因变量缺失下部分线性变系数模型的稳健估计
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摘要
本文讨论因变量缺失下部分线性变系数模型在误差项和解释变量都含有异常点时的稳健估计问题。首先用局部加权线性光滑方法得到非参数部分的稳健估计,然后再得到参数部分的估计,并证明参数和非参数估计量的渐近正态性。最后模拟研究有限样本下估计量的表现。
In this paper, we consider the robust least-squares technique for estimating the parameter component and non-parameter component in the partially linear varying-coefficient models with missing response. Our proposal is resistant against outliers in the residuals and in the carriers as well. Using local weighted linear smoothing method to get the robust estimators of the non-parameter component firstly, and then obtain the estimation of the parameter part, the asymptotic distribution and asymptotic properties of the proposed estimators are discussed. Finally, a simulation study is conducted to examine the finite sample properties of the proposed estimators.
In this paper, we consider the robust least-squares technique for estimating the parameter component and non-parameter component in the partially linear varying-coefficient models with missing response. Our proposal is resistant against outliers in the residuals and in the carriers as well. Using local weighted linear smoothing method to get the robust estimators of the non-parameter component firstly, and then obtain the estimation of the parameter part, the asymptotic distribution and asymptotic properties of the proposed estimators are discussed. Finally, a simulation study is conducted to examine the finite sample properties of the proposed estimators.
引文
[1] Zhang W,Lee S,Song X. Local polynomial fitting in semivarying-coefficient model[J]. Multivar Anal, 2002, 82:166-188.
[2] Fan J Q, Huang T. Profile likelihood inferences on semi-parametric varying-coefficient partially linear models[J]. Bernoulli, 2005, 11:1031-1057.
[3] Xia Y, Zhang W, Tong H. Efficient estimation for semivarying-coefficient models[J]. Biometrika,2004, 91:661-681.
[4] Zhou X, You J. Wavelet estimation in varying coefficient partially linear regression models[J]. Statist Probab Lett, 2004, 68:91-104.
[5]张波,郭海兵.部分线性变系数模型的一种新的轮廓(Profile)最小二乘估计[J].数理统计与管理,2015,34(2):275-283.
[6]杨随根,冯三营,薛留根.部分函数线性变系数模型的样条估计及其应用[J].数理统计与管理,2015,34(5):831-839.
[7] Little R, Rubin D. Statistical Analysis with Missing Data[M]. New York:John Wiley, 2002.
[8] Cheng P. Nonparametric estimation of mean functionals with data missing at random[J]. Journal of the American Statistical Association, 1990, 89:81-87.
[9] Chu C, Cheng P. Nonparametric regression estimation with missing data[J]. Journal of Statistical Planning&Inference, 1995, 48:85-99.
[10] Wang Q, Linton O, Hardle W. Semiparametric regression analysis with missing response at random[J]. Journal of the American Statistical Association, 2004, 99:334-345.
[11] Wang Q, Sun Z. Estimation in partially linear models with missing responses at random[J]. Journal of Multivariate Analysis, 2007, 98:1470-1493.
[12]孙志猛,张忠占,杜江.缺失数据下半参数单调回归模型的估计[J].数理统计与管理,2011,30(6):979-988.
[13] Wei C H. Statistical inference in partially linear varying-coefficient models with missing responses at random[J]. Communications in Statistics—Theory and Methods, 2012, 41:1284-1298.
[14]魏传华.因变量缺失下部分线性变系数变量含误差模型的估计[J].数学物理学报,2010, 30A(4):1042-1054.
[15] Bravo F. Partially linear varying coefficient models with missing at random responses[J]. Annals of the Institute of Statistical Mathematics, 2013, 65:721-762.
[16]邱贻涛,吴刘仓,马婷.缺失数据下联合均值与方差模型的参数估计[J].数理统计与管理,2015, 34(4):621-627.
[17] Bianco A, Boente G. Robust estimators in semi-parametric partly linear regression models[J]. Journal of Statistical Planning and Inference, 2004, 122:229-252.
