部分线性分位回归模型估计的MM算法
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摘要
近年来,关于部分线性分位回归模型的估计方法的研究得到了较多的关注.但由于目标函数的非光滑性,估计程序的实现是比较具有挑战性的.文章将采用MM(Majorization Minimization)算法计算部分线性分位数回归模型的估计.其基本原理是首先找到目标函数的优化函数,然后借助优化函数的最小化过程,逐步迭代至目标函数的解.数值模拟和实证研究表明该算法具有较好的稳定性和较强的数值计算能力.
The estimation of the partial linear quantile regression model has obtained much attention in recent years.However,the realizations of the estimating methods are very challenging since the objective functions are non-smooth.In this paper,we employ the MM(Majorization Minimization)algorithm to calculate the estimators of the regression coefficient of the partial linear quantile regression model.This algorithm first finds surrogate functions that minimize the objective functions.Then via optimizing the surrogate function,the solutions can be obtained which minimize the objective functions.To validate the performance of the proposed algorithm,extensive simulation studies are conducted.It is shown that the proposed algorithm is robust and computationally competitive with the common-used interior algorithm.
The estimation of the partial linear quantile regression model has obtained much attention in recent years.However,the realizations of the estimating methods are very challenging since the objective functions are non-smooth.In this paper,we employ the MM(Majorization Minimization)algorithm to calculate the estimators of the regression coefficient of the partial linear quantile regression model.This algorithm first finds surrogate functions that minimize the objective functions.Then via optimizing the surrogate function,the solutions can be obtained which minimize the objective functions.To validate the performance of the proposed algorithm,extensive simulation studies are conducted.It is shown that the proposed algorithm is robust and computationally competitive with the common-used interior algorithm.
引文
[1]Yu D,Kong L,Mizera I.Partial functional linear quantile regression for neuroimaging data analysis.Neurocomputing, 2016,195(C):74-87.
[2]Sherwood B,Wang L.Partially linear additive quantile regression in ultra-high dimension.Annals of Statistics,2016,44(1):288-317.
[3]Hu Y,Zhao K,Lian H.Bayesian quantile regression for partially linear additive models.Statistics and Computing,2015,25(3):651-668.
[4]翁云妹.半参数变系数分位数回归模型及其两阶段估计:以波士顿房价应用为例.硕士论文,厦门大学,2008.(Weng Y M.Semiparametric functional coefficient quantile regression and its two-step estimation procedure:An application to Boston housing prices data.Master Thesis,Xiamen University,2008.)
[5]李红梅.基于半参数分位数回归的居民个人收入研究.数学的实践与认识,2012,42(8):55-61.(Li H M.Study on the individual income based on the semi-parametric quantile regression model.Mathematics in Practice and Theory, 2012,42(8):55-61.)
[6]关静.分位数回归理论及其应用.博士论文,天津大学,2009.(Guan J.The theory of quantile regression and applications.Doctoral Thesis,Tianjin University,2009.)
[7]姜成飞.分位数回归方法综述.科技信息,2013,(25):185.(Jiang C F.A review of quantile regression method.Science&Technology Information, 2013,(25):185.)
[8]朱平芳,张征宇.无条件分位数回归:文献综述与应用实例.统计研究,2012,29(3):88-96.(Zhu P F,Zhang Z Y.Unconditional quantile regression:Literature review and empirical example.Statistical Research,2012,29(3):88-96.)
[9]陈建宝,丁军军.分位数回归技术综述.统计与信息论坛,2008,23(3):89-96.(Chen J B,Ding J J.A review of technologies on quantile regression.Statistics&Information Forum,2008,23(3):89-96.)
[10]Hunter D,Kenneth L.Quantile regression via an MM algorithm.Journal of Computational and Graphical Statistics,2000,9(1):60-77.
[11]Sun Y.Semiparametric efficient estimation of partially linear quantile regression models.Annals of Economics and Finance, 2005,6(1):105-127.
[12]孙志华,尹俊平,陈菲菲.非参数与半参数统计.北京:清华大学出版社,2016.(Sun Z H,Yin J P,Chen F F.Nonparametric and Semiparametric Statistics.Beijing:Tsinghua University Press,2016.)
