Estimation in Partially Linear Single-Index Models with Missing Covariates

In this article, we consider a partially linear single-index model Y = g(Z τθ0) + X τβ0 + ϵ when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.

[1]  D. Horvitz,et al.  A Generalization of Sampling Without Replacement from a Finite Universe , 1952 .

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

[3]  A. A. Weiss,et al.  Semiparametric estimates of the relation between weather and electricity sales , 1986 .

[4]  H. Ichimura.  Estimation of single index models , 1987 .

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

[6]  J. Robins,et al.  Estimation of Regression Coefficients When Some Regressors are not Always Observed , 1994 .

[7]  Jiti Gao,et al.  Asymptotic theory for partly linear models , 1994 .

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

[9]  Clive W. J. Granger,et al.  Semiparametric estimates of the relation between weather and electricity sales , 1986 .

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

[11]  Nicole A. Lazar,et al.  Statistical Analysis With Missing Data , 2003, Technometrics.

[12]  ARROLL,et al.  Estimation in Partially Linear Models With Missing Covariates , 2004 .

[13]  Hua Liang,et al.  Partially linear single-index measurement error models , 2005 .

[14]  Lixing Zhu,et al.  Empirical likelihood confidence regions of the parameters in a partially linear single-index model , 2005 .

[15]  A. Tsiatis.  Semiparametric Theory and Missing Data , 2006 .

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

[17]  W. Härdle,et al.  Semi-parametric estimation of partially linear single-index models , 2006 .

[18]  Riquan Zhang.  Tests for nonparametric parts on partially linear single index models , 2007 .

[19]  Shuangge Ma.  Book review: Tsiatis, A.A. 2006: Semiparametric Theory and Missing Data. Springer , 2008 .

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

[21]  Hua Liang,et al.  Generalized Partially Linear Models With Missing Covariates. , 2008, Journal of multivariate analysis.

[22]  Dong Wang,et al.  EMPIRICAL LIKELIHOOD FOR ESTIMATING EQUATIONS WITH MISSING VALUES , 2009, 0903.0726.

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

[24]  Qihua Wang.  Statistical estimation in partial linear models with covariate data missing at random , 2009 .

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

[26]  Xiaohui Liu,et al.  Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models , 2011 .

[27]  Thomas M. Stoker,et al.  Investigating Smooth Multiple Regression by the Method of Average Derivatives , 2015 .