A Note on Quantile Coupling Inequalities and Their Applications

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1 A Not o Quatil Couplig Iqualitis ad Thir Applicatios Harriso H. Zhou Dpartmt of Statistics, Yal Uivrsity, Nw Hav, CT 06520, USA. huibi.zhou@yal.du Ju 2, 2006 Abstract A rlatioship btw th larg dviatio ad quatil couplig is studid. W apply this rlatioship to th couplig of th sum of i.i.d. symmtric radom variabls with a ormal radom variabl, improvig th classical quatil couplig iqualitis (th ky part i th clbratd KMT costructios) with a rat = p for radom variabls with cotiuous distributios, or th sam rat modulo costats for th gral cas. Applicatios to th asymptotic quivalc thory ad oparamtric fuctio stimatio ar discussd. Kywords: Quatil Couplig; Larg dviatio; KMT/Hugaria costructio; Asymptotic quivalc; Fuctio stimatio RUNNING TITLE: Quatil Couplig Iqualitis.

2 Itroductio Th KMT/Hugaria costructio i Komlós, Major, ad Tusády (975) is cosidrd o of th most importat statistics ad probability rsults of th last forty yars. It has b widly applid i may aras of statistics ad probability (cf. Shorack ad Wllr (986)). Th quatil couplig of th sum of i.i.d. Broulli(=2) with a ormal radom variabl lis at th hart of KMT/Hugaria costructio for mpirical procss. I this papr, w study th couplig of th sum of i.i.d. symmtric radom variabls with a ormal radom variabl, ad improv th classical quatil couplig bouds with a rat = p for radom variabls whos distributios ar absolutly cotiuous with rspct to a Lbsgu masur, or th sam rat modulo costats for th gral cas. This papr ca b rgardd as a gralizatio of Cartr ad Pollard (2004), which studid th couplig of Biomial(; =2) ad a ormal radom variabl ad improvd th classical quatil couplig bouds (calld Tusády s Lmma) with a rat = p modulo costats. Th KMT costructio playd a ky rol i th progrss of th asymptotic quivalc thory i th last dcad. Nussbaum (996), a brakthrough of asymptotic quivalc thory, stablishd th asymptotic quivalc of dsity stimatio ad Gaussia whit ois udr a Höldr smoothss coditio. A major stp toward th proof of this quivalc rsult is th fuctioal KMT costructio for mpirical procss by Koltchiskii (994), whr lyig at th hart of th costructio is Tusády s Lmma. Th impact of this rsult is that a asymptotically optimal rsult i o of ths oparamtric modls automatically yilds a aaous rsults i th othr modl. Startig from Dooho ad Johsto (995), Bsov smoothss costrait bcam a stadard assumptio i th oparamtric stimatio. Rctly, Brow, Cartr, Low ad Zhag (2004) xtdd th rsult of Nussbaum (996) udr a sharp Bsov smoothss costrait via th improvd Tusády s iquality by Cartr ad Pollard (2004). This asymptotic quivalc rsult is cosidrd a importat progrss i this ara. It is might b worthwhil to mtio that th classical Tusády s iquality may ot b su cit to stablish asymptotic quivalc udr th coditios statd i th papr of Brow, cartr, Low ad Zhag (2004). Gral quatil couplig iqualitis (s Sakhako (984) ad Komlós, Major, ad Tusády (975)) ld to a xtsio of asymptotic quivalc thory i Nussbaum (996) to gral oparamtric stimatio modls (s Grama ad Nussbaum (998, 2002a, 2002b)). Amog thos modls a importat o is th spctral dsity stimatio modl. I Zhou (2004) or Golubv, Nussbaum ad Zhou (2005), w applid a sharp quatil couplig boud btw a Bta ad a ormal radom variabl (a spcial cas of gral rsults i this papr) to stablish th asymptotic quivalc of th spctral dsity stimatio ad Gaussia whit 2

