Asymptotic properties of wavelet estimators for derivative function in heteroscedastic regression model

Abstract

This paper considers a wavelet approach to derivative function estimation in heteroscedastic regression model. A linear wavelet estimator is constructed by using projection operator. The convergence rate via the mean integrated squared error of linear wavelet estimator is proved with some mild conditions. It should be pointed out that this convergence rate is same as the optimal convergence rate for nonparametric wavelet estimations. However, It is unreasonable that the definition of the linear estimator depends on the smooth parameter of the unknown derivative function. In order to overcome this shortage, a nonlinear wavelet estimator is proposed by classical hard thresholding method. This nonlinear estimator not only can obtain the convergence rate as the linear estimator, but also only depends on the observed data. Finally, good performances of those wavelet estimators are presented.