Abstract: The problem is to estimate a smooth regression function for the case of heteroscedastic nonparametric regression. Fixed and random design models are studied simultaneously. Neither smoothness of the estimated regression function nor nuisance functions, that is variance of errors and design density of predictors, are supposed to be known. For this setting we suggest an asymptotically sharp data driven estimate which has minimal constant and maximal rate of local minimax Mean Integrated Squared Error convergence as sample size tends to infinity. The analysis is based on recent results on nonparametric local asymptotic normality and equivalence, in the sense of Le Cam's deficiency distance, between a heteroscedastic regression and corresponding signal-in-noise model. A simplified adaptive estimator is suggested for the case of small sample sizes. This estimator is analyzed via Monte Carlo simulations and compared with an optimal pseudo local linear estimator whose variable bandwidth is based on both underlying regression function and nuisance functions.
Key words and phrases: Heteroscedastic nonparametric regression, adaptation, sharp optimality, small samples.