单指标纵向数据模型的估计
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摘要
纵向数据是一类重要的相关性数据,广泛出现在诸多科研领域。单指标模型是多元非参数回归中重要的降维方法,在纵向数据下研究单指标模型是统计研究的热点问题。针对纵向数据单指标模型,提出惩罚改进二次推断函数方法来讨论模型的参数和非参数估计问题。该方法利用多项式样条回归方法逼近模型中的未知联系函数,将联系函数的估计转化为回归样条系数的估计,然后构造关于样条回归系数和单指标系数的惩罚改进二次推断函数,最小化惩罚改进二次推断函数便可得到模型的估计。理论结果显示,估计结果具有相合性和渐近正态性,最后得到了较好的数值模拟结果和实例数据分析结果,结果显示该方法适用于半参数纵向模型的参数和非参数估计问题。
Longitudinal data is a kind of important correlated data,which appears in many scientific research fields.Single index model is an important dimension reduction method in multivariate nonparametric regression.Research on single index model based on longitudinal data is a hot issue in statistical research.In this paper,we propose a penalized improved method for single-index models with longitudinal data.We first approximate the unknown link function using regression splines method and construct the penalized improved quadratic inference function of regression parameter of spline and coefficient of single index,then get the estimator by minimizing the objective function.Theoretical results show that the estimator has consistency and asymptotic normality and efficiency.Furthermore,we get good numerical simulation results and real data analysis results,which show that the proposed method is suitable for parameter and non-parameter estimation of semi-parametric longitudinal model.
Longitudinal data is a kind of important correlated data,which appears in many scientific research fields.Single index model is an important dimension reduction method in multivariate nonparametric regression.Research on single index model based on longitudinal data is a hot issue in statistical research.In this paper,we propose a penalized improved method for single-index models with longitudinal data.We first approximate the unknown link function using regression splines method and construct the penalized improved quadratic inference function of regression parameter of spline and coefficient of single index,then get the estimator by minimizing the objective function.Theoretical results show that the estimator has consistency and asymptotic normality and efficiency.Furthermore,we get good numerical simulation results and real data analysis results,which show that the proposed method is suitable for parameter and non-parameter estimation of semi-parametric longitudinal model.
引文
[1]李高荣,杨宜平.纵向数据半参数模型[M].北京:科学出版社,2015.
[2]田萍.纵向数据半参数回归模型的估计理论[M].郑州:郑州大学出版社,2008.
[3]赵明涛,许晓丽.半参数纵向模型的惩罚二次推断函数估计[J].统计与信息论坛,2014(8).
[4] Tian P,Yang L,Xue L.Asymptotic Properties of Estimators in Partially Linear Single-index Model for Longitudinal Data[J].Acta Mathematica Scientia,2010,30(3).
[5]郑强,冉光和.中国双向FDI的绿色生产率溢出效应——基于动态面板模型的实证检验[J].统计与信息论坛,2018,33(6).
[6] Hardle W.Nonparametric and Semiparametric Models[M].Berlin:Springer Verlag,2004.
[7] Liang K Y,Zeger S L.Longitudinal Data Analysis Using Generalized Linear Models[J].Biometrika,1986,73(1).
[8] Qu A,Lindsay B G,Li B.Improving Generalised Estimating Equations Using Quadratic Inference Functions[J].Biometrika,2000,87(4).
[9] Fan J,Lin X.New Developments in Biostatistics and Bioinformatics[M].Beijing:World Scientific,2009.
[10]Qu A,Li R.Quadratic Inference Functions for Varying-coefficient Models with Longitudinal Data[J].Biometrics,2006(2).
[11]Qu A,Lindsay B G.Building Adaptive Estimating Equations when Inverse of Covariance Estimation is Difficult[J].Journal of the Royal Statistical Society:Series B(Statistical Methodology),2003,65(1).
[12]Qu A,Song P X K.Assessing Robustness of Generalised Estimating Equations and Quadratic Inference Functions[J].Biometrika,2004,91(2).
[13]Qu A.Assessing the Validity of Weighted Generalized Estimating Equations[J].Biometrika,2011,98(1).
[14]Bai Y,Fung W K,Zhu Z Y.Penalized Quadratic Inference Functions for Single-index Models with Longitudinal Data[J].Journal of Multivariate Analysis,2009,100(1).
[15]Lai P,Li G,Lian H.Quadratic Inference Functions for Partially Linear Single-index Models with Longitudinal Data[J].Journal of Multivariate Analysis,2013,118(5).
[16]Ma S,Liang H,Tsai C L.Partially Linear Single-index Models for Repeated Measurements[J].Journal of Multivariate Analysis,2014,130(8).
