Local generalized empirical estimation of regression

Let be the density of a design variable and the regression function. Then , where . The Dirac Æ-function is used to define a generalized empirical function for whose expectation equals . This generalized empirical function exists only in the space of Schwartz distributions, so we introduce a local polynomial of order approximation to which provides estimators of the function and its derivatives. The density can be estimated in a similar manner. The resulting local generalized empirical estimator (LGE ) of is exactly the Nadaraya-Watson estimator at interior points when , but on the boundary the estimator automatically corrects the boundary effect. Asymptotic normality of the estimator is established. Asymptotic expressions for the mean squared errors are obtained and used in bandwidth selection. Boundary behavior of the estimators is investigated in details. We use Monte Carlo simulations to show that the proposed estimator with compares favorably with the Nadaraya-Watson and the popular local linear regression smoother.