and its extensions. Bivariate Poisson distribution is provided by Holgate (1964), its joint distribution can be gener-ated in many ways(see Kocherlakota & Kocherlakota(1992), here, we use the trivari-ate reduction method which have been provided in Holgate (1964),Kocherlakota &Kocherlakota(1992). If we let Wi,i=1,2,3 are independent random variables and Wi～Poisson(λi), i=1,2,3,
Let N1=W1+W3 and N2=W2+W3, then the random variables N1,N2 follow jointly a Bivariate Poisson distribution: (N1,N2)～BP(λ1,λ2,λ3),
We can give the joint probability function
It have some useful properties:
(1). From marginal means we known the marginal random variables are all to be Poisson with parametersλi+λ3,i=1,2.
(2). Because soλ3 is a measure of the dependence for random count variables.
(3). Let N=N1+N2, we can get the variance is and the mean is E(N)=E(N1+N2)=λ1+λ2+2λ3, so Var(N)> EN, and it is overdispersion.
We can summarize the improvement of Bivariate Poisson distribution repalacing the independence assumption:
(1). Bivariate Poisson distribution considered the correlation of count random variables;
(2). Because most of the observed data is overdispersion, and from the properties of Bivariate Poisson distribution we know that Bivariate Poisson distribution is more suitable for data than double Poisson(it is equaldispersion);
(3). Bivariate Poisson distribution can improve model fit and predictions of out-comes.
Otherwise, this article also summarizes other properties, for instance, development of Bivariate Poisson, multivariate distributions, estimation of the parameters and so on.
As the extension of Bivariate Poisson distribution, Zero-Inflated Bivariate Poisson distribution considered the large proportion of (0,0), it can give a useful describing for this data.Taken from Wang et al(2003),we obtain its joint probability function is where n1, n2 can't both equal to zero and 0< p< 1.
From Kocherlakota & Kocherlakota(2001) we know, If we introduce the Bivari-ate Poisson and its extension to Model, we need to make regression for the param-eters. we follow the regressive method from Denuit(2007),like univariate Poisson regression, loglinear function of explanatory variables. we learn from Kocherlakota & Kocherlakota(2001),Morata(2009),Bivariate Poisson Regression Model is where i=1,2,3,..n,βj= (β1j,β2j,…βpj)′,j=1,2,3 are corresponding vectors of regression coefficients. and are vectors of explanatory variables.
Learn from Walhin(2001),Wang et al(2003),Zero-Inflated Bivariate Regres-sion is
(1). Take a logistic function of covariates Z with coefficients a as the parameter p and the Model is
(2). we suppose that the logistic component concerning p is fixed and the Poisson parameterλi are related to the set of covariates, so the Model is
this article summarize the methods of estimated parameters of Regression Mod-els. About the estimations of the parameters of Regression Models, there have many methods. Numerical methods for maximizing the likelihood, Newton-Raphson itera-tive, have been used in many articles(see Jung & Winkelmann(1993),Kocherlakota & Kocherlakota(2001)); Ho & Singer(2001)used a generalized least squares method; EM algorithm to obtain maximum likelihood estimations are also used to estimate the parameters(see Karlis& Ntzoufras(2003)). otherwise, From a Bayesian point of view, MCMC method are also used to estimated parameters(see Karlis & Meligkotsidou(20 05)).
In the last part, we summarized the applications of Regression Models in many fields,such as sports data analysis, ruin theory, labor mobility, especially in insurance.
引文
[1]Denuit M, Marechal X, Pitrebois S, Walhin J F. Actuarial Modeling of Claim Counts. London:John Wiley & Sons,2007.
[2]Holgate P. Estimation for the bivariate Poisson distribution. Biometrika,1964, 51(1/2):241-245.
[3]Jung R C, Winkelmann R. Two aspects of labor mobility:A bivariate Poisson regression approach. Empirical Economics,1993,18(3):543-556.
[4]Karlis D, Ntzoufras I. Analysis of sports data using bivariate Poisson models. Jour-nal of the Royal Statistical Society (Statistician).2003,52:381-393.
[5]Karlis D, Ntzoufras I. Bivariate Poisson and diagonal inflated bivariate Poisson regression models in R. Journal of Statistical Software.2005,14 (10):1-36.
