Large and Moderate Deviation Principles for Kernel Distribution Estimator

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1 Iteratoal Mathematcal Forum, Vol. 9, 2014, o. 18, HIKARI Ltd, Large ad Moderate Devato Prcples for Kerel Dstrbuto Estmator Yousr Slaou Uversté de Poters Laboratore de Mathématques et Applcato Futuroscope Chasseeul, Frace Copyrght c 2014 Yousr Slaou. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract I ths paper we prove large ad moderate devatos prcples for the kerel estmator of a dstrbuto fucto troduced by Nadaraya Some ew estmates for dstrbuto fuctos. Theory Probab. Appl. 9, We provde results both for the potwse ad the uform devatos. Mathematcs Subject Classfato: 62E20, 60F10 Keywords: Dstrbuto estmato; Large ad Moderate devatos prcples 1 Itroducto Let X 1,...,X be depedet, detcally dstrbuted of radom varables, ad let f ad F deote respectvely the probablty desty of X 1 ad the dstrbuto fucto of X 1. Nadaraya (1964) troduce a kerel K (that s, a fucto satsfyg R K(x)dx = 1), a fucto K (that s, a fucto defed by K (z) = z K (u) du), ad a badwdth (h ) (that s, a sequece of postve real umbers that goes to zero). The estmator proposed by Nadaraya (1964) to estmate the dstrbuto fucto F at the pot x s gve by F (x) = 1 ( ) x Xk K. (1) h

2 872 Yousr Slaou Some theoretcal propertes of the estmator F have bee vestgated (see amog may others, Nadaraya (1964), Ress (1981), ad Hll (1985)). Ress (1981) ad Falk (1983) showed that the kerel dstrbuto estmator (1) have a asymptotcally better performace tha emprcal dstrbuto fucto, whch does ot take to accout the smoothess of F. Recetly, large ad moderate devatos results have bee proved for the wellkow orecursve kerel desty estmator troduced by Roseblatt (1956) (see also Parze, 1962). The large devatos prcple has bee studed by Loua (1998) ad Worms (2001). Gao (2003) ad Mokkadem et al. (2005) exted these results ad provde moderate devatos prcples. The purpose of ths paper s to establsh large ad moderate devatos prcples for the orecursve dstrbuto estmator (1). Let us frst recall that a R m -valued sequece (Z ) 1 satsfes a large devatos prcple (LDP) wth speed (ν ) ad good rate fucto I f : 1. (ν ) s a postve sequece such that lm ν = ; 2. I : R m 0, has compact level sets; 3. for every borel set B R m, f x B I (x) lm f ν 1 log P Z B lm sup ν 1 log P Z B f I (x), x B where B ad B deote the teror ad the closure of B respectvely. Moreover, let (v ) be a oradom sequece that goes to fty; f (v Z ) satsfes a LDP, the (Z ) s sad to satsfy a moderate devatos prcple (MDP). The frst am of ths paper s to establsh potwse LDP for the kerel dstrbuto estmator (1). We show that usg the badwths defed as h = h () for all, where h s a regularly varg fucto wth expoet ( a), a 0, 1. We prove that the sequece (F (x) F (x)) satsfes a LDP wth speed () ad the rate fucto defed as follows: { ( ) f F (x) 0, Ix : t F (x) I 1+ t F (x) (2) f F (x) =0, I x (0) = 0 ad I x (t) =+ for t 0. where I (t) = sup u R {ut ψ (u)} ψ (u) = exp (u) 1.

3 Large ad moderate devato prcples 873 Our secod am s to provde potwse MDP for the dstrbuto estmator defed by (1). For ay postve sequece (v ) satsfyg lm v v 2 = ad lm =0 ad geeral badwdths (h ), we prove that the sequece v (F (x) F (x)) satsfes a LDP of speed (/v 2) ad rate fucto J x (.) defed by { f f (x) 0, J x : t t2 2F (x) f f (x) =0, J x (0) = 0 ad J x (t) =+ for t 0. (3) Fally, we gve a uform verso of the prevous results. More precsely, let U be a subset of R; we establsh large ad moderate devatos prcples for the sequece (sup F (x) F (x) ). 2 Assumptos ad ma results We defe the followg class of regularly varyg sequeces. Defto 1. Let γ R ad (v ) 1 be a oradom postve sequece. We say that (v ) GS(γ) f lm 1 v 1 = γ. (4) + v Codto (4) was troduced by Galambos ad Seeta (1973) to defe regularly varyg sequeces (see also Bojac ad Seeta, 1973). Typcal sequeces GS (γ) are, for b R, γ (log ) b, γ (log log ) b, ad so o. 2.1 Potwse LDP for the Nadaraya s dstrbuto estmator To establsh potwse LDP for F, we eed the followg assumptos. (L1) K : R R s a bouded ad tegrable fucto satsfyg R K (z) dz =1, ad R zk (z) dz =0. (L2) (h ) GS( a) wth a 0, 1.

