Empirical likelihood inference for parameters in a partially linear errors-in-variables model

In this paper, we consider the application of the empirical likelihood method to a partially linear model with measurement errors in the non-parametric part. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter by using the empirical log-likelihood ratio function, and the resulting estimator is shown to be asymptotically normal. Some simulations and an application are conducted to illustrate the proposed method.

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

[2]  H. Cui,et al.  Empirical likelihood inference for partial linear models under martingale difference sequence , 2008 .

[3]  Bing-Yi Jing,et al.  Empirical likelihood for partial linear models with fixed designs , 1999 .

[4]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .

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

[6]  Xuming He,et al.  Quantile Regression Estimates for a Class of Linear and Partially Linear Errors-in-Variables Models , 1997 .

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

[8]  Jianqing Fan,et al.  Nonparametric regression with errors in variables , 1993 .

[9]  Y. Qin.  Empirical Lilelihood Ratio Confidence Regions in A Partly Linear Model , 1999 .

[10]  Hengjian Cui,et al.  A Semi‐parametric Regression Model with Errors in Variables , 2003 .

[11]  Hua Liang,et al.  Empirical-likelihood-based semiparametric inference for the treatment effect in the two-sample problem with censoring. , 2005, Biometrika.

[12]  Hua Liang,et al.  Asymptotic Normality of Parametric Part in Partially Linear Models with Measurement Error in the Nonparametric Part , 2000 .

[13]  Jian Shi,et al.  Empirical Likelihood for Partially Linear Models , 2000 .

[14]  R. Carroll,et al.  Deconvolving kernel density estimators , 1987 .

[15]  Hengjian Cui,et al.  Empirical Likelihood Confidence Region for Parameters in Semi‐linear Errors‐in‐Variables Models , 2006 .

[16]  Cui Hengjian,et al.  On Parameter Estimation for Semi-linear Errors-in-Variables Models , 1998 .

[17]  Song Xi Chen,et al.  Empirical Likelihood Confidence Intervals for Linear Regression Coefficients , 1994 .

[18]  W. Härdle,et al.  Estimation in a semiparametric partially linear errors-in-variables model , 1999 .

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

[20]  J. N. K. Rao,et al.  Empirical likelihood-based inference in linear errors-in-covariables models with validation data , 2002 .

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