Asymptomatic property of wavelet estimators of partial derivative functions in a regression model with multiplicative noise

Abstract

This paper investigates nonparametric estimations of partial derivatives of a regression function in multiplicative noise regression model. Some new theoretical contributions are obtained as follows. Firstly, we establish a wavelet estimator of the partial derivatives functions based on wavelet bases, and derive a convergence rate over global Lp(1≤p<+∞) error in Besov spaces Br,qs(Rd). It turns out that this wavelet estimator can attain optimal rate of convergence in the case of r≥p. Secondly, in order to overcome the shortage of this above wavelet estimator in other opposition case r<p and get an adaptive estimator, an adaptive estimator is proposed by using the means of hard thresholding algorithm. Finally, it should be noted that the convergence rates of this adaptive estimator in all cases are consistent with the optimal rate of convergence in nonparametric wavelet estimation.