Linear programming problems for $L_1$ optimal frontier estimation
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Linear programming problems for L1 optimal frontier estimation Stéphane Girard — Anatoli Iouditski — Alexander Nazin. N° 5466 Janvier 2005. N 0249-6399. ISRN INRIA/RR--5466--FR+ENG. Thème NUM. apport de recherche.
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(15) . pLf. N. akgxyPckPuKc. h→0. ura. N →∞. urfw ckPRT)gfxDcejpLf . Z. g(x) =. x/h. K(t) dt (x−1)/h. !−1. ,. x ∈ [0, 1] ,. G~. r3. x(prieikT>xDcya2ckPT¢<urakjsxX rT(iefRT" K ucckPRT¢4prgRfwuKiejT>a(¥/jG Trs¥bPRT"f x ∈ [0, h) pLi x ∈ (1 − h, 1] Í fwRT(T"w,¥ prf ¥RPRjsT PRT(f x ∈ [0, h) pri x ∈ (1 − h, 1] g(x) ≡1 p fTS u$_¤T>urakxjs_¤∈ pL[h,¢a¡1T"ik−LTh]cePuc g(x) > 1 Z. 1. Kh (x, u) du = 1. ∀ x ∈ [0, 1]. d53. urfw'¥Rx(prfakT"l/gRT"fLce_L¥RwbgRT)cep jsf/ckT"ikvRmurx(jf©cePRT jsfLceT(LieurFurfw ckPRTwbT"ikjsucejLTr¥ Z. 1. 0. ∂ Kh (x, u) du = 0 ∂x. ∀ x ∈ [0, 1] .. O 3. VXpKceTr¥2cePucUT"l/guKckjsprfOr3Su$_ï¢4TrT"ikjT"w wbjieT"xcks_r¥ouLa j¨cUjsaUwbT"SUprfa¡ckiyuceT"w#jsf cePRT Q Rv v4T(fwRj '¥ ]bgR¢akT"xDcejpLf~~ 9r ZT(fpKckT K , max K(t) ¥ g , max g(x) ¥<ura&T(s,uragRfxckjsprfursa max. 0. max. ,. Cβ (ϕ). TuKmakp wbT"fRpKceT ¢`_. 0. Z. 1. |t|β |ϕ(t)| dt ,. ϕ ∈ C 0 ([−1, 1]) ,. −1. Cβ (K, K ) , gmax Kmax Cβ (K) + Cβ (K 0 ) .. yr3. ". f9 r3. uU,jsvaexyPRjck xpLfadcyuKf/c&pri&gRf<xDckjsprf ϕ : R → R ¥bckPuKcjsa jckP L < ∞. 9r9-3 |ϕ(s) − ϕ(t)| ≤ L |s − t| O&PT jsfwbjmx(uKckpLiZgfxDcejpLfïjma wbT"fRpKceT"w7¢/_ 1{·} PRjsxyPïT"l/gursa 1 jcePRTuKiergRSUT(f/c xpLfwbjckjsprfïPRprmwRa ceikgTr¥RuKf<w 0 prckPRT"ikjma¡TL Q a&j¨cjmavRieprT>w¤¢4T(sp jsf9,T(SUS; u 9ckPRTakgRik§urxT)pr¦ckPT T"a¡ckjsSUuKckT>wakgRvRv4prikc 9$5 3 Sb , {(x, y) : 0 ≤ x ≤ 1 , 0 ≤ y ≤ fb (x)} Su$_¤¢4TuKvvRikp bjsSuckT>w¤urapLspa Lϕ. ϕ. ϕ. N. N. . Z. î!î¹ ÐyÓ. 1 0. fbN (x) dx =. N X i=1. αi + O(h) .. >. 9 r3.