[18] Luo S H, Zhang C Y, Xu F M. The local linear M-estimation with missing response data[J]. Journal of Applied Mathematics, 2014,(2):1-10.
[19]彭佳,李长青,王晓燕.基于Walsh平均的非参数回归模型的稳健估计[J].数理统计与管理,2015, 34(4):636-646.
[20]谢振中.基于M估计的稳健单指数投资组合模型[J].数理统计与管理,2015, 34(2):349-356.
[21]富伯亭,郑刘锁.半变系数模型的M估计[J].工程数学学报,2009, 26(4):609-617.
[22] Huber P. Robust Statistics[M]. New York:Wiley, 1981.
[23] Yohai V. High breakdown point and high efficiency robust estimates for regression[J]. The Annals of Statistics, 1987, 15:642-656.
[2] Fan J Q, Huang T. Profile likelihood inferences on semi-parametric varying-coefficient partially linear models[J]. Bernoulli, 2005, 11:1031-1057.
[3] Xia Y, Zhang W, Tong H. Efficient estimation for semivarying-coefficient models[J]. Biometrika,2004, 91:661-681.
[4] Zhou X, You J. Wavelet estimation in varying coefficient partially linear regression models[J]. Statist Probab Lett, 2004, 68:91-104.
[5]张波,郭海兵.部分线性变系数模型的一种新的轮廓(Profile)最小二乘估计[J].数理统计与管理,2015,34(2):275-283.
[6]杨随根,冯三营,薛留根.部分函数线性变系数模型的样条估计及其应用[J].数理统计与管理,2015,34(5):831-839.
[7] Little R, Rubin D. Statistical Analysis with Missing Data[M]. New York:John Wiley, 2002.
[8] Cheng P. Nonparametric estimation of mean functionals with data missing at random[J]. Journal of the American Statistical Association, 1990, 89:81-87.
[9] Chu C, Cheng P. Nonparametric regression estimation with missing data[J]. Journal of Statistical Planning&Inference, 1995, 48:85-99.
[10] Wang Q, Linton O, Hardle W. Semiparametric regression analysis with missing response at random[J]. Journal of the American Statistical Association, 2004, 99:334-345.
[11] Wang Q, Sun Z. Estimation in partially linear models with missing responses at random[J]. Journal of Multivariate Analysis, 2007, 98:1470-1493.
[12]孙志猛,张忠占,杜江.缺失数据下半参数单调回归模型的估计[J].数理统计与管理,2011,30(6):979-988.
[13] Wei C H. Statistical inference in partially linear varying-coefficient models with missing responses at random[J]. Communications in Statistics—Theory and Methods, 2012, 41:1284-1298.
[14]魏传华.因变量缺失下部分线性变系数变量含误差模型的估计[J].数学物理学报,2010, 30A(4):1042-1054.
[15] Bravo F. Partially linear varying coefficient models with missing at random responses[J]. Annals of the Institute of Statistical Mathematics, 2013, 65:721-762.
[16]邱贻涛,吴刘仓,马婷.缺失数据下联合均值与方差模型的参数估计[J].数理统计与管理,2015, 34(4):621-627.
[17] Bianco A, Boente G. Robust estimators in semi-parametric partly linear regression models[J]. Journal of Statistical Planning and Inference, 2004, 122:229-252.
[18] Luo S H, Zhang C Y, Xu F M. The local linear M-estimation with missing response data[J]. Journal of Applied Mathematics, 2014,(2):1-10.
[19]彭佳,李长青,王晓燕.基于Walsh平均的非参数回归模型的稳健估计[J].数理统计与管理,2015, 34(4):636-646.
[20]谢振中.基于M估计的稳健单指数投资组合模型[J].数理统计与管理,2015, 34(2):349-356.
[21]富伯亭,郑刘锁.半变系数模型的M估计[J].工程数学学报,2009, 26(4):609-617.
[22] Huber P. Robust Statistics[M]. New York:Wiley, 1981.
[23] Yohai V. High breakdown point and high efficiency robust estimates for regression[J]. The Annals of Statistics, 1987, 15:642-656.