Convergence Rates of Density Estimation in Besov Spaces
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2 H Y WANG 59 d The Beov ce c be dcretzed by the euece orm of wveet coeffcet My uefu wveet re eerted by c fucto More recey f c fucto wth x h x the : x h x defe wveet [3] Cery whe comcty uorted d cotuou the correod wveet h the me roerte A orthoorm wveet b of L R eerted from dto d trto of c fucto d t correod wveet e x x : : x x Z Athouh wveet b re cotructed for L R mot of them cottute ucodto be for L R A c fucto ced t reur f h cotuou dervtve of order t d t correod wveet h vh momet of order t e x x d x t The foow emm [3] y mortt roe th er Lemm Let be comcty uorted t reur orthoorm c fucto wth the correod wveet d t If f L R : f d : f d the the foow two codto re euvet: ) f B R ; ) Furthermore d f d B Before troduc Fo Lemm we eed the otto of Kubc-Leber dtce [4] Let P d Q wth P be boutey cotuou wth reect to Q (deoted by P Q) The the Kubc-Leber dtce defed by x K PQ : x dx x Where d re dety fucto of P Q reectvey Lemm (Fo Lemm [4]) Let ( P ) be robbty meurbe ce d A m If A Av for v the wth A c td for the comemet of A d m : f K P Pv vmm u P m v ex 3 c A m e m m By Lemm d we c how the foow reut: Theorem Let f B r R L wth r d r If f etmtor of f wth d rdom me the / r/ / r u E f f mx fb r R L where Br R L : f Br R f L d f B r h comct uort}; The otto x y me x Cy wth cott C Remr Note tht for mx r / r/ / r/ / r / r d for r / r/ / r mx The theorem reformute of the ower boud () By u the de of referece [5] we how th theorem the ext two ecto Proof of Theorem Frty we rove u f B R L E f f / r/ / r r uch tht B r R L Oe eed cotruct d u E f / r/ / r Let be comcty uorted t t reur d orthoorm c fucto be the correod wv et wth u N Here d fter e Coyrht ScRe
3 6 H Y WANG N deote the et of otve teer The there ext comcty uorted dety fucto (e x dx ) tfy x d x Br R d x c Let : The the umber of eemet deoted by # / / Motvted by [ 5] oe defe : r d wth I : Obv- ouy x : x x I : f I ee x x d d / r xc for re whch me tht dety fucto for ech By the umto of the wveet comcty uorted d t tme dfferetbe Therefore B r Rt d B r R Bec ue // r C d o Br due to Lemm Hece x B r For / / r : B r R L Cery // r () due to : Fur- thermore A : f tfe A A for Rec tht # By Lemm c 3 u P A m ex Here d e fter P f td for the robbty meure correod- to the dety fucto f x: f x f x f x It ey to ee tht P P from the cotructo of Sce f etmtor of dety wth d rdom me c E f P f P A The c u E f u P A Next oe how 3 m ex e () c : Rec tht f x x f f f f x d f x fx f x K P P : f x dx The x x f x d f x K P P f x x K P P f K P P f x x d : d u u f Note tht for Hece u The f x f x d f x f x K P P f x dx f x x f x f x f x d x : f K P P K P P Moreover v v v x x x dx (3) Accord to the defto of u d x c o Thu x x x dx c x dx c x c by the orthoormty of The (3) reduce to / r c (4) r Te The Now oe c chooe C uch tht C d C4 r c Therefore 4/ r c Cc e e d () reduce to the dered foow from d / r Now we rove E f u C u fb r The / / r by () R L roof deed o other emm [4] E f f Our Coyrht ScRe
4 H Y WANG 6 Lemm (Vrhmov-Gbert) Let : m The there ext ubet M of wth uch tht m m/8 m M d M 8 It uffcet to cotruct M uch tht B R L d r u E f (5) A roved b ove et be comcty uorted tt reur d orthoorm c fuct o be the correod wveet wth u N Aume B r R L d [ ] c Defe / : : d x: x x wth (ote tht ) Sce r oe ow tht d By Lemm B r / r / / r r Hece B R L r Br C d o Note th t the uort of for re mutuy dot The x c c me for b Th wth x d x x d x tht dety fucto for ech Accord to Lemm there ext uch M 3 tht M d 3 (6) Becue u u for oe ow tht Th wth (6) d 3 d ed to (7) / 8 : Cery the et A f M tfy A A for The Fo Lemm yed M c M u P A m M ex 3 e (8) O the other hd t foow M c from the mr rumet to the roof of (4) Te The ( ) chooe cott C uch tht Hece oe c 4 4 c c C M Me e e c Therefore (8) reduce to u M P A C d u f ME u P C f M Th wth (7) d 3 Acowedemet yed (5) The uthor Huy W rtefu to the referee for ther vube commet d th her dvor Profeor Youm Lu for h hefu udce Th wor uorted by the Nto Ntur Scece Foudto of Ch (No 87) d Ntur Scece Foudto of Be (No 83) 4 Referece [] G Kerychr d D Pcrd Dety Etmto Beov Sce Stttc & Probbty Letter Vo 3 No do:6/67-75(9)93-s [] D L Dooho I M Johtoe G Kerychr d D Pcrd Dety Etmto by Wveet Threhod The A of Stttc Vo 4 No do:4/o/ [3] W Härde G Kerychr D Pcrd d A B Ty- Aroxmto d Stttc A- bov Wveet cto Srer-Ver New Yor 997 [4] A B Tybov Itroducto to Normetrc Etmto (Eh) Reved d Exteded from the 4 Frech Or Trted by Vdmr Zt Srer Sere Stttc Srer New Yor 9 [5] P Bd G Kerychr D Mrucc d D Pcrd Adtve Dety Etmto for Drecto Dt U Needet The A of Stttc Vo 37 No 6A Coyrht ScRe
5 6 H Y WANG do:4/ 9-AOS68 [6] C Chrtohe Rereo wth Rdom De: A Mmx Study Stttc & Probbty Letter Vo 77 No do:6/6 5 [7] A B Tybov Otm Rte of Areto COLT/ Kere 3 Lecture Note Artfc Iteece 777 Srer Hedeber Coyrht ScRe
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