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(79) > 9. ]bjfx(T. . αi. urfw ckPRT"ikT(prieTr¥. 0. 0. W
(80) w . . urfw rT(iefRT" K urikT)fRpLfbªàfRT(/uckjsrTL¥LjcprsspackPuKc §}/|. Z N N X X 1 1 x − Xi x − Xi αi K dx ≤ αi dx = αi K h h h h i=1 i=1 i=1. Z1 X N. Z1. . fbN (x) dx −. 53. R. N X. αi. ≤. i=1. h Z1 Z N X gmax − 1 1{|x − Xi | ≤ h}dx r3 αi + Kmax h i=1. }/~. 0. N X. ≤ (gmax − 1)Kmax. 1−h. }. :dW3. αi 1{0 ≤ Xi ≤ 2h}. i=1. +. N X. αi 1{1 − 2h ≤ Xi ≤ 1}. i=1. !. ≤ (gmax − 1)Kmax 4Cα h .. §} §}ryr3 rOr3. O&PTjfT"l/guKsjcd_ } y53&pLspa&ieprS 9PdW3akjfx(T ¢<prckP¬jsf/ckT"ikuKmajsfK}:d 33$}rOr3uKieT pKckPTsT(fRrckP 2h urfw ckP`ga Su$_¬¢4T¤xpLT(ieT"w¬¢`_9cdpikT"suKckT>w jsf/ckT(ieuKma)pKckPRTUpLikS [(j − 1)/m , j/m ] jsfN 9Pd53D pLfakT"l/gRT(f/ce_L¥/TPu$rTZvRikpLT"wUcePRT)gRvRv4T(i¢4prgRf<w¤pLicePRT wbj~4T"ikT"fxTjf ckPTsTËcPurfwa¡jmwbTH}L~ 3D O&PT©spT"iX¢4prgRf<w jmaXvikpLT"w9jsf¬cePRTakurS T©SuKfRfRT"i" Íf<wbT(T>w'¥,wbT"x(prSUv4pLakj¨cejpLn f }`W 33$}r } 3XjsSUvRsjT>a(¥ akjsfxTckPT)ckT(ie S §}rr} 3jma&fRpLfbª;fT(LuKckjsrTL¥ h. Z1 0. fbN (x) dx −. N X. αi. ≥ −. i=1. N X i=1. . αi . ≥ −Kmax. Z0. +. . 1 K x − Xi dx h h. 1+h Z 1. −h. N X. αi 1{0 ≤ Xi ≤ h}. i=1. +. N X i=1. ≥ −Kmax 2Cα h .. O&Pjsax(prSUvRsTckT>ackPRT vikp`pK%pK2,T"S Su;9r. h. αi 1{1 − h ≤ Xi ≤ 1}. !. G|r. r3. G|. ^9-3. ;|r ;|Kr3 53. Éæ%î,ÉËÊ.
(81) 9r9. L1
(82) . ( ³,³*))³ ) ¯ E + tT"SUjfw ckP<uc)TUuraea¡gS T;}a9C3D_¬urvRvRs_/jsfRuKgbjsjmuKie_9,T(SUSu ~uKf<w ¦T(SUSuHO©TiyadcXurikiejLTZuKc G|K } 3 max |fb (x)| G|L5| 3 = max max |fb (x)| ;|Kr~ 3 log N 1 (X − X ) max |fb (x)| ≤ L g C (K, K ) + max 0 N. x∈[0,1]. 0 N. 1≤i≤N +1 x∈[Xi−1 ,Xi ] f,β. ≤ Lf,β. jckP. max. 0. β. i. N h2. i−1. 2. x∈[Xi−1 ,Xi ] 8 1≤i≤N +1 2 log N log N 1 gmax Cβ (K, K 0 ) CX max |fbN000 (x)| , + 2 Nh 8 N x∈[0,1]. CX > 4Cf /fmin. 000 N. O&PRT Su bjsSgRS ceT(ieS jsfs;|rd53jsa&¢4prgRf<wbT"wurapLspa pLiuKf`_ . |fbN000 (x)|. G|. dW3. x ∈ [0, 1].
(83) 3
(84)
(85) d
(86) αi
(87)
(88) 3 Kh (x, Xi )
(89)
(90) dx i=1
(91) X
(92) 3
(93) N
(94) ∂ αi 1{|x − Xi | ≤ h} ≤ sup
(95)
(96) 3 Kh (v, u)
(97)
(98) · u,v ∂v i=1. ≤. N X. −4 ≤ gmax LK · 3Cα h , e 00 h. akjsfxT §akT(T¦T(SUSu|pLi&ckPRTwRTceurjsT"wwbT"SUprfa¡ckiyucejpLfa3. ^9-3. 2 4 3 gmax + 6Kmax gmax . LKe 00 , LK 00 + 3LK 0 Kmax gmax + LK gmax 3LK + 10Kmax. ]bgR¢a¡ckjckgbcejfR ; |WO53¥"§~Lr3jsf/ckpYG| d 3_`jT"swRa max |fbN0 (x)|. x∈[0,1]. 2 log N log N 3 C + g L C h X max K e 00 α N h2 8 N h2 log N gmax Cβ (K, K 0 ) N h2. ≤ Lf,β gmax Cβ (K, K 0 ) ≤ 2Lf,β. gfwbT(iuLwRwbjckjsprfur,uraea¡gS vRckjsprf[PRjmxyPPRprmw¤ceikgT)priuKs,akgb£¤xjsT(f/cks_ muKierT h≥. O&PTieT"akgRcprspa" î!î¹ ÐyÓ. 3CX2 Cα LKe 00 log N . 8Lf,β Cβ (K, K 0 ) N. G| G~rr3 5yr3. G~.
(99) 3
(100)
(101) ∂
(102) −4
(103) sup
(104) 3 Kh (v, u)
(105)
(106) ≤ gmax LK e 00 h ∂v u,v. PT(ieT. ;|. WOr3. G~L. 53. G~r G~K} 3. r3. 3. N . G~L|. 53.
(107) $ 9. . $ fbN & " J ∗ P ¯ 9">%T A &
(108) ' %^
(109) +% j N' %T . . . . W
(110) w . . .