[13]余平,杜江,张忠.占部分函数型线性可加分位数回归模型.系统科学与数学,2017,37(5):1335-1350.(Yu P,Du J,Zhang Z.Partial function linear addictive quantile regression model.Journal of Systems Science and Mathematical Sciences, 2017,37(5):1335-1350.)
[14]Peng Q Y,Zhou J J,Tang N S.Varying coefficient partially functional linear regression models.Statistical Papers,2016,57(3):1-15.
[15]Zou Q,Zhu Z.M-estimators for single-index model using B-spline.Metrika,2014,77(2):225-246.
[16]Chen X,Sun J,Liu L.Semiparametric partial linear quantile regression of longitudinal data with time varying coefficients and informative observation times.Statistica Sinica, 2016,25(4):1437-1458.
[17]Aneiros-Perez G,Vieu P.Semi-functional partial linear regression.Statistics&Probability Letters,2006,76(11):1102-1110.
[2]Sherwood B,Wang L.Partially linear additive quantile regression in ultra-high dimension.Annals of Statistics,2016,44(1):288-317.
[3]Hu Y,Zhao K,Lian H.Bayesian quantile regression for partially linear additive models.Statistics and Computing,2015,25(3):651-668.
[4]翁云妹.半参数变系数分位数回归模型及其两阶段估计:以波士顿房价应用为例.硕士论文,厦门大学,2008.(Weng Y M.Semiparametric functional coefficient quantile regression and its two-step estimation procedure:An application to Boston housing prices data.Master Thesis,Xiamen University,2008.)
[5]李红梅.基于半参数分位数回归的居民个人收入研究.数学的实践与认识,2012,42(8):55-61.(Li H M.Study on the individual income based on the semi-parametric quantile regression model.Mathematics in Practice and Theory, 2012,42(8):55-61.)
[6]关静.分位数回归理论及其应用.博士论文,天津大学,2009.(Guan J.The theory of quantile regression and applications.Doctoral Thesis,Tianjin University,2009.)
[7]姜成飞.分位数回归方法综述.科技信息,2013,(25):185.(Jiang C F.A review of quantile regression method.Science&Technology Information, 2013,(25):185.)
[8]朱平芳,张征宇.无条件分位数回归:文献综述与应用实例.统计研究,2012,29(3):88-96.(Zhu P F,Zhang Z Y.Unconditional quantile regression:Literature review and empirical example.Statistical Research,2012,29(3):88-96.)
[9]陈建宝,丁军军.分位数回归技术综述.统计与信息论坛,2008,23(3):89-96.(Chen J B,Ding J J.A review of technologies on quantile regression.Statistics&Information Forum,2008,23(3):89-96.)
[10]Hunter D,Kenneth L.Quantile regression via an MM algorithm.Journal of Computational and Graphical Statistics,2000,9(1):60-77.
[11]Sun Y.Semiparametric efficient estimation of partially linear quantile regression models.Annals of Economics and Finance, 2005,6(1):105-127.
[12]孙志华,尹俊平,陈菲菲.非参数与半参数统计.北京:清华大学出版社,2016.(Sun Z H,Yin J P,Chen F F.Nonparametric and Semiparametric Statistics.Beijing:Tsinghua University Press,2016.)
[13]余平,杜江,张忠.占部分函数型线性可加分位数回归模型.系统科学与数学,2017,37(5):1335-1350.(Yu P,Du J,Zhang Z.Partial function linear addictive quantile regression model.Journal of Systems Science and Mathematical Sciences, 2017,37(5):1335-1350.)
[14]Peng Q Y,Zhou J J,Tang N S.Varying coefficient partially functional linear regression models.Statistical Papers,2016,57(3):1-15.
[15]Zou Q,Zhu Z.M-estimators for single-index model using B-spline.Metrika,2014,77(2):225-246.
[16]Chen X,Sun J,Liu L.Semiparametric partial linear quantile regression of longitudinal data with time varying coefficients and informative observation times.Statistica Sinica, 2016,25(4):1437-1458.
[17]Aneiros-Perez G,Vieu P.Semi-functional partial linear regression.Statistics&Probability Letters,2006,76(11):1102-1110.