纵向数据部分线性模型的惩罚广义矩方法
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摘要
纵向数据是指对每一个个体在不同的时间点进行观测而得到的由截面和时间序列融合在一起的数据,因此它既可以分析出个体随时间的变化趋势,又可以分析总体的变化趋势。对于半参数部分线性模型,通常的做法是用样条方法或者核方法逼近非参数部分,然后再用线性模型的估计方法去估计参数部分。参数部分的估计可以采用传统的广义线性模型方法(GLM)、广义估计方程方法(GEE)或者二次推断函数方法(QIF),然而,对于某些依赖于时间的协变量而言,上述的估计方法的效率是不足的。
本文在引入Lai和Small(2007)对依赖于时间的协变量的分类基础上,使用p-样条拟合非参数函数,对不同的矩条件用不同的广义矩方法对模型的参数和非参数进行估计,并且给出了估计量的大样本性质;我们用计算机模拟和算例证明了当模型中存在不同的矩条件时,采用不同的惩罚广义矩方法可以显著地提高估计精度。
Longitudinal data is referred to data in which individuals are measured repeatedly over time,so it combines elements of cross-sectional data and time-series data.It can not only analyze effectively the change of individuals over time, but also be of use in prediction of the population.In this paper,we focused on partial linear model,which is semi-parametrical.The general procedure is to make an approximation of the nonparametric part by kernel estimation or spline estimation in the first step,followed by the parametric estimation with traditional method,including Generalized Linear Model(GLM),Generalized Estimating Equation(GEE),or Quadratic Inference Function(QIF).However,with some specified covariates,these methods will result in a loss in efficiency.
In this paper,on the basis of the classification of the time-dependent covariates given by Lain and Small(2007),we fit the nonparametric part with P-spline, and estimate the parametrical and nonparametric part with different Generalized method of moments estimation for different moment conditions,implemented by the proof of the asymptotical properties for the estimator,which is also been proved by simulation and illustrative example,from which we can also find out that different penalized general method of moments estimations for different moment conditions perform more efficiently.
本文在引入Lai和Small(2007)对依赖于时间的协变量的分类基础上,使用p-样条拟合非参数函数,对不同的矩条件用不同的广义矩方法对模型的参数和非参数进行估计,并且给出了估计量的大样本性质;我们用计算机模拟和算例证明了当模型中存在不同的矩条件时,采用不同的惩罚广义矩方法可以显著地提高估计精度。
Longitudinal data is referred to data in which individuals are measured repeatedly over time,so it combines elements of cross-sectional data and time-series data.It can not only analyze effectively the change of individuals over time, but also be of use in prediction of the population.In this paper,we focused on partial linear model,which is semi-parametrical.The general procedure is to make an approximation of the nonparametric part by kernel estimation or spline estimation in the first step,followed by the parametric estimation with traditional method,including Generalized Linear Model(GLM),Generalized Estimating Equation(GEE),or Quadratic Inference Function(QIF).However,with some specified covariates,these methods will result in a loss in efficiency.
In this paper,on the basis of the classification of the time-dependent covariates given by Lain and Small(2007),we fit the nonparametric part with P-spline, and estimate the parametrical and nonparametric part with different Generalized method of moments estimation for different moment conditions,implemented by the proof of the asymptotical properties for the estimator,which is also been proved by simulation and illustrative example,from which we can also find out that different penalized general method of moments estimations for different moment conditions perform more efficiently.
引文
[1]Bai Yang,Zhu Zhongyi and Fung Wing K.(2008).Partial linear models for longitudinal data based on quadratic inference functions.Scandinavian Journal of Statistics35,104-118.
[2]Bhargava,A.(1994).Modelling the health of Filipino children.J.R.Statist.Soc.A,157,417-432.
[3]Billingsley,P.(1995).Probability and Measure,3rd edn.New York:John Wiley &Sons.
[4]Chen,L.C.,Huq,E.and Huffman,S.(1981).A prospective study of the risk of diarrheal diseases according to the nutritional status of children.Am.J.Epidem.,114,284-292.
[5]Crowder,M.(1995).On the use of a working correlation matrix in using generalised linear models for repeated measures.Biometrika 82,407-410.
[6]Diggle,P.J.,Heagerty,P.J.,Liang K-Y,and Zerger,S.L.(2002).The analysis of longitudinal data,2nd edn.New York:Oxford University Press.
[7]Eichenbaum,M.,Hansen,L.P.and Singelton,K.S.(1988).A time series analysis of representative agent models of consumption and leisure under uncertainty.Q.J.Econ.,103,51-78.
[8]Fan,J.(1992).Design-adaptive nonparametric regression.J.Amer.Stat.Assoc 87,998-1004.
[9]F.R.Hampel(1974).The infulence curve and its role in robust estimation.J.Am.Statist.Assoc.,69,383-393.