(111) 2 . γ > Lf,β gmax Cβ (K). §~L~. r3. ? W ω ∈ Ω ) %^ w=Hr N1 = N1 (ω, γ) %;>%T T
(112) _ N ≥ N1 ) %^ ? 9V
(113) <XI,R Y*BIX V P ~ dW3 ∗ β iW e . G~ ( ³,³*)o³*)¯ #+ pLfakjswbT"iouKie¢RjckiyuKie_ j¨ceP N (ω) ikpLS ¦T(SUSu~RÍf/ckiepbwbgxT gfxDcejpLf f (u) = f (u) + γh uKfw vakT(gwRpKªàT"Na¡ck≥jsSUNuKckpL(ω) iea JP ≤ Cf + γh . 0. β. γ. 1 + δi1 α ei = 2. Z. Xi. 1 + δiN fγ (u) du + 2 Xi−1. Z. 0. Xi+1. fγ (u) du ,. i = 1, . . . , N. §~. rOr3. Xi. PT(ieT adcyuKfwRapri )ieprfT"xy rT"ia¡_`S¢4pr; T(sp TwbT(SUpLfadceieuKckT)cePucZx(prfwRj¨cejpLfK§~r~r3T(f<a¡gRieT"a cePRTrT"xδckpLipKZva¡T"gwbprª;T>adcejSucepriya αe = (eα . . . , αe ) cep ¢4Turf urwbSUjmakakjs¢RTv4prjsfLc©pri©ckPRT,n f9Cy53Y;K5 3¥bpriurf/_ a¡gb£¤x(jT"fLce_ surikLT O&PjsajsSUvRsjT>aa¡pLuK¢jsj¨cd_ pK%ckPRT¦n1f9Cy53Y GK 3&uKfw N . ij. 1. JP∗. ¦Tc. ≤. N X. 1. T. (f (u) + γhβ ) du = Cf + γhβ .. G~. 5yr3. 0. µ pLi&ckPRTaeuK LTZpr%akjSUvRsjsx(j¨cd_L¥RTjsSUv<p/a¡T)cePRTurwwbj¨cejpLfuK,uLakakgRSUvbcejpLfa i=1. CX > 4Cf /fmin. α ei =. Z. N. log N ≤ min h ≤ ρN h β. PjsxyPPRprmw ckiegRTpriur N muKierTT(fRpLgRrP,. . fmax 1 , γ ρCX. . ,. . d r3. Éæ%î,ÉËÊ.
(114) > 9. L1
(115) . L µ jsiyadc>¥RT)vRieprT)xpLfa¡ckiyuKjsfLcyaV 9>|53gRfwbT"i 9. feN (x). ,. N X i=1. =. N +1 Z Xi X i=1. +. =. α ei Kh (x, Xi ). 1 2. 1 + 2 Z 1. fγ (u) du. Xi−1. Z Z. ¥. αi = α ei i = 1, . . . , N. µ priuKie¢RjckiyuKie_. x ∈ [0, 1]. Kh (x, Xi ) + Kh (x, Xi−1 ) 2. X1. fγ (u) du (Kh (x, X1 ) − Kh (x, 0)) 0 1. fγ (u) du (Kh (x, XN ) − Kh (x, 1)) XN. +. i=1 X. +. 1 2. 1 + 2. Z Z. i−1. fγ (u). K. d 53. K. d r3. }. d 3. r|. 0. ZXi. dJ9-3. d 53. fγ (u) Kh (x, u) du. N +1 X. ¥. Kh (x, Xi )+Kh(x, Xi−1 ) − Kh (x, u) du 2. X1. fγ (u) du (Kh (x, X1 ) − Kh (x, 0)) 0 1. fγ (u) du (Kh (x, XN ) − Kh (x, 1)) .. K~. d r3 drdW3 dWOr3. XV p Ta¡T"vuKiyuceT(s_¢4prgfw T"urxyP pK<ckPRTakgRSUSuKfwabdK|533 d O53!ieprS ¢<T"p XgTcepid 3¥KcePRTSuKjsf ceT(ieS dr|53jma&¢<pLgRfwbT>wuraprsspa XN. . Z. î!î¹ ÐyÓ. 1 0. fγ (u) Kh (x, u) du = f (x) + γhβ +. Z. 1. (f (u) − f (x)) Kh (x, u) du 0. ≥ f (x) + (γ − Lf,β gmax Cβ (K))hβ .. dWyr3. r. O r3.
(116) "} 9. . . W
(117) w . . O&PT ª;ckPakgRSUSuKfw©ieprS d~53%jmawbT>xpLS v4pLakT"wUurfw©ckPRT"f ¢4prgfwbT"wU¢<urakjfR)pLf©ceieurv<T"(jsgRSzpLikS©gRmu T"ikiepri&i urapLspa . Kh (x, Xi ) + Kh (x, Xi−1 ) − Kh (x, u) du fγ (u) 2 Xi−1 Z Xi Kh (x, Xi ) + Kh (x, Xi−1 ) − Kh (x, u) du ≥ fγ (x) 2 Xi−1 Z. . Xi. −. ZXi. Xi−1. OT9-3. /. O 53.
(118)
(119)
(120) Kh (x, Xi ) + Kh (x, Xi−1 )
(121)
(122) |fγ (u) − fγ (x)|
(123) − Kh (x, u)
(124)
(125) du 2. L. O r3. r}.
