Adaptive And Optimal Estimations Over Pointwise Risk For Locally Regular Densities By Wavelets

As an important research direction in Statistics,nonparametric estimation includes density estimation,regression estimation and censoring estimation,in which density estimation is the basis.The Lp risk estimations of density functions have made great achievements,while the pointwise risk estimation is relatively less.It is unnatural to require global smoothness of an estimated function,when discussing pointwise risk estimation.In this dissertation,we define a local Holder space and study the adaptive wavelet optimal estimations of density functions over pointwise risk on that spaceMotivated by the work of Rebelles(G.Rebelles.Pointwise adaptive estima-tion of a multivariate density under independence hypothesis.Bernoulli,2015,21(4):1984-2023),we prove firstly linear wavelet estimator attains optimality on pointwise risk estimation,although that estimation is not adaptive.For the iso-tropic density functions,the nonlinear wavelet estimator defined by thresholding method provides an adaptive and nearly-optimal estimation.However,it does not work for anisotropic density functions.Fortunately,the data driven estimator overcomes the difficulty and gives a better convergence rate than the nonlinear wavelet one.In addition,we show a better convergence rate for density functions with independent structure,so that the dimension disaster is eliminatedBecause the errors occur in the measurement data,the additive noise model is concerned.This paper descusses the deconvolution density estimation with severely ill-posed noises over a local Holder space in Chapter 4.After giving a lower bound estimation,we define an estimator by truncating a linear wavelet estimator,then prove that it attains the adaptive and optimal convergence rate.This is different with the classical model,where the linear estimator is optimal but not adaptive.Finally,we consider a generalized deconvolution model,which includes the classical and deconvolution models as special examples.For that model,Lepski and Willer study the adaptive and optimal density estimation on Lp risk over Nikol'skii spaces by kernel methods(O.Lepski and T.Willer.Lower bounds in the convolution structure density model.Bernoulli,2017,23(2):884-926;O.Lepski and T.Willer.Oracle inequalities and adaptive estimation in the convolution structure density model.Ann.Statist,2019,47(1):233-287).Chapter 5 discusses the adaptive and optimal pointwise estimations for anisotropic density functions over local Holder spaces by wavelets.Most of the results are similar to Chapter 3.The biggest difference:we show that the data driven estimator not only attains adaptivity and nearly-optimality,but also does the optimality among adaptive estimations.