Composite Quantile Regression for Varying Coefficient Models with Response Data Missing at Random

Composite quantile regression (CQR) estimation and inference are studied for varying coefficient models with response data missing at random. Three estimators including the weighted local linear CQR (WLLCQR) estimator, the nonparametric WLLCQR (NWLLCQR) estimator, and the imputed WLLCQR (IWLLCQR) estimator are proposed for unknown coefficient functions. Under some mild conditions, the proposed estimators are asymptotic normal. Simulation studies demonstrate that the unknown coefficient estimators with IWLLCQR are superior to the other two with WLLCQR and NWLLCQR. Moreover, bootstrap test procedures based on the IWLLCQR fittings is developed to test whether the coefficient functions are actually varying. Finally, a type of investigated real-life data is analyzed to illustrated the applications of the proposed method.

[1]  Liugen Xue.  Empirical Likelihood Confidence Intervals for Response Mean with Data Missing at Random , 2009 .

[2]  Jianqing Fan,et al.  Statistical Estimation in Varying-Coefficient Models , 1999 .

[3]  Raymond J Carroll,et al.  Multiple imputation in quantile regression. , 2012, Biometrika.

[4]  M. Kenward,et al.  Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls , 2009, BMJ : British Medical Journal.

[5]  Zongwu Cai,et al.  Nonparametric Quantile Estimations for Dynamic Smooth Coefficient Models , 2008 .

[6]  Lan Wang,et al.  Weighted quantile regression for analyzing health care cost data with missing covariates , 2013, Statistics in medicine.

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

[8]  Yoon-Jae Whang,et al.  SMOOTHED EMPIRICAL LIKELIHOOD METHODS FOR QUANTILE REGRESSION MODELS , 2004, Econometric Theory.

[9]  Chaohui Guo,et al.  Weighted composite quantile regression estimation and variable selection for varying coefficient models with heteroscedasticity , 2015 .

[10]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[11]  Chin-Tsang Chiang,et al.  Smoothing Spline Estimation for Varying Coefficient Models With Repeatedly Measured Dependent Variables , 2001 .

[12]  J. N. K. Rao,et al.  Empirical likelihood-based inference under imputation for missing response data , 2002 .

[13]  Zhihua Sun,et al.  Estimation in partially linear models with missing responses at random , 2007 .

[14]  Chunfeng Huang,et al.  Smoothing spline estimation in varying‐coefficient models , 2004 .

[15]  Keith Knight,et al.  Limiting distributions for $L\sb 1$ regression estimators under general conditions , 1998 .

[16]  Heung Wong,et al.  On locally weighted estimation and hypothesis testing of varying-coefficient models with missing covariates , 2009 .

[17]  Yujie Gai,et al.  Weighted local linear composite quantile estimation for the case of general error distributions , 2013 .

[18]  Jianhua Z. Huang,et al.  Varying‐coefficient models and basis function approximations for the analysis of repeated measurements , 2002 .

[19]  Nonparametric $$M$$M-type regression estimation under missing response data , 2016 .

[20]  H. Zou,et al.  Composite quantile regression and the oracle Model Selection Theory , 2008, 0806.2905.

[21]  R. Tibshirani,et al.  Varying‐Coefficient Models , 1993 .

[22]  Shu Yang,et al.  Imputation methods for quantile estimation under missing at random , 2013 .

[23]  Rui Li,et al.  Smoothed empirical likelihood analysis of partially linear quantile regression models with missing response variables , 2013 .

[24]  Peter B Gilbert,et al.  Quantile Regression for Competing Risks Data with Missing Cause of Failure. , 2012, Statistica Sinica.

[25]  Qingguo Tang.  Robust estimation for spatial semiparametric varying coefficient partially linear regression , 2015 .

[26]  Chin-Tsang Chiang,et al.  A Two-Step Smoothing Method for Varying-Coefficient Models with Repeated Measurements , 2000 .

[27]  Nageswara S. V. Rao,et al.  Nadaraya-Watson estimator for sensor fusion , 1997 .

[28]  Mi-Ok Kim,et al.  Quantile regression with varying coefficients , 2007, 0708.0471.

[29]  D. Rubin.  INFERENCE AND MISSING DATA , 1975 .

[30]  Yong Zhou,et al.  Efficient Quantile Regression Analysis With Missing Observations , 2015 .

[31]  Oliver Linton,et al.  Semiparametric Regression Analysis With Missing Response at Random , 2003 .

[32]  Kai,et al.  NEW EFFICIENT AND ROBUST ESTIMATION IN VARYING-COEFFICIENT MODELS WITH HETEROSCEDASTICITY , 2012 .

[33]  Zongwu Cai,et al.  Semiparametric quantile regression estimation in dynamic models with partially varying coefficients , 2012 .

[34]  Runze Li,et al.  NEW EFFICIENT ESTIMATION AND VARIABLE SELECTION METHODS FOR SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS. , 2011, Annals of statistics.