Related Topics in Partially Linear Models: Semi-parametric Regression, Measurement Errors,Missing Data, Single-index Models, Regression Calibration

Various effects have been made to remedy the curse ofdimensionality for high-dimensional data. Partiallylinear models, as an effective dimensional reductiontechnique, have been intensively studied inliterature. We develop methodology for the estimationof regression parameters in partially linear modelswhen the covariates are measured with errors or maybe missing. We are particularly concerned with twocases where we observe a surrogate of the covariate.The second case focuses on the linear covariatebeing incompletely observable. We give thecorresponding solutions for the above problems. Thefirst solution employs the technique of correctingfor attenuation. The second is proposed using inverseweight probability. The resulting estimators areproven to be asymptotically normal. The model is usedto analyze a data set from the Framingham Heart Studyfor the purpose of illustrating the methods. We alsoinvestigate the semiparametric partially linearsingle index errors-in-variables models, for whichtwo classes of estimators are proposed, and thecorresponding theoretical properties are derived andcompared.