Variable Selection and Empirical Likelihood based Inference for Measurement Error Data

Using nonconvex penalized least squares, we propose a class of variable selection procedures for linear models and partially linear models when the covariates are measured with additive error. The rate of convergence and the asymptotic normality of the resulting estimate are established. We further demonstrate that, with proper choice of penalty functions and the regularization parameter, the resulting estimate performs as well as an oracle procedure. A robust standard error formula is derived using a sandwich formula, and empirically tested. Local polynomial regression techniques are used to estimate the baseline function in the partially linear model. To avoid to estimate the asymptotic covariance in establishing confidence region of the parameter of interest, we further develop a statistic based on the empirical likelihood principle, and show that the statistic is asymptotically chi-squared distributed. Finite sample performance of the proposed inference procedures is assessed by Monte Carlo simulation studies. We further illustrate the proposed procedures by an application.