3 ois udr a Bsov smoothss costrait. O possibly itrstig applicatio of our rsult is couplig a mdia statistic with a ormal radom variabl. W obtai a sharp quatil couplig iquality which also improvs th classical quatil couplig bouds with a rat = p udr crtai smoothss coditios for th distributio fuctio (s sctio 5). It icluds th Cauchy distributio as a spcial cas. This couplig rsult may b of idpdt itrst bcaus of th fudamtal rol of mdia i statistics. Th papr is orgaizd as follows. I sctio 2, w list basic rsults for th quatil couplig of th sum of i.i.d. symmtric radom variabl. I sctio 3, w giv a gral assumptio to obtai a quatil couplig iquality with a improvd rat = p, which immdiatly implis a sharp quatil couplig rsult for th sum of i.i.d. symmtric radom variabl with cotiuous distributio. Sctio 4 givs a gral assumptio to obtai a quatil couplig iquality with a improvd rat modulo costats. Som applicatios of th couplig rsults ar discussd i sctio 5. 2 Basic Rsults Th quatil couplig of th sum of i.i.d. Broulli(=2) (or Biomial(; =2)) with a ormal radom variabl is a ky stp i KMT/Hugaria couplig of th mpirical distributio with a Browia bridg i Komlós, Major, ad Tusády (975). Th tight quatil couplig boud for Biomial(; =2) i Tusády (977) is formulatd as follows: thr is a radom variabl X distributd Biomial(; =2) ad a Y = =2 + p Z=2 distributd N(=2; =4) such that jx Y j C + C jxj2 ; wh jxj "p for som C; " > 0. S Massart (2004) for possibl xplicit valus of C ad ", although w do t d thm i stablishig asymptotic quivalc rsults. Th proof of this boud was rst sktchd i Komlós, Major, ad Tusády (975) ad dtaild i svral paprs,.g., Maso ad va Zwt (987), Brtagoll ad Massart (989), Dudly (2000), Major (2000), Maso (200), Lawlr ad Trujillo Frrras (2005), tc. Cartr ad Pollard (2004) improvd that classical quatil bouds for Biomial(; =2) with a rat = p modulo costats. Mor spci cally, thy showd that for th couplig btw a X distributd Biomial(; =2) ad a Y = =2 + p Z=2 distributd 3

4 N(=2; =4), for som C; " > 0. jx Y j C + C jxj3 ; wh jxj "p 2 Th couplig bouds for gral radom variabls ad th dtaild proofs ca b foud i Sakhako (984, 996). I this sctio, w xtd th rsult of Cartr ad Pollard (2004) to gral symmtric radom variabls, i.., sharps th boud i Sakhako (984, 996) (or Komlós, Major, ad Tusády (975)) for th sum of symmtric radom variabls. Th followig propositio is th classical quatil couplig rsult (cf. Sakhako (996) or Lmma i Komlós, Major, ad Tusády (975)). Lmma 2 i Propositio Lt X, X 2,..., X b i.i.d. radom variabls such that EX = 0, EX 2 = P, E xp ft jx jg < for som t > 0. Lt S = p i= X i, ad Z b a stadard ormal radom variabl. Th for vry, thr is a radom variabl S with L S = L (S ) such that S Z C p + C p S 2 for S " p, whr C ; " > 0 do ot dpd o. I may practical situatios, th radom variabls ar symmtric. W hav a improvmt o th classical quatil couplig rsult with a rat = p for radom variabls with cotiuous distributios. Thorm I additio to th assumptios i Propositio suppos that EX 3 = 0 ad th charactristic fuctio v (t) satis s lim sup jtj! jv (t)j <. Th for vry, thr is a radom variabl S with L S = L (S ) such that for S Z C + C S 3 S p ", whr C; " > 0 do ot dpd o. If th absolutly cotiuous compot of th radom variabl X is ozro, th assumptio lim sup jtj! jv (t)j < i Thorm is satis d. Without that assumptio for th charactristic fuctio v (t), w hav a improvmt o th classical quatil couplig boud with a rat = p modulo costats. 4

