On 'Fit the Short Curve' Principle for Smoothing Nonparametric Estimators,

Abstract : Let (X, Y) be a bivariate random vector with EY < oo. The nonparametric regression problem is to estimate the regression function r(x) = E(Y/X = x) (1) based on a random sample (Xi, Yi), i=l,..,n from (X, Y). The Nadaraya-Watson (NW), the Nearest Neighbor (NN), and the Optimal Quantile (OQ) kernel type estimators of r(x) defined in (2)-(4) depend on smoothing parameters h, k and p, respectively. The asymptotic optimal form of these smoothing parameters is known, see Coulomb (1977) and Mack (1981).