Wavelet Pointwise Estimations Of Regression Functions Based On Strong Mixing Noisy Data

Regression estimation with noisy data is one of important research problems in nonparametric statistics,and has extremely significant application value in statistics,sociology,big data processing and other fields.Wavelets has been widely used in regression estimations by the unique local time frequency analysis property.Inspired by the work of Donoho,Chaubey,Chesneau,this paper focuses on wavelet pointwise estimation of regression function based on strong mixing noisy data.Firstly,a linear wavelet estimator is constructed by projection method,and the convergence rate over lp(1≤p<∞)risk is given in isotropic Besov spaces Bp,q s(H)based on strong mixing sample.When 1≤p≤2,the convergence rate is consistent with the optimal convergence rate under independent sample;The convergence rate gets worse due to the complexity of strong mixing sample in the case of p>2.Secondly,because the construction of linear estimator depends on the smooth parameter s of the unknown function,the linear estimator is not adaptive.In order to overcome this shortage of the linear estimator,a nonlinear wavelet estimator is defined by thresholding algorithm,and the convergence rate of pointwise risk is analyzed.Compared with the linear estimator,those two wavelet estimators have the same convergence rate up to a ln n factor.Finally,this paper assumes that regression functions have anisotropic characteristic for more practical significance,and discusses the convergence rates of linear wavelet estimator under independent and strong mixing conditions in anisotropic Besov space Bp,qs(H)respectively.More importantly,in order to obtain adaptability,a data driven wavelet estimator is constructed.A selection rule is established to select the parameters of the estimator,and the pointwise convergence rate is analyzed under independent condition.It should be pointed out that the convergence rate is same as the optimal convergence rate when the anisotropy index si=s(i=l,…,d).