5 Thorm 2 I additio to th assumptios i Propositio suppos that EX 3 = 0. Th for vry, thr is a radom variabl S with L S = L (S ) such that for S Z p C + C S 3 S p ", whr C; " > 0 do ot dpd o. Th assumptios of Thorm 2 ar satis d for X =Broulli(=2) =2. Thorm 2 is th a atural xtsio of Cartr ad Pollard (2004). 3 Quatil Couplig for Cotiuous cas I this sctio, w giv a gral assumptio to obtai a quatil couplig iquality with a improvd rat. W th apply this iquality to th sum of idpdt radom variabls with vaishig third momt to obtai Thorm which icluds th couplig of th sum of symmtric radom variabls as a spcial cas. A basic iquality for Mill s ratio will b dd to driv th quatil couplig iquality. Lmma For x > 0, w hav ' (x) _ (x) > mi x; 2 p x + 2 p Th followig thorm givs th rlatioship btw th xistc of a crtai typ of larg dviatio rsult ad a sharp quatil couplig iquality. That typ of larg dviatio is oft calld Ptrov xpasio. Actually, th xpasio w us i this papr is v mor prcis tha that of Ptrov (s Rmark 2). Mayb it is bttr to call it Saulis xpasio (s pag 249 i Ptrov (975)). Thorm is just a immdiat cosquc of th followig thorm ad Propositio 2. I this papr, w us a otatio O (x), which mas a valu btw som C > 0. Cx ad Cx for Thorm 3 Lt Z b a stadard ormal radom variabl. Lt S b a radom variabl with a distributio fuctio G (x) = P (S x). Assum that thr is a positiv that for all, P (S < x) = ( x) xp O x 4 + ; P (S < x) = (x) xp O x 4 + ; 5 " such

6 whr G (x) = G (x), ad (x) = (x), ad O ( x 4 + ) is uiform o th itrval x 2 [0; " p ]. Ad th xpasio abov holds wh < is rplacd by. Th for vry, thr is a radom variabl S with L S = L (S ) such that for S Z C + C S 3 S p ", whr C ; " > 0 do ot dpd o. () Rmark Th d itio of distributio fuctio hr is di rt from that i Ptrov (975), or Major (2000), or Maso (200), tc. Thy d G (x) = P (S < x). But w us th mor stadard d itio G (x) = P (S x). Rmark 2 Lt a (; x) = =2 x 3 + =2 x + =2. Th Ptrov xpasio is rplacig O ( x 4 + ) i th Thorm by O (a (; x)) (s Thorm i Chaptr VIII of Ptrov (975), or Thorm A i Komlós, Major, ad Tusády (975)). But th corrspodig couplig iquality will b S Z C p + C p 2 S (s Sakhako (984, 996)). Th dviatio trm O ( x 4 + ) improvs O (a (; x)) with a rat = p for x i a costat lvl, so is th corrspodig quatil couplig iquality. Th followig is a dtaild proof of Thorm 3. It is a modi catio of th proof for th classical cas, which was sktchd i Komlós, Major, ad Tusády (975). whr Proof: D such that L S bcaus th drivatio for S = G (Z) (2) G (x) = if fu; G (u) xg ; = L (S ). Without loss of grality, w assum that 0 S " p, C " p S 0 is similar. Th quatio () is quivalt to + + S 3 S Z C S 3 6

7 i.., S C + S 3 (Z) S + C + S 3. D G S = P (S < x). From th d itio of S i (2) w hav G S (Z) G S, th w d oly to show S C G S + G S S 3 S + C + S 3. i.. x C x3 )! (x) G (x ) G (x) (x) (x) x + C ( + x3 )! (x) wh 0 x " p. From th assumptio i th thorm, w kow max G (x) (x) ; G (x ) (x) C x 4 + for som C > 0. Thus it is ough to show thr is C > 0 such that x C ( + x3 )! C x 4 + (3) (x) x + C ( + x3 )! (x) W oly show th rst part of th iquality abov du to th symmtry of th quatio. It is asy to s that th rst part of th quatio abov is satis d udr th coditio x C + jxj3 0 (w will s latr th valu of C 2 ca b spci d as 8 p 2C). It implis x C = for su citly larg udr a assumptio that C " 2, which holds choosig su citly small ". Th for 0 x C = ad su citly larg, w 7

