Nonparametric regression estimation in the heteroscedastic errors-in-variables problem

In the classical errors-in-variables problem, the goal is to estimate a regression curve from data in which the explanatory variable is measured with error. In this context, nonparametric methods have been proposed that rely on the assumption that the measurement errors are identically distributed. Although there are many situations in which this assumption is too restrictive, nonparametric estimators in the more realistic setting of heteroscedastic errors have not been studied in the literature. We propose an estimator of the regression function in such a setting and show that it is optimal. We give estimators in cases in which the error distributions are unknown and replicated observations are available. Practical methods, including an adaptive bandwidth selector for the errors-in-variables regression problem, are suggested, and their finite-sample performance is illustrated through simulated and real data examples.