Estimation for partially varying-coefficient single-index models with distorted measurement errors

In this paper, we study partially varying-coefficient single-index model where both the response and predictors are observed with multiplicative distortions which depend on a commonly observable confounding variable. Due to the existence of measurement errors, the existing methods cannot be directly applied, so we recommend using the nonparametric regression to estimate the distortion functions and obtain the calibrated variables accordingly. With these corrected variables, the initial estimators of unknown coefficient and link functions are estimated by assuming that the parameter vector $$\beta $$ β is known. Furthermore, we can obtain the least square estimators of unknown parameters. Moreover, we establish the asymptotic properties of the proposed estimators. Simulation studies and real data analysis are given to illustrate the advantage of our proposed method.

[1]  Yong Zhou,et al.  Corrected local polynomial estimation in varying‐coefficient models with measurement errors , 2006 .

[2]  Lixing Zhu,et al.  The EFM approach for single-index models , 2011, 1211.5220.

[3]  Jianbo Li,et al.  Penalized Spline Varying-Coefficient Single-Index Model , 2010, Commun. Stat. Simul. Comput..

[4]  Zhensheng Huang.  Efficient inferences on the varying-coefficient single-index model with empirical likelihood , 2012, Comput. Stat. Data Anal..

[5]  Shuang Dai,et al.  Estimation for varying coefficient partially nonlinear models with distorted measurement errors , 2019, Journal of the Korean Statistical Society.

[6]  D. Rubinfeld,et al.  Hedonic housing prices and the demand for clean air , 1978 .

[7]  D. Ruppert,et al.  Penalized Spline Estimation for Partially Linear Single-Index Models , 2002 .

[8]  Qi Li,et al.  Efficient estimation of a semiparametric partially linear varying coefficient model , 2005, math/0504510.

[9]  Lixing Zhu,et al.  Partial linear single index models with distortion measurement errors , 2013 .

[10]  Lixing Zhu,et al.  Empirical likelihood confidence regions in a partially linear single‐index model , 2006 .

[11]  Zhensheng Huang,et al.  TESTS FOR VARYING-COEFFICIENT PARTS ON VARYING-COEFFICIENT SINGLE-INDEX MODEL , 2010 .

[12]  Lu Lin,et al.  Covariate-Adjusted Partially Linear Regression Models , 2010 .

[13]  Tizheng Li,et al.  Estimation and inference for varying coefficient partially nonlinear models , 2013 .

[14]  Riquan Zhang,et al.  Empirical likelihood for the varying‐coefficient single‐index model , 2010 .

[15]  W. Härdle,et al.  Optimal Smoothing in Single-index Models , 1993 .

[16]  Jianqing Fan,et al.  Profile likelihood inferences on semiparametric varying-coefficient partially linear models , 2005 .

[17]  Damla Şentürk,et al.  Covariate-adjusted regression , 2005 .

[18]  Jianbo Li,et al.  Partially varying coefficient single index proportional hazards regression models , 2011, Comput. Stat. Data Anal..

[19]  Fan Feng,et al.  Efficient penalized estimating method in the partially varying-coefficient single-index model , 2013, J. Multivar. Anal..

[20]  L. Xue,et al.  Variable selection for partially varying coefficient single-index model , 2013 .

[21]  Jianqing Fan,et al.  Generalized Partially Linear Single-Index Models , 1997 .

[22]  Jialiang Li,et al.  A Functional Varying-Coefficient Single-Index Model for Functional Response Data , 2017, Journal of the American Statistical Association.

[23]  K. Fang,et al.  Asymptotics for kernel estimate of sliced inverse regression , 1996 .

[24]  Lixing Zhu,et al.  Covariate-adjusted nonlinear regression , 2009, 0908.1849.

[25]  Liugen Xue,et al.  Bias-corrected statistical inference for partially linear varying coefficient errors-in-variables models with restricted condition , 2014 .

[26]  Jinhong You,et al.  Estimation of a semiparametric varying-coefficient partially linear errors-in-variables model , 2006 .

[27]  Damla Şentürk,et al.  Covariate Adjusted Correlation Analysis via Varying Coefficient Models , 2005 .

[28]  Jane-ling Wang,et al.  Estimation for a partial-linear single-index model , 2009, 0905.2042.

[29]  Qihua Wang,et al.  Statistical inference in partially-varying-coefficient single-index model , 2011, J. Multivar. Anal..

[30]  Statistical inference for a varying-coefficient partially nonlinear model with measurement errors , 2016 .

[31]  Lixing Zhu,et al.  On an asymptotically more efficient estimation of the single-index model , 2010, J. Multivar. Anal..

[32]  Riquan Zhang,et al.  Varying-coefficient single-index model , 2008, Comput. Stat. Data Anal..

[33]  Wen-Xin Zhou,et al.  Nonparametric covariate-adjusted regression , 2016, 1601.02739.

[34]  Riquan Zhang,et al.  Efficient empirical-likelihood-based inferences for the single-index model , 2011, J. Multivar. Anal..

[35]  Runze Li,et al.  ESTIMATION AND TESTING FOR PARTIALLY LINEAR SINGLE-INDEX MODELS. , 2010, Annals of statistics.

[36]  B. Silverman,et al.  Weak and strong uniform consistency of kernel regression estimates , 1982 .

[37]  H. Müller,et al.  Local Polynomial Modeling and Its Applications , 1998 .