8 hav x C x3 )! (x) = 2 x C ( + x3 )! (0) + 2 x C x C + x3 + x3 whr th last iquality follows from th fact ( + y) y=2 for 0 y. Writ x C 2 + x3 = C + x3 x (0). Sic C ( + x3 ) 2 ad ' (x) 9 p 2 implis C + x3 x (0), for 0 x 2, th itrmdiat valu thorm C + x3 x 9 p 2 9 p 2 C 2 + x3 C 8 p 2 x 4 + which is mor tha C ( x 4 + ) wh C 8 p 2C. Thus th quatio (3) is stab- lishd i th cas of x C + jxj Now w cosidr th cas x C + jxj3 0. Th itrmdiat valu thorm tlls 2 C us thr is a umbr btw x ad x ( + 4 x3 ) such that x C ( + x3 )! (x) C x 4 ( + x3 )! (x) = C 4 + x3 ' () (). From th lmma (), w hav C x 4 ( + x3 )! (x) C 4 + x3 C x x3 + p 2 2 C 4 + x3 x p 2 2 C x4 + C. 8

9 wh C 6C. Puttig all togthr, w stablish (3) ad prov th thorm. I som applicatios, it is mor covit to us th followig corollary. Th boud ivolvs oly th ormal radom variabl. I Zhou (2004), w usd th couplig of Bta distributio with a ormal to stablish asymptotic quivalc of Gaussia variac rgrssio ad Gaussia whit ois with a drift, ad w foud that it was much asir to us th followig boud i momts calculatios. Corollary Udr th assumptio of Thorm 3, for vry thr is a radom variabl S with L S = L (S ) such that S Z C + jzj3 ; wh jzj " p for som C; " > 0. Proof: Obviously th iquality () still holds, wh S p " for 0 < " ". Lt s choos " small ough such that C" 2 < =2. WhS p ", w hav from (), which implis by th triagl iquality, i.., so w hav for som C > 0. S S S Z C + C Z C + S 2, C jzj + S 2 S 2C + 2 jzj, (4) 3 2C + 2 jzj C + jzj3 Wh S = " p > 0 for ay " with 0 < " ", w kow Z 0 from th d itio of quatil couplig, ad from (4) w hav Z " p I th d itio of quatil couplig, w s that S is a icrasig fuctio of Z. So w hav S p ", wh Z " 2C. Similarly w may show S ", wh p Z " + 2C. Thus w hav S p 2C ", wh jzj " p. (5) 2C. 9

10 Lt " 2 = " =2. W hav " 2 p < " p from (5), so w hav jzj "2 p 2C jzj " p for > 2C " 2 2=3, th w kow 2C S p o " S Z C + jzj3 p 2=3 2C ; wh jzj " 2 ad >. " 2 Thus w hav S Z C + jzj3 ; wh jzj " 2 p. A applicatio of Thorm 3 ad Corollary is th couplig of th sum of idpdt radom variabls with a ormal radom variabl. Assum that thos radom variabls hav it xpotial momt ad vaishig third momt (.g. symmtric radom variabl). Th followig is th Saulis xpasio (S pag 249 i Ptrov (975), or pag 88 i Saulis ad Statulvicius (99)), which givs a sharp approximatio to th tail probability of th sum of thos radom variabls. Th proof of th this rsult ca also b drivd basd o similar argumts i Sctio 8.2 i Ptrov (975). Propositio 2 Lt X ; X 2 ;..., X b i.i.d. radom variabls for which EX = 0; EX 2 = ; EX 3 = 0; E xp (a jx j) < for som a > 0; ad lim sup xp E (itx ) < : jtj! D S = p P i= X i. Th thir xists positiv costats C ad such that i th itrval 0 x. P S < x = ( x) xp C x 4 + C P (S < x) = ( x) xp C x 4 C Not that th xpasio abov holds for Y i = X i. This implis th xpasio abov holds wh < is rplacd by. This propositio ad Thorm 3 immdiatly imply th followig corollary ad Thorm. 0