(126) 2
(127)
(128) ∂ Kh (x, u)
(129) O 3 (Xi − Xi−1 )3
(130)
(131) 1{|x − Xi | ≤ 2h} ≥ −(fmax + γh ) max
(132) 12 u∈[0,1]
(133) ∂u2 Z Xi O 53 gmax LK −Lf,β [(u − Xi−1 ) + (Xi − u)] du. |u − x|β 1{|x − Xi | ≤ 2h} 2 2h Xi−1 β. /|. _ urvRvRs_`jfR¦T(SUSu ~¥`ckPRT)<iea¡cckT"ikSjma¢4prgRf<wbT"wurapLspa K( } ) ≥ − C log N N ru fw ckPRTakT"x(prfw prfRT)jma¢4prgRfwRT"w¢`_ O. 2. fmax gmax LK 0 (Xi − Xi−1 ) 1{|x − Xi | ≤ 2h}, 6h3. X. g L| ( )≥−. O. max Lf,β. LK. 2h2. . log N CX 1{|x − Xi | ≤ 2h} N. Y¬prieT(prT"i"¥LieprS ¦T(SUSu ~¥bprfRT x"uKf9a¡PpiyadccePuc urfwa¡T>xprf<w¤cePuc. N +1 X. 1{|x − Xi | ≤ 2h}(Xi − Xi−1 ) ≤ 4h +. i=1. N +1 X. 1{|x − Xi | ≤ 2h}. ≤ ≤ ≤. Z. Xi. Z. Xi. |u − x|β du .. Xi−1. CX log N , N. OrOr3. |u − x|β du. Oryr3. |u − x|β du. x−2h−CX (log N )/N. . 4h + CX. OJdW3. Xi−1. i=1 x+2h. Z. L~. O r3. log N N. log N 4h + CX N. . max. v∈[−2h−CX (log N )/N,2h]. . log N 2h + CX N. β. .. |v|β. r. y r3 y^9-3. Éæ%î,ÉËÊ.
(134) $| 9. L1
(135) . O&P`ga"¥bT)urikiejsrT)uc&ckPRT ¢4prgfw pri&ckPTa¡gRS dK~53&urapLspa . . Kh (x, Xi ) + Kh (x, Xi−1 ) − Kh (x, u) du 2 i=1 Xi−1 CX fmax LK 0 log N log N log N 4 + CX ≥ −gmax CX Nh Nh 6 Nh β ! Lf,β LK β log N + h 2 + CX 2 Nh 2 5 log N ≥ − gmax CX CX fmax LK 0 + 3β+1 ρ−1 Lf,β LK . 6 Nh N +1 Z Xi X. fγ (u). /. y 53. r. y r3. K}. y 3. L| Q cmura¡c"¥bjcjsaakjsS jsmuKie_¤wbT"SUprfa¡ckiyuceT"w ckPuKc&¢4pKceP9a¡gRSUSuKf<wRa*drd 3&uKfwsdWO53&uKieT¢<pLgRfwbT>w ur¢<pLT ¢`_ O((log N/(N h)) ) µ pLijfa¡ceurfxTL¥`pri_ d5d 3¥RprfRT)pL¢bceurjfa. y 53. 2.
(136) Z
(137)
(138) X1
(139)
(140)
(141) fγ (u) du (Kh (x, X1 ) − Kh (x, 0))
(142) ≤ (fmax + γhβ )X1 |Kh (x, X1 ) − Kh (x, 0)|
(143)
(144) 0
(145) 2 y r3 log N . ≤ 2fmax gmax LK CX Nh. r~. O&P`ga"¥`ieprS. r~ jcprspapLiT"urxyP. dJ9-3Yy 53. j = 1, . . . , N. ckPuKc. feN (Xj ) ≥ f (Xj ) + (γ − Lf,β gmax Cβ (K))h + O β. pLia¡gR£x(jT"f/cks_muKierT γ−Lf,β. N ≥ N0 (ω). . log N N h1+β/2. 2 . . CX fmax LK 0. R]`jSUjssuriks_r¥'x(prfa¡ckiyuKjsf/ceaX 9"~r3PRpLsw¬ckiegRT gRfwbT"i α = αe ¥ ¥`T fRp P<u$rTZckpU¢<pLgRfw ckPTuK¢akprsgbckT)uKsgRT pr x ∈ [0, 1] i. jsfa¡ckT>urw pKbdJ9-3D g T"ikT. N X i=1. i. 2 !. N. α ei. 12LK + 5. i = 1, . . . , N. ≥ Yj. y dW3. . +3. β+1. . Lf,β LK . ρ y5Or3. Í fwbT"T"w'¥FpLiurik¢j¨ceieurik_. X d e h (x, Xi ) Kh (x, Xi ) = α ei K dx i=1. e h (x, u) , ∂ Kh (x, u) K ∂x. î!î¹ ÐyÓ. log N Nh. PRT"f©¢4pKcePjfRT>l/guKsj¨cejT>abdK532urfwcePRT&prsspjfR)pLfRT&PRprmw©ckiegRT. 5 gmax Cβ (K) ≥ gmax CX 6. feN0 (x) =. . y5yr3. >r. 9 r3.
