Pointwise Wavelet Estimation of Regression Function Based on Biased Data

Abstract

In this paper, we establish wavelet estimations on pointwise $$l^{p}(1\le p<\infty )$$ risk for multivariate regression functions based on biased data. We firstly introduce a linear wavelet estimator and discuss the convergence rate of this estimator. In order to obtain an adaptive estimator, a nonlinear wavelet estimator is constructed by the hard thresholding method. It should be pointed out that the convergence rate of linear and nonlinear wavelet estimators coincide with the optimal convergence rate of pointwise nonparametric estimation.