11 Corollary 2 Udr th assumptio i Propositio 2, for vry, thr is a radom variabl S with L S = L S such that for S Z mi C + C S 3 C ; + C jzj3 S " p or jzj " p, whr C; " > 0 do ot dpd o. 4 Quatil Couplig for Gral Cas I this sctio, w giv a gral assumptio to obtai a quatil couplig iquality with a improvd rat modulo costats. O applicatio of th rsult is sharpig classical quatil couplig iquality with a rat modulo costats for th sum of idpdt symmtric radom variabls. So this rsult is a gralizatio of Cartr ad Pollard (2004), whr thy cosidrd couplig for Biomial(; =2). Th followig thorm ad Lmma 2 imply Thorm 2. Thorm 4 Lt Z b a stadard ormal radom variabl. Lt S b a radom variabl with a distributio fuctio G (x). Assum that thr is a positiv " such that for all, P (S < x) = ( x) xp O x 4 + =2 ; P (S < x) = (x) xp O x 4 + =2 ; whr G (x) = G (x), ad (x) = (x), ad O x 4 + =2 is uiform o th itrval x 2 [0; " p ] with " > 0. Ad th xpasio abov holds wh < is rplacd by. Th for vry, thr is a radom variabl S with L S = L (S ) such that for S Z C p + C S 3 S p ", whr C ; " > 0 do ot dpd o. Th proof of Thorm 4 is similar to that of Thorm 3, so w skip th proof. Similar to th proof of Corollary, w hav Corollary 3 Udr th assumptio of Thorm 4, for vry thr is a radom variabl S with L S = L (S ) such that S whr C; " > 0 do ot dpd o. Z p C + C jzj3 ; wh jzj " p

12 A applicatio of Thorm 4 ad Corollary 3 is th couplig of th sum of idpdt radom variabls with a ormal radom variabl. Assum that thos radom variabls hav it xpotial momt ad vaishig third momt (.g. symmtric radom variabl). A approximatio to th tail probability of th sum of thos radom variabls is giv i th followig lmma. Th proof of th approximatio is basd o similar argumts i Sctio 8.2 i Ptrov (975). It is a xtsio of Thorm i Cartr ad Pollard (2004). Lmma 2 Lt X ; X 2 ;..., X b i.i.d. radom variabls for which EX = 0; EX 2 = ; EX 3 = 0; E xp (a jx j) < for som a > 0: Th thir xists positiv costats " such that i th itrval 0 x ", whr P (S < x) = ( x) xp O x 4 + =2 (6) P (S < x) = (x) xp O x 4 + =2 (7) S = p X X i, ad th xpasio abov holds wh < is rplacd by. Proof: From Thorm 2 of Sctio 8.2 i Ptrov (975), w kow P (S < x) = ( x) xp C x 4 + C (x + ) =2 (Our otatio is di rt from that of Ptrov. Our x hr is thir z = x= p ). Udr th assumptio of EX 3 = 0, w ca rplac th trms + O (z) of quatios (2.37) ad (2.38) i Sctio 8.2 i Ptrov (975) by + O (z 2 ). I th sam sctio, from quatio (2.35) w ca rplac th trm + O (z) of quatio (2.40) by + C= p. W kp vrythig ls i th proof Thorm 2 of Sctio 8.2 i Ptrov (975). Th w stablish th followig approximatio of th tail probability P (S < x) = ( x) xp C x 4 + C x 2 = + =2 Not that x 2 = ( x 4 + ) 2 2 argumt for quatio (7) is similar. Th xpasio holds if rplacig X i by wh < is rplacd by. i= x 4 + =2. Th w obtai quatio (6). Th 2 X i. This implis th xpasio abov holds