(146) >~ 9. . W
(147) w . . . Ga¡T"T]`gR¢akT"xckjsprf9~Rl9C3jckPckPRT)pLspjsfR©gvRv<T"i¢4prgRfw . > wRT"wbgx(T"w prieS f9(}/| 3¥" 9"}L~53 gT"fxTL¥RprfRT)Su$_¤ieT(v4T"uKccePRTuKiergS T"f/ceapKbdrW33$d Or3¢/_ xyPuKfRLjf pLi O&PRT(ieTpLikTL¥$ur/cePRTiyuceT"a¦ikpLSOT9C33$y dW3¦akPRprgsw¢<T&wbjs/jmwbT>w¢`_ ¥PjsTocePRTuK¢akprsgbceT K urgRT)prK!ecePRT SuKjsf ceT(ieS pKwbT"x(prSUv4pLakj¨cejpLf,¥RwbgRT)cepqOr3¥Rjsa¢<pLgRfwbT>wuLaprshpa
(148) Z
(149)
(150) Z
(151)
(152)
(153)
(154)
(155) 9>Lr 3
(156) e (x, u) du
(157) =
(158) (f (u) − f (x)) ∂ K (x, u) du
(159) f (u) K
(160)
(161)
(162)
(163) ∂x 9>r 3 ≤ L g C (K, K )h , jsfa¡ckT>urwprB d y53YOL53D2tT"SUjfwckPT&wRTfRjckjsprfqy 33$ 9"r 3,pLi C (K, K ) PRjsxyP©prsspa,ieprS f9"T 9C3D O&P`ga"¥`priakgb£¤xjsT(f/ce_ surikLT N ≥ N (ω) urfw priT"uLxyP X T)urikiejsrTZuKc
(164)
(165) }3 log N 9>KJ log N
(166) e
(167) g C (K, K )h +O ≤L g C (K, K ) .
(168) f (X )
(169) ≤ L N h ρN h VZuKSUT(s_r¥`jsfRT"l/gurjcd_ f9"K5} 3PRpLswackiegRTurSUp/adca¡gikT"_UpLi&ur4cePRpLakT N ≥ N (ω) a¡gxyP ckPuK_c dK5 3jsa LT(iej¨<T"wuKfw 9>Lr| 3 log N log N 5 L C (K, K ) −1 ≥ g C h N 6 Nh L L 9>r 12L ~3 L + · C f +3 5 ρ PT(ieHT Ga¡T"T ,T(SUSu¤|pri&cePRTwbTcyuKjsT>w wRT(SUprfa¡ckiyucejpLaf 3 9> d53 K . K , L ,L +L g L ,L +L g
(170)
(171)
(172)
(173)
(174)
(175)
(176) 0 x−u
(177)
(178) x − u
(179)
(180)
(181)
(182) e −2
(183)
(184)
(185) K ≤ h g K K (x, u) + g K
(186)
(187) h max
(188) max max
(189)
(190)
(191) . h h. 9 ^9P3. h. h. 1. 1. γ. h. h. 0. 0. max. f,β. β−1. 0. β. 0. β. 0. j. 2. 0 N. j. max. f,β. β−1. 0. β. max. f,β. 2 3. 0. β. 2. 0. 2. f,β. 0. β. max. β+1. X. 1+β/2. X max. K. K0. e K. max. max. e0 K. e K. e0 K. K 00. K0. max. β+1. f,β. e K. max. µ jsfurs_r¥bcePRTxpLfa¡ckiyuKjsfLcyaV 9PdW3j¨ceP. > ursakpPRpLswceikgTgRfwRT(i = αe ¥ i = 1, . . . , N ÍfwbT"T"w'¥4¢`_¦T(SUSu¤~¤ckPRTprspjsfRjsfRT"l/gurjckjsT"a Pprmw u a"%priXur N ≥ Nα (ω) urfw priT>urxyP j = 1, . . . , m ¥RPRT"ikT m = bh c. 9 5O 3. Cα ≥ 6fmax. i. 0. N X. i. h. h. . −1. . α ei 1{(j − 1)/mh ≤ Xi < j/mh } ≤ (fmax + γhβ ) 1/mh + 2CX. log N N. . >. 9 5y 3. " gfwbT(i)urwwbj¨cejpLfuKuraeakgRSUvbckjsprfaXdK53DO&P`ga"¥'xpLfadceieurjf/cyaw 9 d 3ZuKieTgRsT>w¬gRfwRT(i 9>5O53ZuKsSUpLa¡c akgRieTr¥`pLiuKf`_ akgb£¤xjsT(f/cks_¤muKierT N }<]`jf<xTuKs αe ≥ 0 ¥Rxprf<adceieurjf/ceVa f9CO53Pprmw ceikgTr¥RuKf<w¦T(SUSuU©jma&vRieprT>w' i=1. ≤ 6fmax h,. 9r9 3. i. Éæ%î,ÉËÊ.
