Adaptive Wavelet Estimations For Deconvolution Density Functions

Density estimation is an important research direction of Statistics.Compared with global estimation,the achievements of pointwise estimation are relatively few.Liu and Wu study adap-tive and optimal wavelet estimation of density function under pointwise risk in the local Holder space,see Y.M.Liu&C.Wu.Point-wise estimation for anisotropic densities[J].Journal of Multivariate Analysis,2019,171,112-125.Because the measured samples usually contain er-rors,this thesis discusses adaptive and optimal wavelet estimation of the deconvolution density function under pointwise risk in the local Holder space by wavelet methodWe first give an upper bound estimation of linear wavelet estimator.To obtain the adap-tivity,the nonlinear wavelet estimation is considered.Because the nonlinear wavelet estimator depends on an upper bound of the unknown parameter,we study data-driven estimation.Finally,a lower bound estimation is given,which shows all of our upper bound estimations optimal or near optimal.It should be pointed out that our results are natural extensions of the correspond-ing theorems of Liu&Wu in isotropic case.