[10]Hall,P.and Opsomer,J.D.(2005).Theory for penalised spline regression.Biometrika 92,105-118.
[11]Hansen,L.P.(1982).Large sample properties of generalized method of moments estimators.Econometrica 50,1029-1054.
[12]H(a|¨)rdle,W.,Mammen,E.,and M(u|¨)ller M.(1998).Testing parametric versus semiparametric modelling in generalized linear models.J.Amer.Star.Assoc 93,1461-1474.
[13]Jarrow,R.,Ruppert,D.,and Yu,Y.(2004).Estimating the interest rate term structure of corporate debt with a semeparametric penalized spline model.Journal of the American Statistical Association 99,57-66.
[14]Kim,I.,Cohen,N.,and Carroll,R.J.(2003).Semiparametric regression in matched case-control studies.Biometrics 59,1158-1169.
[15]Kossmann,J.,Nestel,P.,Herrera,M.G.,El Amin,A.and Fawzi,W.W.(2000).Undernutrition in relation to childhood infections:a prospective study in the Sudan.Eur.J.Clin.Nutrn,54,463-472
[16]Lai,T.L.and Small,D.(2007).Marginal regression analysis of longitudinal data with time-dependent covariates:a generalized method-of-moments approach.J.R.Statist.Soc.B 69,Part 1,79-99.
[17]Laird,N.M.& Ware,J.H.(1982).Random effects models for longitudinal data.Biometrics 38,963-74.
[18]Liang,K-Y and Zerger,S.L.(1986).Longitudinal data analysis using generalized linear models.Biometrika 73,13-22.
[19]Little,R.J.and Rubin,D.B.(2002).Statistical analysis with missing data,2nd edn.New York:Wiley.
[20]Marlene Miiller.(2001).Estimation and testing in generalized partial linear models-A comparative study.Statistics and computing 11,299-309.
[21]McCullagh,P.& Nelder,J.A.(1983).Generalized linear models.London:Chapman and Hall.
[22]Pepe,M.S.and Anderson,G.L.(1994).A cautionary note on inference for marginal regression models with longtidinal data and general correlated response data.Communs Statist.Simuln Computn 23,939-951.
[23]Qu,A.,Lindsay,B.G.,and Li,B.(2000).Improving generalised estimationg equations using quadratic inference functions.Biometrika 87,823-836.
[24]Qu,A.and Li,R.(2006).Quadratic inference functions for varying-coefficient models with longitudinal data.Biometrics 62,379-391.
[25]Ruppert,D.and Carroll,R.J.(2000).Spatially-adaptive penalties for spline fitting.Australian and New Zealand Journal of Statistics 42,205-223.
[26]Ruppert,D.(2002).Selecting the number of knots for penalized splines.Journal of Computational and Graphical Statistics 11,735-757.
[27]Ruppert,D.,Wand,M.P.,and Carroll,R.J.(2003).Semiparametric regression.New York:Cambridge University Press.
[28]Severini T.A.and Wong W.H.(1992).Generalized profile likelihood and conditionally parametric models.Annals of Statistics 20,1768-1802.
[29]Tonglet,R.,Lembo,E.M.,Zihindula,P.M.,Wodon,A.,Dramaix,M.and Hennart,P.(1999).How useful are anthropometric,clinical and dietary measurements of nutritional status as predictors of morbidity of young children in central Africa.Trop.Med.Int.Hlth.,4,120-130.
[30]Wand,M.P.(1999).On the optimal amount of smoothing in penalized spline regression.Biometrika 86,936-940.
[31]Ware,J.H.(1985).Linear models for the analysis of longitudinal studies.Am.Statistician 39,95-101.
[32]Yu,Y.and Ruppert,D.(2002).Penalized spline estimation for partially linear single-index models.Journal of the American Statistical Association 97,1042-1054.
[33]Yu,Y.and Ruppert,D.(2004).Root-n consistency of penalized spline estimator for partially linear single-index models under general Eucilidean space.Statistica Sinica 14,449-456.
[34]Zerger,S.L.and Liang,K-Y(1991).Feedback models for discrete and continuous time series.Statist.Sin 1,51-64
[35]Zerger,S.L.and Liang,K-Y(1992).An overview of methods for the analysis of longitudinal data.Statistics in Medicine 11,1825-1839.
[2]Bhargava,A.(1994).Modelling the health of Filipino children.J.R.Statist.Soc.A,157,417-432.
[3]Billingsley,P.(1995).Probability and Measure,3rd edn.New York:John Wiley &Sons.
[4]Chen,L.C.,Huq,E.and Huffman,S.(1981).A prospective study of the risk of diarrheal diseases according to the nutritional status of children.Am.J.Epidem.,114,284-292.
[5]Crowder,M.(1995).On the use of a working correlation matrix in using generalised linear models for repeated measures.Biometrika 82,407-410.
