Estimation and empirical likelihood for single-index models with missing data in the covariates

The estimation and empirical likelihood for single-index models with missing covariates are studied. A generalized estimating equations estimator for index coefficients with missing covariates is constructed, and its asymptotic distribution is obtained. The local linear estimator for link function achieves optimal convergence rate. By using the bias-correction and inverse selection probability weighted methods, a class of empirical likelihood ratios is proposed such that each of our class of ratios is asymptotically chi-squared. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths (areas) of confidence intervals (regions). An example of a real data set is illustrated.

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

[2]  Thomas M. Stoker,et al.  Semiparametric Estimation of Index Coefficients , 1989 .

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

[4]  J. N. K. Rao,et al.  Empirical Likelihood‐based Inference in Linear Models with Missing Data , 2002 .

[5]  A. Owen.  Empirical Likelihood Ratio Confidence Regions , 1990 .

[6]  Jeng-Min Chiou,et al.  Quasi-Likelihood Regression with Unknown Link and Variance Functions , 1998 .

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

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

[9]  Junshan Shen,et al.  Estimation of single index model with missing response at random , 2010 .

[10]  Ker-Chau Li,et al.  Slicing Regression: A Link-Free Regression Method , 1991 .

[11]  H. Tong,et al.  Article: 2 , 2002, European Financial Services Law.

[12]  Lixing Zhu,et al.  Empirical Likelihood Inference in Nonlinear Errors-in-Covariables Models With Validation Data , 2007 .

[13]  Thomas M. Stoker.  Consistent estimation of scaled coefficients , 2011 .

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

[15]  B. Jørgensen,et al.  The theory of linear models , 1995 .

[16]  Lixing Zhu,et al.  Empirical likelihood for single-index models , 2006 .

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

[18]  Liugen Xue.  Empirical likelihood for linear models with missing responses , 2009, J. Multivar. Anal..

[19]  Peter Hall,et al.  Smoothed empirical likelihood confidence intervals for quantiles , 1993 .

[20]  Biao Zhang,et al.  Empirical‐likelihood‐based inference in missing response problems and its application in observational studies , 2007 .

[21]  Jiahua Chen,et al.  Empirical likelihood estimation for ?nite populations and the e?ective usage of auxiliary informatio , 1993 .

[22]  Wei Lin,et al.  Identifiability of single-index models and additive-index models , 2007 .

[23]  Ker-Chau Li.  Sliced inverse regression for dimension reduction (with discussion) , 1991 .

[24]  Lixing Zhu,et al.  Empirical Likelihood for a Varying Coefficient Model With Longitudinal Data , 2007 .

[25]  Liugen Xue,et al.  Empirical likelihood for semiparametric regression model with missing response data , 2011, J. Multivar. Anal..

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

[27]  H. Tong,et al.  An adaptive estimation of dimension reduction space, with discussion , 2002 .

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

[29]  J. Lawless,et al.  Empirical Likelihood and General Estimating Equations , 1994 .

[30]  A. Owen.  Empirical likelihood ratio confidence intervals for a single functional , 1988 .

[31]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[32]  S. R. Searle,et al.  The theory of linear models and multivariate analysis , 1981 .

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

[34]  Ker-Chau Li,et al.  Sliced Inverse Regression for Dimension Reduction , 1991 .

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

[36]  Liugen Xue,et al.  Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data , 2010 .

[37]  Lixing Zhu,et al.  Empirical Likelihood Semiparametric Regression Analysis for Longitudinal Data , 2007 .