(192) 9 d. L1
(193) . ¯ b5. . 2XA58 2 Z / 9, X< : 5 W W W $ & 30
(194) <XI h W } ,2WH. W ) %^b l ? : PS YX#I,R YX*"$V :
(195) *D I,R
(196) #D 5 9, X, ? Z/ A r& e . >% S^W ν ∈ (1, 2) W =)%^W.
(197). . N +1 X. . 3. 1{|x − Xi | ≤ 2h}(Xi − Xi−1 ) = o. i=1. . h logν N N2. . 9r959P3. %T x)%^w
(198) *)%^w Y" <i / e / ? l . >% T )%^W &
(199) C" H x %J 2 #/ T W w >T% q ,2s P >%T; L
(200) CX$&YX*"$[P . P L
(201) C
(202) *D ,. IPZ/ . & :%^ Z/ s % >%T K
(203) >%T ' %^ ' Ke^% WH . %02 . A7 P* >%T 0/ S)%J ? e32:%T 0. 0. $ fbN ¯ 5)%^ & . .
(204) ' %T <
(205) x
(206) e W ω ∈ Ω )%^ =Hr ? q % ) ^ % W &
(207) 2 W N ≥ N (ω) N2 (ω) x ∈ [0, 1]
(208) 2. . )% W ( ³,³*)³*) o ¯ . C4 (β) fbN (x) ≥ f (x) − h2. C4 (β) 5T i . CB ! . a¡gxyP ckP<uc. N ≥ N6 (ω) ik ∈ {1, . . . , N }. urfw. log N N. >. 2+β 1+β. 9r9 r3. + ,T(c&g<a&ceur rT)gakTpr¦T(SUSuidurfwj¨cya prieprssurik_¤ jsf/ckiep`wRgxjsfR δy = Lf,β δxβ ,. O&P`ga"¥4pri)uKf`_. . urfw urf`_. δx ,. 2Cf log N fmin Lf,β N. 1 ! 1+β. .. ". 9r9 3. ckPT(ieT©T`jma¡ceaXjckP vikpL¢uK¢Rjssj¨cd_prfRTP3Xurf¬jsfLceT(LT(i 9r9(J} 3 |x − X | ≤ δ. x ∈ [0, 1]. ik. x. VXp ¥`ckPRT T>adcejSuKckjsprfT(ieikpLi&uKcu©v4prjsf/c x x(urf ¢4T TbvurfwbT"wura Yik ≥ f (Xik ) − δy .. f (x) − fbN (x). = [f (x) − f (Xik )] i h + f (Xik ) − fbN (Xik ) h i + fbN (Xik ) − fbN (x) .. >|. 9r9 r3. "~. 9r9 3 9r9Pd53 9r9CO 3. O&PTZceT(ieS jsfckPRT iejLP/c&P<uKfwa¡jmwbT 9r9"~53Su$_¤¢4T ¢4prgRfwRT"wurapLspa. |f (x) − f (Xik )| ≤ Lf,β |x − Xik |β ≤ Lf,β δxβ ,. î!î¹ ÐyÓ. 9r9Cy 3.
(209) 9-O. . uLa&T",uLackPRT)ceT(ieS. . W
(210) w . . 9r9CO53. $K.
(211)
(212)
(213)
(214) b
(215) fN (Xik ) − fbN (x)
(216) ≤ LfbN |x − Xik | ≤ LfbN δx ,. jckP©u,jsvaexyPRj¨ceoxpLfa¡ceuKf/c L pLi,ckPTgRfxckjsprfT>adcejSuKckpri wRgRT)ckYp 9$W| 3pr_i ;K5 3O&P`ga(¥, 959$|W3jSUvRsjsT"a fbN. fbN (x). 9 3. %tT"S jsfw)cePuc. f (Xik ) − fbN (Xik ) ≤ (Yik + δy ) − Yik = δy .. pLS¢RjsfRjsfRuKsFckPRT>a¡T ¢4prgfwRa&TpL¢bceurjfieprS. >~ ckPuKc&priuKs. f959 53. N ≥ N6 (ω). f (x) − fbN (x) ≤ δy + Lf,β δxβ + LfbN δx .. O&PT(ieTprieTr¥!uKvRvRs_`jf¬,T(SUSu9uKfwakgR¢a¡ckjckgbcejf¬TbvRieT"aea¡jsprf<aXf959>53Zpri sT"uLw¤cep ckPRT spT"i&¢<pLgRfw. δx. fbN (Xik ) ≥ Yik. $. 9 :9P3. ¥ uKfw. $r jsf/ckpK 9$rW3. 9 r3. δy. ≥ f (x) − 2Lf,β δxβ + LfbN δx. fbN (x). $K. 9 3. $} pLiurf/_ a¡gR£x(jT"f/cks_ muKierT N GadcyuKikckjsfR ieprS ieurfwbpLS uR a"¦fj¨ceTXjsfLceT(LT(i>¥rPRjmxyP wbp`T"aofRpKcwbT(v4T(fw pLf x3DO&PRTiea¡cjfRT>l/guKsj¨cd_ jsfGK}r3!PuLa%¢<T"T(fUuKvRvRsjsT"wPT(ieTjsfpLiewbT"i¦cepakjsS vj_ ckPTspT(i%¢<pLgRfw' ¦T(SUSu } jmavRieprT>w' C4 (β) ≥ f (x) − h2. . log N N. 2+β 1+β.
