Wavelet Estimations Of L~p Risk For High Dimensional And Uncompactly Supported Density Functions

Density estimation is an important research direction for nonparametric statistics.The es-timations for compactly supported density functions have already made a great achievements.There exists relatively less study for uncompactly supported density functions.By using the wavelet method,Cao and Liu(K.K.Cao,Y.M.Liu.On the Reynaud-Bouret-Rivoirard-Tuleau-Malot problem.International Journal of Wavelets Multiresolution and Information Processing.2018,16(5):1850038)provide an upper bound of LP(2 ? p<?)risk estimation for uncom-pactly supported density functions in Besov space Br,qs(R).This thesis tries to extend their results to high dimensional cases.Motivated by the work of Kerkyacharian and Picard(G.Kerkyacharian,D.Picard.Density estimation in Besov spaces.Statistics and Probability Letters.1992,13(1):15-24),we provide Lp(Rd)(1 ?p<?)risk estimation for uncompactly supported density functions by using a linear wavelet estimator;Inorder to obtain the adaptivity,we show the upper bound of Lp(2sr/d+r ?p<?)risk estimation by using a nonlinear wavelet estimator;Finally,we discuss L1 risk estimation on a set of special density functions.When d=1,our results reduce to the corresponding theorems of Cao and Liu.