Estimation of partial linear error-in-variables models for ρ−-mixing dependence data

摘要

Consider the partly linear regression model Y = xβ + g(t) + e where the explanatory x is erroneously measured, and both t and the response Y are measured exactly, the random error e is ρ−-mixing. Let [(x)tilde]tilde{x} be a surrogate variable observed instead of the true x in the primary survey data. Assume that in addition to the primary data set containing N observations of {(Y_j,[(x)tilde]_j,t_j)_j=n+1n+N}big{(Y_{j},tilde{x}_{j},t_{j})_{j=n+1}^{n+N}big}, which is ρ−-mixing data sets, an independent validation data containing n observations of {(x_j,[(x)tilde]_j,t_j)_j=1n}big{(x_{j},tilde{x}_{j},t_{j})_{j=1}^{n}big} is available. The exact observations on x may be obtained by some expensive or diffcult procedures for only a small subset of subjects enrolled in the study. In this paper, inspired by Berberan-Santos et al. [J. Math. Chem. 37 (2005)101], a semiparametric method with the primary data is employed to obtain the estimators of β and g(·) based on the least squares criterion with the help of validata. The proposed estimators are proved to be strongly consistent.