(217)
(218) 9L]`jf<xT ¥`ckPT L ªàfRprieSpK%T"a¡ckjsSuckjsprfT"ikiepri&x(urf ¢4T TbvurfwbT"wuLa |u| = u − 2u1{u < 0} . 9 J3. . 1. kfbN − f k1. =. Z. + 2. 1 0. Z. 0. h 1. i fbN (x) − f (x) dx h. i n o f (x) − fbN (x) 1 fbN (x) < f (x) dx.. $r| 9$K~r3 9 53. R Q Rv vRs_`jf¤,T(SUSuraV9 urfw cep ckPRT iejLP/c&P<uKfwa¡jmwbTif9>r|53_/jsT(mwRa lim sup h. −β. Z. 1. h. i b fN (x) − f (x) dx ≤ γ + 4Cα (gmax − 1)Kmax 1{β = 1} a.s.. >. f9 dW3. VXpKceTr¥`ckP<ucprfTSu$_ γ = 2L g C (K) ¥bprijsfa¡ceuKf<xTr ÍfpLiewRT(ickppL¢bceurjfuUa¡jsSUjmuKiieT"akgRcpLi&ckPRT)ceT(ieSf9>K~W3¥RfRprckT)ckP<ucX,T"SUSUu }UjsS vjsT"a N →∞. 0. f,β. max. β. h i bN (x) ≤ C4 (β) < ∞ a.s. ζN (x, ω) , ε−1 (N ) f (x) − f LB. Éæ%î,ÉËÊ.
(219) 9-y. L1
(220) . gfRj¨pLikSUs_¤jckPikT>a¡v4T"xc&ckp¢4pKckP. x ∈ [0, 1]. uKfw. N ≥ N2 (ω). ¥Rj¨ceP. $ gXT(fx(Tr¥,pLfRT SUu$_¬urvRvRs_ µ uKckpLg sT(SUSuR¥'cyuK `jf jsf/ckpurx(x(prgRf/cckPuKc jma u x(prf/ckjsf`gRprga"¥ u1{u > 0} SUpLfRpKceprfRTgRfxckjsprf Z h o i n 9$W yr3 lim sup ε (N ) f (x) − fb (x) 1 fb (x) < f (x) dx 1 εLB (N ) , 2 h. . log N N. 2+β 1+β. 9 WO 3. .. 1. N →∞ 1. −1 LB. N. N. 0. "L 9"T9-3 ≤ C (β) < ∞ a.s. }<9O&P/g<a(¥%ckPT pr¢RceuKjsfRT>wikT"suKckjsprfa)ceprLTckPT(ijckP f9>L|W3urfwN 9$K~53)jsSUvRs_ GL|W3 O&PRT"prieT(S 9 jsa vikpLT"w' Z. ≤. f9 r3. lim sup ζN (x, ω) 1{ζN (x, ω) > 0} dx N →∞. 0. 4. ï Íf ]`gR¢akT"xckjsprf©~~9oTT"a¡ceuK¢jma¡P©a¡pLS TvRieprv4T(ikckjsT"a,ieT(muceT"w)cepcePRTx(prieikT>xDceT"w) rT"ikfT(;%]`gR¢<a¡T>xDckjsprf©~Rº vikT>a¡T"fLcya,akprSUTuKg bjsjmuKie_XsT(SUSura¦PRjmxyP Pu$rT¢4T(T(fga¡T>wckpvikpLTO&PT(prieT(M S 9r µ jsfurs_r¥$TxprssT"xDc jsf¬]`gR¢akT"xckjsprf~R Ua¡pLS T)sT(SUSurawbT>wbjmx(uceT"w ckpUcePRT vRiep/pr!pr%tT"SUurik |`
(221)
(222) *& # ,# ¦Tc&ckPT ¢urakjsxZ rT"ikfT(4gRfxckjsprf K ¢<T wbT(fRT"wuLajsf¬]`T"xckjsprf}R¥urfw¤cePRT)¢uKf<wbjswbckP h ∈ (0, 1/2) tT(SUjf<w cePRT T"a¡ckjsSucepri fb wbT(fRT>w js[f G5| 3&uLaprspa . . . N. N X b fN (x) = αi Kh (x, Xi ) i=1 αi ≥ 0, i = 1, . . . , N,. PT(ieTZcePRT rT"ikfT(4gRfxckjsprf. Kh (x, t) = h−1 K((x − t)/h). PjsT. Kh (x, t) = h. urfw. Kh (x, t) = h. î!î¹ ÐyÓ. −1. −1. K((x − t)/h). K((x − t)/h). Z. Z. ∀ x ∈ (h, 1 − h). x/h. K(t) dt −1. 1. K(t) dt (x−1)/h. >L. 9 r3. !−1. !−1. ∀ x ∈ [0, h]. ∀ x ∈ [1 − h, 1] .. "r. f9 3. >K}. 9 3. "/|. 9 53.
