Wavelet Estimations Of Regression Model With Mixed Noise

Nonparametric regression estimation has important theoretical significance and practical application value in statistics,econometrics and big data processing.Wavelet analysis is widely used in regression estimation base on its unique time-frequency analysis characteristics.Inspired by the work of Chesneau,Donoho and Masry,this paper studies estimation problems of regression model with mixed noise by wavelet method.Firstly,this paper constructs a linear wavelet estimator by wavelet method,and proves the Lp(1≤p<∞)consistency of wavelet estimator without any smooth conditions of the regression function.The conclusion shows that the linear estimator can approximate the regression function when the sample size is sufficiently large.Furthermore,the strong convergence rate of the linear estimator is discussed in Besov space.This convergence rate is consistent with the optimal strong convergence rate of conventional nonparametric estimation.Secondly,in order to obtain an adaptive estimator,a nonlinear wavelet estimator is constructed by hard thresholding method,and the convergence rates over Lp(1≤p<∞)risk of the linear and nonlinear wavelet estimators in Besov space are discussed respectively.When p>pd/(2s+d),the convergence rates of those two estimators are the same up to a ln n factor.The convergence rate of the nonlinear estimator is better than the linear estimator in the case of p≤pd/(2s+d).Finally,for pointwise error of wavelet estimators of regression function with mixed noise,this paper discusses the pointwise error of linear and nonlinear wavelet estimators in local Holder space.The results show that two estimators can effectively approximate the regression function when the sample size is sufficiently large,and have the same convergence rate up to a ln n factor.In addition,the effectiveness of those two wavelet estimators are validated by numerical experiments.