Estimating functionals of the error distribution in parametric and nonparametric regression

We consider estimation of linear functionals of the error distribution for two regression models: parametric and nonparametric, and for two types of errors: independent of the covariate and centered (type I), and conditionally centered given the covariate (type II). We show that the residual-based empirical estimators for the nonparametric type I model remain efficient in the type II model. For the parametric type I regression model, efficient estimators are obtained by correcting the empirical estimator using that the errors are centered, and using an efficient estimator for the regression parameter. Since such efficient parameter estimators do not remain consistent in the parametric type II model, neither does the empirical estimator. We construct efficient estimators for linear functionals of the error distribution in the parametric type II regression model, starting from residual-based empirical estimators, correcting it for the fact that the errors are conditionally centered, and using an appropriate efficient weighted least squares estimator for the regression parameter.