The Adaptive Wavelet Estimations For Not Necessarily Compactly Supported Density Functions

Density estimation is an important research direction for nonparametric statistics,which plays a fundamental role for regression and censoring estima-tions.The estimation for compactly supported density functions have already made a great achievements,see the work of Donoho and etc(D.L.Donoho,I.M.Johnstone,G.Kerkyacharian,D.Picard.Density estimation by wavelet thresh-olding.Ann.Statist.,1996,24(2):508-539).There exists relatively less study for non-compactly supported density functions.Motivated by the work of Donoho,Ju-ditsky,Goldenshluger and Lepski,this dissertation studies the Lp(1?p<?)risk estimations for not necessarily compactly supported density functions on Besov space Br,qs(R)by adaptive wavelet estimators.Based on biorthogonal wavelet,Reynaud-Bouret and etc(P.Reynaud-Bouret,V.Rivoirard,C.Tuleau-Malot.Adaptive density estimation:a curse of support.J.Stat.Plann.Inference,2011,141(1):115-139)investigate optimal estimation of L2 risk for not necessarily compactly supported density functions on Besov space Br,qs(R).They pose an open problem:Can their estimation be extended to Lp(1 ?p<?)risk?This dissertation firstly answers the question positively by using the classical wavelet hard thresholding estimator for p>2sr+r;When 2<p<2sr+r,we borrow the biorthogonal wavelet estimator from Juditsky&Lambert-Lacroix(A.Juditsky,S.Lambert-Lacroix.On minimax density estimation on R.Bernoulli,2004,10(2):187-220),and give the corresponding Lp risk estimation.In particular,our result with p=2 coincides with the theorem of Reynaud-Bouret and etc.Secondly,motivated by the work of Goldenshluger&Lepski(A.Goldensh-luger,O.Lepski.On adaptive minimax density estimation on Rd.Probab.The-ory Relat.Fields.,2014,159(3-4):479-543),we construct a completely adaptive wavelet estimator by a data driven method.Moreover,the convergence rates of Lp risk estimation are provided for 1?p<?.Unfortunately,the index of convergence rate reduces to zero for p=1,which means the estimator missing consistency.This is natural,because Goldenshluger&Lepski have pointed out that the smoothness assumption doesn't imply the existence of a uniform estima-tor,or consistency of estimatorFinally,we define a set of density functions F?(M)(0 ?(0,1],M>0 is a constant).It reflects the decay property of a density function f in some sense the smaller 0 corresponds to the faster decay of the function.For this class of density functions,we provide a convergence rate of L1 risk estimation by using the classical wavelet hard thresholding estimator.The result can be considered as an extension of Juditsky&Lambert-Lacroix's work...