13 Rmark 3 I Lmma 2, w assum that thos radom variabls ar idtically distributd, but it ca b xtdd to o-idtically cas similar to Thorm 2 of Sctio 8.2 i Ptrov (975). Thorm 4 ad Lmma 2 imply th followig corollary (or basically Thorm 2). It xtds Thorm 2 of Cartr ad Pollard (2004). Corollary 4 Udr th assumptio i Lmma 2, for vry, thr is a radom variabl S with L S = L (S ) such that for S Cp Z mi + C S 3 C ; p + C jzj3 S p p " or jzj ", whr C; " > 0 do ot dpd o. 5 Som Exampls I this sctio, w discuss som applicatios of rsults i prvious sctios to asymptotic quivalc thory ad oparamtric fuctio stimatio. Exampl : Asymptotic quivalc of dsity stimatio ad Gaussia whit ois: E : y(); :::; y(); i.i.d. with dsity f o [0; ] F : dy t = f =2 (t) dt + 2 =2 dw t Th asymptotic quivalc rsult abov was stablishd i Brow, Low, Cartr ad Zhag ( 2004) udr a Bsov smoothss costrait. Th ky ida of that papr is applyig th classical KMT costructio. W th d a couplig for Biomial radom variabl ad a ormal radom variabl. Lt X ; X 2 ;..., X b i.i.d. Broulli(=2). Th Corollary 4 tlls us for vry thr is a radom variabl S with L S = L S such that S Cp Z mi + C S 3 C ; p + C jzj3 for S p p " or jzj ", whr C; " > 0 do ot dpd o (s also Cartr ad Pollard (2004)). This rsult was usd i th KMT costructio to stablish th asymptotic quivalc udr th Bsov smoothss coditio, compact i Bsov balls of B =2 2;2 ad B =2 4;4. If o applis th classical Tusády s iquality, a strogr smoothss coditio would b dd to stablish th asymptotic quivalc. 3

14 ois: Exampl 2: Asymptotic quivalc of spctral dsity stimatio ad Gaussia whit E : y(); :::; y(); a statioary ctrd Gaussia squc with spctral dsity f F : dy t = f (t) dt + 2 =2 =2 dw t whr f has support o [ ; ]. This asymptotic quivalc btw Gaussia spctral dsity, Gaussia variac rgrssio ad Gaussia whit ois i Golubv, Nussbaum ad Zhou( 2005) udr a Bsov smoothss costrait. I that papr, w usd a dyadic KMTtyp costructio, but di rt from th classical KMT costructio. I th KMT papr, thy usd a complicat coditioal quatil couplig for highr rsolutios. It is asy to obsrv that L(XjX + Y ) = L((X + Y )B ) for two idpdt ad idtically distributd radom variabls X ad Y with law 2, th w ca avoid th coditioal quatil couplig by cosidrig th couplig for a Bta radom variabl. Th followig couplig iquality is th usd. Lt Z b a stadard ormal radom variabl. For vry, thr is a mappig T : R 7! R such that th radom variabl B = T (Z) has th Bta (=2; =2) law ad (=2 B =2 ) 2 Z Cp mi + C jb 2 =2j 3 ; C + C jzj3 for jb =2j ", whr C; " > 0 do ot dpd o (cf. Zhou (2004)). Exampl 3: Quatil couplig of Mdia statistics. Lt X ; X 2 ; : : : ; X i.i.d. with dsity f (x). For simplicity, lt = 2k + with som itgr k, ad assum that f (0) > 0, f(0) = 0, ad f 2 C 3. Lt Z b a stadard ormal radom variabl. For vry, thr is a mappig T : R 7! R such that th radom variabl X md = T (Z) has dsity f (x) ad p 4f (0) X md Z C + jzj3 ; wh jzj " p whr C; " > 0 do ot dpd o. Dtails ad mor gral discussios will b prstd i Brow, Cai ad Zhou (workig papr). I this papr, w apply this quatil couplig boud to oparamtric locatio modl with Cauchy ois ad cosidr wavlt rgrssio. Dooho ad Yu (2000) cosidrd a similar problm, but miimax proprty is uclar for thir procdur. I wavlt rgrssio sttig, Hall ad Patil (996) studid oparamtric locatio modls ad achivd th optimal miimax rat, but udr a assumptio of th xistc of it forth momt. W do t d ay momt coditio, ad th ois ca b gral ad ukow, but achiv optimal miimax rat of covrgc. Without th 4