[17]Li G,Lai P,Lian H.Variable Selection and Estimation for Partially Linear Single-index Models with Longitudinal Data[J].Statistics and Computing,2015,25(3).
[18]Han P,Song P X K.A Note on Improving Quadratic Inference Functions Using a Linear Shrinkage Approach[J].Statistics&Probability Letters,2011,81(3).
[19] Westgate P M,Braun T M.An Improved Quadratic Inference Function for Parameter Estimation in the Analysis of Correlated Data[J].Statistics in Medicine,2013,32(19).
[20]Khan N M,Heenaye M.A Computationally Efficient Algorithm to Solve Generalized Method of Moments Estimating Equations Based on Secant-vector Divisions Procedure[J].International Journal of Scientific and Statistical Computing,2011,2(1).
[21]Hansen L P.Proofs for Large Sample Properties of Generalized Method of Moments Estimators[J].Journal of Econometrics,2012,170(2).
[22] Wang Y G,Lin X,Zhu M.Robust Estimating Functions and Bias Correction for Longitudinal Data Analysis[J].Biometrics,2005,61(3).
[2]田萍.纵向数据半参数回归模型的估计理论[M].郑州:郑州大学出版社,2008.
[3]赵明涛,许晓丽.半参数纵向模型的惩罚二次推断函数估计[J].统计与信息论坛,2014(8).
[4] Tian P,Yang L,Xue L.Asymptotic Properties of Estimators in Partially Linear Single-index Model for Longitudinal Data[J].Acta Mathematica Scientia,2010,30(3).
[5]郑强,冉光和.中国双向FDI的绿色生产率溢出效应——基于动态面板模型的实证检验[J].统计与信息论坛,2018,33(6).
[6] Hardle W.Nonparametric and Semiparametric Models[M].Berlin:Springer Verlag,2004.
[7] Liang K Y,Zeger S L.Longitudinal Data Analysis Using Generalized Linear Models[J].Biometrika,1986,73(1).
[8] Qu A,Lindsay B G,Li B.Improving Generalised Estimating Equations Using Quadratic Inference Functions[J].Biometrika,2000,87(4).
[9] Fan J,Lin X.New Developments in Biostatistics and Bioinformatics[M].Beijing:World Scientific,2009.
[10]Qu A,Li R.Quadratic Inference Functions for Varying-coefficient Models with Longitudinal Data[J].Biometrics,2006(2).
[11]Qu A,Lindsay B G.Building Adaptive Estimating Equations when Inverse of Covariance Estimation is Difficult[J].Journal of the Royal Statistical Society:Series B(Statistical Methodology),2003,65(1).
[12]Qu A,Song P X K.Assessing Robustness of Generalised Estimating Equations and Quadratic Inference Functions[J].Biometrika,2004,91(2).
[13]Qu A.Assessing the Validity of Weighted Generalized Estimating Equations[J].Biometrika,2011,98(1).
[14]Bai Y,Fung W K,Zhu Z Y.Penalized Quadratic Inference Functions for Single-index Models with Longitudinal Data[J].Journal of Multivariate Analysis,2009,100(1).
[15]Lai P,Li G,Lian H.Quadratic Inference Functions for Partially Linear Single-index Models with Longitudinal Data[J].Journal of Multivariate Analysis,2013,118(5).
[16]Ma S,Liang H,Tsai C L.Partially Linear Single-index Models for Repeated Measurements[J].Journal of Multivariate Analysis,2014,130(8).
[17]Li G,Lai P,Lian H.Variable Selection and Estimation for Partially Linear Single-index Models with Longitudinal Data[J].Statistics and Computing,2015,25(3).
[18]Han P,Song P X K.A Note on Improving Quadratic Inference Functions Using a Linear Shrinkage Approach[J].Statistics&Probability Letters,2011,81(3).
[19] Westgate P M,Braun T M.An Improved Quadratic Inference Function for Parameter Estimation in the Analysis of Correlated Data[J].Statistics in Medicine,2013,32(19).
[20]Khan N M,Heenaye M.A Computationally Efficient Algorithm to Solve Generalized Method of Moments Estimating Equations Based on Secant-vector Divisions Procedure[J].International Journal of Scientific and Statistical Computing,2011,2(1).
[21]Hansen L P.Proofs for Large Sample Properties of Generalized Method of Moments Estimators[J].Journal of Econometrics,2012,170(2).
[22] Wang Y G,Lin X,Zhu M.Robust Estimating Functions and Bias Correction for Longitudinal Data Analysis[J].Biometrics,2005,61(3).