Dimension reduction in partly linear error-in-response models with validation data

Consider partial linear models of the form Y = X^τβ + g(T) + e with Y measured with error and both p-variate explanatory X and T measured exactly. Let Ỹ be the surrogate variable for Y with measurement error. Let primary data set be that containing independent observations on (Ỹ, X, T) and the validation data set be that containing independent observations on (Y, Ỹ, X, T), where the exact observations on Y may 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 equations and distribution assumption of Y given Ỹ, a semiparametric dimension reduction technique is employed to obtain estimators of β and g(·) based the least squared method and kernel method with the primary data and validation data. The proposed estimators of β are proved to be asymptotically normal, and the estimator for g(·) is proved to be weakly consistent with an optimal convergent rate.