Letf(x) be the density of a design variableX andm(x) = E[Y∣X = x] the regression function. Thenm(x) = G(x)/f(x), whereG(x) = rn(x)f(x). The Dirac δ-function is used to define a generalized empirical functionG n(x) f...Letf(x) be the density of a design variableX andm(x) = E[Y∣X = x] the regression function. Thenm(x) = G(x)/f(x), whereG(x) = rn(x)f(x). The Dirac δ-function is used to define a generalized empirical functionG n(x) forG(x) whose expectation equalsG(x). This generalized empirical function exists only in the space of Schwartz distributions, so we introduce a local polynomial of orderp approximation toG n(.) which provides estimators of the functionG(x) and its derivatives. The densityf(x) can be estimated in a similar manner. The resulting local generalized empirical estimator (LGE ) ofm(x) is exactly the Nadaraya-Watson estimator at interior points whenp = 1, 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 withp = 1 compares favorably with the Nadaraya-Watson and the popular local linear regression smoother.展开更多
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.10001004 and 39930160)by the US NSF(Grant No.DMS-9971301).
文摘Letf(x) be the density of a design variableX andm(x) = E[Y∣X = x] the regression function. Thenm(x) = G(x)/f(x), whereG(x) = rn(x)f(x). The Dirac δ-function is used to define a generalized empirical functionG n(x) forG(x) whose expectation equalsG(x). This generalized empirical function exists only in the space of Schwartz distributions, so we introduce a local polynomial of orderp approximation toG n(.) which provides estimators of the functionG(x) and its derivatives. The densityf(x) can be estimated in a similar manner. The resulting local generalized empirical estimator (LGE ) ofm(x) is exactly the Nadaraya-Watson estimator at interior points whenp = 1, 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 withp = 1 compares favorably with the Nadaraya-Watson and the popular local linear regression smoother.