[6]Diggle,P.J.,Heagerty,P.J.,Liang K-Y,and Zerger,S.L.(2002).The analysis of longitudinal data,2nd edn.New York:Oxford University Press.
[7]Eichenbaum,M.,Hansen,L.P.and Singelton,K.S.(1988).A time series analysis of representative agent models of consumption and leisure under uncertainty.Q.J.Econ.,103,51-78.
[8]Fan,J.(1992).Design-adaptive nonparametric regression.J.Amer.Stat.Assoc 87,998-1004.
[9]F.R.Hampel(1974).The infulence curve and its role in robust estimation.J.Am.Statist.Assoc.,69,383-393.
[10]Hall,P.and Opsomer,J.D.(2005).Theory for penalised spline regression.Biometrika 92,105-118.
[11]Hansen,L.P.(1982).Large sample properties of generalized method of moments estimators.Econometrica 50,1029-1054.
[12]H(a|¨)rdle,W.,Mammen,E.,and M(u|¨)ller M.(1998).Testing parametric versus semiparametric modelling in generalized linear models.J.Amer.Star.Assoc 93,1461-1474.
[13]Jarrow,R.,Ruppert,D.,and Yu,Y.(2004).Estimating the interest rate term structure of corporate debt with a semeparametric penalized spline model.Journal of the American Statistical Association 99,57-66.
[14]Kim,I.,Cohen,N.,and Carroll,R.J.(2003).Semiparametric regression in matched case-control studies.Biometrics 59,1158-1169.
[15]Kossmann,J.,Nestel,P.,Herrera,M.G.,El Amin,A.and Fawzi,W.W.(2000).Undernutrition in relation to childhood infections:a prospective study in the Sudan.Eur.J.Clin.Nutrn,54,463-472
[16]Lai,T.L.and Small,D.(2007).Marginal regression analysis of longitudinal data with time-dependent covariates:a generalized method-of-moments approach.J.R.Statist.Soc.B 69,Part 1,79-99.
[17]Laird,N.M.& Ware,J.H.(1982).Random effects models for longitudinal data.Biometrics 38,963-74.
[18]Liang,K-Y and Zerger,S.L.(1986).Longitudinal data analysis using generalized linear models.Biometrika 73,13-22.
[19]Little,R.J.and Rubin,D.B.(2002).Statistical analysis with missing data,2nd edn.New York:Wiley.
[20]Marlene Miiller.(2001).Estimation and testing in generalized partial linear models-A comparative study.Statistics and computing 11,299-309.
[21]McCullagh,P.& Nelder,J.A.(1983).Generalized linear models.London:Chapman and Hall.
[22]Pepe,M.S.and Anderson,G.L.(1994).A cautionary note on inference for marginal regression models with longtidinal data and general correlated response data.Communs Statist.Simuln Computn 23,939-951.
[23]Qu,A.,Lindsay,B.G.,and Li,B.(2000).Improving generalised estimationg equations using quadratic inference functions.Biometrika 87,823-836.
[24]Qu,A.and Li,R.(2006).Quadratic inference functions for varying-coefficient models with longitudinal data.Biometrics 62,379-391.
[25]Ruppert,D.and Carroll,R.J.(2000).Spatially-adaptive penalties for spline fitting.Australian and New Zealand Journal of Statistics 42,205-223.
[26]Ruppert,D.(2002).Selecting the number of knots for penalized splines.Journal of Computational and Graphical Statistics 11,735-757.
[27]Ruppert,D.,Wand,M.P.,and Carroll,R.J.(2003).Semiparametric regression.New York:Cambridge University Press.
[28]Severini T.A.and Wong W.H.(1992).Generalized profile likelihood and conditionally parametric models.Annals of Statistics 20,1768-1802.
[29]Tonglet,R.,Lembo,E.M.,Zihindula,P.M.,Wodon,A.,Dramaix,M.and Hennart,P.(1999).How useful are anthropometric,clinical and dietary measurements of nutritional status as predictors of morbidity of young children in central Africa.Trop.Med.Int.Hlth.,4,120-130.
[30]Wand,M.P.(1999).On the optimal amount of smoothing in penalized spline regression.Biometrika 86,936-940.
[31]Ware,J.H.(1985).Linear models for the analysis of longitudinal studies.Am.Statistician 39,95-101.
[32]Yu,Y.and Ruppert,D.(2002).Penalized spline estimation for partially linear single-index models.Journal of the American Statistical Association 97,1042-1054.
[33]Yu,Y.and Ruppert,D.(2004).Root-n consistency of penalized spline estimator for partially linear single-index models under general Eucilidean space.Statistica Sinica 14,449-456.
[34]Zerger,S.L.and Liang,K-Y(1991).Feedback models for discrete and continuous time series.Statist.Sin 1,51-64
[35]Zerger,S.L.and Liang,K-Y(1992).An overview of methods for the analysis of longitudinal data.Statistics in Medicine 11,1825-1839.