(223) r. . . W
(224) w . . &O P`ga"¥'cePRT rT"ikfRT"%gRfxDcejpLf K (x, t) jmawbT(fRT>w¬pLiuKf`_ (x, t) ∈ [0, 1] × R ¥!uKf<w¬cePRTT"a¡ckjsSuckpLi f9"/W3jmawbTfT"wpLiurf`_ `jmu cePRTZ LT(iefRT( K (x, t) xprieieT"xDceT"wuKccePRT¡¢4prgRfwuKiejT>aK p fRT Su$_¤T>urakjs_pr¢<a¡T"ikLTXcePuc x ∈ [0, 1] Z f9"Lr~ 3 K (x, u) du = 1 ∀ x ∈ [0, 1] urfw'¥Rx(prfakT"l/gRT"fLce_L¥RwbgRT)cep T RxyPuKfRLjfcePRT jf/ceT(riyuK'urfw ckPRTwbT"ikjsucejLTr¥ Z f9"J dW3 ∂ K (x, u) du = 0 ∀ x ∈ [0, 1] ∂x VXpKceTr¥bcePucXT>l/gucejpLsf f9" dW3SUu$_uKma¡pU¢4T rT"ikjT"wwbjsieT"xDce_L µ pLijf<adcyuKfx(Tr¥RpLf cePRTsTËcX¢4prgfwRuKie_r¥ j; Trpri x ∈ [0, h] ¥bT)Pu$LT h. h. 1. h. 0. 1. h. 0. Kh (x, t) = h. −1. K((x − t)/h)g(x) ,. g(x) =. ZT(fpKckjsfR. Z. x/h. K(t) dt −1. !−1. .. >. 9 5yr3. e h (x, u) = ∂ Kh (x, u) , K ∂x. TckP`gaP<u$rT g 0 (x) = −. urfw. Z. x/h. K(t) dt −1. !−2. >. 9 5O 3. h−1 K(x/h) = −g 2 (x)h−1 K(x/h). e h (x, u) = h−1 K((x − u)/h)g 0 (x) + g(x)h−2 K 0 ((x − u)/h) K x−u x−u −1 −1 0 = g(x)h −K Kh (x, 0) . h K h h. "}L. 9 r3. (} 9"}/r3 9 a9-3. XT(fx(Tr¥bcePRT)jf/ceT(riyuK g. "}L T>l/guKma"T(iep9pri x ∈ [0, h] Q a¡jsSUjmuKi©vRiep/prS jsrP/c ¢<TikT"v<T>uceT"wpLi x ∈ [1 − h, 1] µ jf<uKs_L¥ T>l/guKsj¨cd_ 9"L~W3PRprmwRa&ceikgTZpLiuKs x ∈ (h, 1 − h) ckp`p¥RakjsfxT g(x) ≡ 1 prT(icePRjsajf/ckT"ikurG ÍfPuKc&prspa"¥bT)gakT)SUprieTLT(fRT"ieur4prieSgRmuraVG|r33$§~r3jsfa¡ckT>urw pKb 9"rOW3D¥bcePucjma Z. 1. 0. 2 e h (x, u) du = − g (x) K x K h2 h. Kh (x, t) = h. −1. Z. K((x − t)/h) g(x) ,. 1. K. 0. . x−u h. g(x) =. Z. . du +. u=1 g(x) x−u −K h h u=0. x/h. K(t) dt (x−1)/h. !−1. ,. x ∈ [0, 1] .. 9 3. "}r}. 9 3. Éæ%î,ÉËÊ.
(225) ^9. L1
(226) . O&PT(ieTprieTr¥buLaprspaieprS f9"ry53D¥"f9(}L}53priuKf`_. x ∈ [0, 1]. ¥. e h (x, u) = h−1 K((x − u)/h)g 0 (x) + g(x)h−2 K 0 ((x − u)/h) K x−u g(x) 1 0 x − u = +K (Kh (x, 1) − Kh (x, 0)) . K h h h h. (}`| 9"}L~ 3 9 53. O&PTZpLspjsfR,T(SUSu vRieprT"a&¦jvaexyPRjck(ª;sj LTx(prfa¡ceurf/cea&jsf[f9"JdP3&uKfw[§~T9C33$§~/W3D ¯ 9" r K r7 ; ,R w)%^i A &
(227) ,R QB,V . W>%. . %TA. . %TA. ? 5T . h. ? 2
(228) <B#XI &' ^% )%^S
(229) . <. Z/. 9 Ke n h ∈ (0, 1/2) ,
(230)
(231)
(232) e
(233) 2 ,
(234) Kh (x, u)
(235) ≤ gmax h−2 LK + gmax Kmax
(236)
(237)
(238) ∂
(239)
(240) e h (x, u)
(241) ≤ gmax h−3 L e , K K
(242) ∂u
(243). Y>T % ? ? r^ : P % 4. "}. 9 Jd53.
(244) 2
(245)
(246) ∂
(247)
(248) e h (x, u)
(249) ≤ gmax h−4 L e 0 , K K
(250) ∂u2
(251). L = L 00 + L 0 g K LK K max max K e = LK 0 + LK gmax Kmax e0 K
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