[6]Karlis D, Meligkotsidou L. Multivariate Poisson regression with covariance struc-ture. Statistics and Computing.2005,15:255-265.
[7]Kocherlakota S, Kocherlakota K. Regression in the bivariate Poisson distribution. Communications in Statistics-Theory and Methods.2001,30:815-827.
[8]Kocherlakota S, Kocherlakota K. Bivariate Discrete Distributions. New York:Mar-cel Dekker,1992.
[9]Li C, Lu J, Park J, Kim K, Peterson J. Multivariate zero-inflated Poisson models and their applications. Technometrics.1999,41:29-38.
[10]Lai C D. Construction of bivariate distributions by a generalised trivariate reduc-tion technique. Statistics & Probability Letters.1995,25:265-270.
[11]L B i Morata. A priori ratemaking using bivariate Poisson regression models. Insurance:Mathematics and Economics.2009,44:135-141.
[12]Tsionas E. Bayesian multivariate Poisson regression. Communications in Statistics-Theory and Method.2001,30:243-255.
[13]Vernic R. On the bivariate generalized Poisson distribution. ASTIN Bulletin. 1997,27 (1):23-31.
[14]Walhin J F. Bivariate ZIP models. Biometrical Journal.2001,43:147-160.
[15]Walhin J F. Bivariate Hofmann distributions. Journal of Applied Statistics.2003, 30 (9):1033-1046.
[16]Wang K, Lee A, Yau K, Carrivick P. A bivariate zero inflated Poisson regression model to analyze occupational injuries. Accident Analysis and Prevention.2003, 35:625-629.
[17]Greene W. Correlation in The Bivariate Poisson Regression Model.New York Uni-versity.2007.
[18]McLachlan G, Krishnan T. The EM algorithm and extensions. London:John Wiley & Sons,1997.
[19]Ho L, Singer J. Generalized least squares methods for bivariate Poisson regression. Communications in Statistics-Theory and Methods.2001,30:263-277.
[20]Partrat C. Compound model for two dependent kinds of claim. Insurance:Math-ematics & Economics.1994,15(2-3):219-231.
[21]Daniel A, Griffith·Jean H P, Paelinck. Specifying a joint space-and time-lag using a bivariate Poisson distribution. J Geogr Syst.2009,11:23-36.
[22]Lee L, Ho-Antonio F B C. Control charts for individual observations of a bivariate Poisson process. Int J Adv Manuf Technol.2009,43:744-755.
[23]Gurmu S, Elder J. Generalized bivariate count data regression models.Economics Letters.2000,68:31-36.
[24]McKendrick A G. Applications of mathematics to medical problems. Proceedings of the Edinburgh Mathematical Society.1926,44:98-130.
[25]Irwin J O. The place of mathematics in medical and biological atatistics. Journal of the Royal Statistical Society.1963,126:1-44.
[26]Lawless J F. Negative Binomial and Mixed Poisson Regression. The Canadian Journal of Statistics/La Revue Canadienne de Statistique.1987, (Vol.15, No.3): 209-225.
[27]Cummins D J, Wiltbank L J. Estimating the total claims distribution using mul-tivariate frequency and severity distributions. Journal of Risk and Insurance.1983, 50:377-403.
[28]Centeno M D L. Dependent risks and excess of loss reinsurance. Insurance:Math-ematics and Economics.2005,37:229-238.
[29]Abramowitz M, Stegun I A. Handbook of Mathematical Functions. New York: Dover Publications.1974.
[30]Demper A P, Laird N, Rubin D B. Maximam likelihood estimation from incomplete data via the EM algorithm (with discussion). J Roy Statist Soc B.1997,39:1-38.
[31]峁诗松,王静龙,濮晓龙.高等数理统计.北京：高等教育出版社.1998.
[32]黄明游,刘播,徐涛.数值计算方法.北京：科学出版社.2005.
[2]Holgate P. Estimation for the bivariate Poisson distribution. Biometrika,1964, 51(1/2):241-245.
[3]Jung R C, Winkelmann R. Two aspects of labor mobility:A bivariate Poisson regression approach. Empirical Economics,1993,18(3):543-556.
