Estimation of Partial Linear Error-in-Variables Models with Validation Data

Consider the partial linear models of the formY=Xτβ+g(T)+e, where thep-variate explanatoryXis erroneously measured, and bothTand the responseYare measured exactly. LetXbe the surrogate variable forXwith measurement error. Let the primary data set be that containing independent observations on (Y, X, T) and the validation data set be that containing independent observations on (X, X, T), where the exact observations onXmay be obtained by some expensive or difficult procedures for only a small subset of subjects enrolled in the study. In this paper, without specifying any structure equation and the distribution assumption ofXgivenX, a semiparametric method with the primary data is employed to obtain the estimators ofβandg(·) based on the least-squares criterion with the help of validation data. The proposed estimators are proved to be strongly consistent. The asymptotic representation and the asymptotic normality of the estimator ofβare derived, respectively. The rate of the weak consistency of the estimator ofg(·) is also obtained.