15 assumptio of f(0) = 0 or f 2 C 3, w may still obtai couplig bouds, but may ot as b tight as th boud abov. Th tightss of th uppr boud a cts th th udrlyig smoothss coditio w d i drivig asymptotic proprtis. Rfrcs [] Brtagoll, J. ad Massart, P. (989). Hugaria costructios from th oasymptotic viw poit, A. Probab. 7 () [2] Brow, L.D., Cai, T. T., Zhou, H. H.. Robust Noparamtric Wavlt Estimatio. I prparatio. [3] Brow, L.D., Cartr, A.V., Low, M.G. ad Zhag, C. (2004). Equivalc thory for dsity stimatio, Poisso procsss ad Gaussia whit ois with drift. A. Statist. 32, [4] Cartr, A. V. ad Pollard, D. (2004). Tusády s iquality rvisitd. A. Statist. 32, [5] Dooho, D. L. ad Johsto, I. M. (995). Adapt to ukow smoothss via wavlt shrikag. J. Amr. Stat. Assoc. 90, [6] Dooho, D. L. ad Yu, T. P.-Y. (2000). Noliar Pyramid Trasforms Basd o Mdia-Itrpolatio. SIAM Joural of Math. Aal., 3(5), [7] Dudly, R. M. (2000). Nots o mpirical procsss. Lctur ots for a cours giv at Aarhus Uiv., August 999. [8] Golubv, G. K., Nussbaum, M. ad Zhou, H. H., Asymptotic quivalc of spctral dsity stimatio ad Gaussia whit ois. Availabl at Submittd. [9] Grama, I. ad Nussbaum, M. (998). Asymptotic quivalc for oparamtric gralizd liar modls. Probab. Thory Rlat. Filds [0] Grama, I ad Nussbaum, M. (2002). Asymptotic quivalc for oparamtric rgrssio. Mathmatical Mthods of Statistics, (), -36. [] P. Hall ad P. Patil (996). O th choic of smoothig paramtr, thrshold ad trucatio i oparamtric rgrssio by wavlt mthods, J. Roy. Statist. Soc. Sr. B, 58,

16 [2] Komlös, J., Major, P. ad Tusády, G. (975). A approximatio of partial sums of idpdt rv s ad th sampl df. I Z. Wahrsch. vrw. Gbit [3] Komlös, J., Major, P. ad Tusády, G. (976). A approximatio of partial sums of idpdt rv s ad th sampl df. II Z. Wahrsch. vrw. Gbit [4] Lawr, G, F. ad Trujillo Frrras, J. A.. Radom walk loop soup. Availabl at [5] Major, P. (2000). Th approximatio of th ormalizd mpirical ditributio fuctio by a Browia bridg. Tchical rport, Mathmatical Istitut of th Hugaria Acadmy of Scics. Nots availabl from [6] Maso, D. M. (200). Nots o th KMT Browia bridg approximatio to th uiform mpirical procss. I Asymptotic Mthods i Probability ad Statistics with Applicatios (N. Balakrisha, I. A. Ibragimov ad V. B. Nvzorov, ds.) Birkhäusr, Bosto. [7] Massart, P. (2002). Tusády s lmma, 24 yars latr. A. Ist. H. Poicaré Probab. Statist [8] Nussbaum, M. (996). Asymptotic quivalc of dsity stimatio ad Gaussia whit ois. A. Statist. 24, [9] Ptrov, V. V. (975). Sums of Idpdt Radom Variabls. Sprigr-Vrlag. (Elish traslatio from 972 Russia ditio). [20] Sakhako, A. (984). Th rat of covrgc i th ivariac pricipl for oidtically distributd variabls with xpotial momts. Limit thorms for sums of radom variabls. Trudy Ist. Matm., Sibirsk. Otdl. AN SSSR (i Russia). [2] Sakhako, A. I. (996). Estimats for th Accuracy of Couplig i th Ctral Limit Thorm. Sibria Mathmatical Joural, Vol. 37. No 4. [22] Saulis, L ad Statulvicius, V.A. (99). Limit thorms for larg dviatios. Kluwr Acadmic Publishrs. [23] Zhou, H. H. (2004). Miimax Estimatio with Thrsholdig ad Asymptotic Equivalc for Gaussia Variac Rgrssio. Ph.D. Dissrtatio. Corll Uivrsity, Ithaca, NY. 6