[4]Karlis D, Ntzoufras I. Analysis of sports data using bivariate Poisson models. Jour-nal of the Royal Statistical Society (Statistician).2003,52:381-393.
[5]Karlis D, Ntzoufras I. Bivariate Poisson and diagonal inflated bivariate Poisson regression models in R. Journal of Statistical Software.2005,14 (10):1-36.
[6]Karlis D, Meligkotsidou L. Multivariate Poisson regression with covariance struc-ture. Statistics and Computing.2005,15:255-265.
[7]Kocherlakota S, Kocherlakota K. Regression in the bivariate Poisson distribution. Communications in Statistics-Theory and Methods.2001,30:815-827.
[8]Kocherlakota S, Kocherlakota K. Bivariate Discrete Distributions. New York:Mar-cel Dekker,1992.
[9]Li C, Lu J, Park J, Kim K, Peterson J. Multivariate zero-inflated Poisson models and their applications. Technometrics.1999,41:29-38.
[10]Lai C D. Construction of bivariate distributions by a generalised trivariate reduc-tion technique. Statistics & Probability Letters.1995,25:265-270.
[11]L B i Morata. A priori ratemaking using bivariate Poisson regression models. Insurance:Mathematics and Economics.2009,44:135-141.
[12]Tsionas E. Bayesian multivariate Poisson regression. Communications in Statistics-Theory and Method.2001,30:243-255.
[13]Vernic R. On the bivariate generalized Poisson distribution. ASTIN Bulletin. 1997,27 (1):23-31.
[14]Walhin J F. Bivariate ZIP models. Biometrical Journal.2001,43:147-160.
[15]Walhin J F. Bivariate Hofmann distributions. Journal of Applied Statistics.2003, 30 (9):1033-1046.
[16]Wang K, Lee A, Yau K, Carrivick P. A bivariate zero inflated Poisson regression model to analyze occupational injuries. Accident Analysis and Prevention.2003, 35:625-629.
[17]Greene W. Correlation in The Bivariate Poisson Regression Model.New York Uni-versity.2007.
[18]McLachlan G, Krishnan T. The EM algorithm and extensions. London:John Wiley & Sons,1997.
[19]Ho L, Singer J. Generalized least squares methods for bivariate Poisson regression. Communications in Statistics-Theory and Methods.2001,30:263-277.
[20]Partrat C. Compound model for two dependent kinds of claim. Insurance:Math-ematics & Economics.1994,15(2-3):219-231.
[21]Daniel A, Griffith·Jean H P, Paelinck. Specifying a joint space-and time-lag using a bivariate Poisson distribution. J Geogr Syst.2009,11:23-36.
[22]Lee L, Ho-Antonio F B C. Control charts for individual observations of a bivariate Poisson process. Int J Adv Manuf Technol.2009,43:744-755.
[23]Gurmu S, Elder J. Generalized bivariate count data regression models.Economics Letters.2000,68:31-36.
[24]McKendrick A G. Applications of mathematics to medical problems. Proceedings of the Edinburgh Mathematical Society.1926,44:98-130.
[25]Irwin J O. The place of mathematics in medical and biological atatistics. Journal of the Royal Statistical Society.1963,126:1-44.
[26]Lawless J F. Negative Binomial and Mixed Poisson Regression. The Canadian Journal of Statistics/La Revue Canadienne de Statistique.1987, (Vol.15, No.3): 209-225.
[27]Cummins D J, Wiltbank L J. Estimating the total claims distribution using mul-tivariate frequency and severity distributions. Journal of Risk and Insurance.1983, 50:377-403.
[28]Centeno M D L. Dependent risks and excess of loss reinsurance. Insurance:Math-ematics and Economics.2005,37:229-238.
[29]Abramowitz M, Stegun I A. Handbook of Mathematical Functions. New York: Dover Publications.1974.
[30]Demper A P, Laird N, Rubin D B. Maximam likelihood estimation from incomplete data via the EM algorithm (with discussion). J Roy Statist Soc B.1997,39:1-38.
[31]峁诗松,王静龙,濮晓龙.高等数理统计.北京：高等教育出版社.1998.
[32]黄明游,刘播,徐涛.数值计算方法.北京：科学出版社.2005.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |