Local polynomial regression and simulation–extrapolation

Summary.  The paper introduces a new local polynomial estimator and develops supporting asymptotic theory for nonparametric regression in the presence of covariate measurement error. We address the measurement error with Cook and Stefanski's simulation–extrapolation (SIMEX) algorithm. Our method improves on previous local polynomial estimators for this problem by using a bandwidth selection procedure that addresses SIMEX's particular estimation method and considers higher degree local polynomial estimators. We illustrate the accuracy of our asymptotic expressions with a Monte Carlo study, compare our method with other estimators with a second set of Monte Carlo simulations and apply our method to a data set from nutritional epidemiology. SIMEX was originally developed for parametric models. Although SIMEX is, in principle, applicable to nonparametric models, a serious problem arises with SIMEX in nonparametric situations. The problem is that smoothing parameter selectors that are developed for data without measurement error are no longer appropriate and can result in considerable undersmoothing. We believe that this is the first paper to address this difficulty.