4 874 Yousr Slaou The followg Theorem gves the potwse LDP for F ths case. Theorem 1 (Potwse LDP for Nadaraya s dstrbuto estmator). Let Assumptos (L1) ad (L2) hold ad assume that F s cotuous at x. The, the sequece (F (x) F (x)) satsfes a LDP wth speed () ad rate fucto defed by (2). 2.2 Potwse MDP for the Nadaraya s dstrbuto estmator Let (v ) be a postve sequece; we assume that (M1) K : R R s a cotuous, bouded fucto satsfyg R K (z) dz =1, ad, R zk (z) dz = 0 ad R z2 K (z) dz <. (M2) (h ) GS( a) wth a 0, 1. (M3) F s bouded, twce dfferetable, ad F (2) (x) s bouded. v (M4) lm v = ad lm 2 =0. The followg Theorem gves the potwse MDP for F. Theorem 2 (Potwse MDP for the kerel dstrbuto estmator (1)). Let Assumptos (M1) (M4) hold ad assume that F s cotuous at x. The, the sequece (F (x) F (x)) satsfes a MDP wth speed (/v) 2 ad rate fucto J x defed (3). 2.3 Uform LDP ad MDP for the Nadaraya s dstrbuto estmator To establsh uform large devatos prcples for the dstrbuto estmator defed by (1) o a bouded set, we eed the followg assumptos: (U1) ) R zk (z) dz = 0 ad R z2 K (z) dz <. ) K s Hölder cotuous. (U2) F s bouded, twce dfferetable, ad, sup x R F (2) (x) <. v (U3) lm 2 log v =0. Set U R; order to state a compact form the uform large ad moderate devatos prcples for the dstrbuto estmator defed by (1) o U, we set: g U (δ) = { ( ) F U, I 1+ δ F U, whe v 1, (L1) ad (L2) hold δ 2 2 F U, whe v, (M1) (M4) hold g U (δ) = m {g U (δ),g U ( δ)}

5 Large ad moderate devato prcples 875 where F U, = sup F (x). Remark 1. The fuctos g U (.) ad g U (.) are o-egatve, cotuous, creasg o 0, + ad decreasg o, 0, wth a uque global mmum 0 ( g U (0) = g U (0) = 0). They are thus good rate fuctos (ad g U (.) s strctly covex). Theorem 3 below states uform LDP o U the case U s bouded, ad Theorem 4 the case U s ubouded. Theorem 3 (Uform devatos o a bouded set for the kerel dstrbuto estmator (1)). Let (U1) (U3) hold. The for ay bouded subset U of R ad for all δ>0, lm 1 v 2 log P sup v F (x) F (x) δ = g U (δ) (5) To establsh uform large devatos prcples for the dstrbuto estmator (1) o a ubouded set, we eed the followg addtoal assumptos: (U4) ) There exsts β>0 such that R x β f (x) dx <. ) F s uformly cotuous. (U5) There exsts τ>0 such that z z τ K (z) s a bouded fucto. (U6) ) There exsts ζ>0 such that R z ζ K (z) dz < ) There exsts η>0 such that z z η F (z) s a bouded fucto. Theorem 4 (Uform devatos o a ubouded set for the estmator defed by (1)). Let (U1) (U6) hold. The for ay subset U of R ad for all δ>0, g U (δ) lm f lm sup 1 v 2 log P sup v F (x) F (x) δ 1 v sup 2 log P v F (x) F (x) δ β β +1 g U (δ) The followg corollary s a straghtforward cosequece of Theorem 4. Corollary 1. Uder the assumptos of Theorem 4, f R x ξ F (x) dx < for all ξ R, the for ay subset U of R, lm 1 v 2 log P sup v F (x) F (x) δ = g U (δ) (6) Commet. Sce the sequece (sup F (x) F (x) ) s postve ad sce g U s cotuous o 0, +, creasg ad goes to fty as δ, the applcato of Lemma 5 Worms (2001) allows to deduce from (5) or (6) that sup F (x) F (x) satsfes a LDP wth speed () ad good rate fucto g U o R+.

6 876 Yousr Slaou 3 Proofs Throught ths secto we use the followg otato: ( ) x Xk Y k, = K h (7) Notg that, vew of (1), we have F (x) E F (x) = 1 (Y k, E Y k, ) Let (Ψ ) ad (B ) be the sequeces defed as Ψ (x) = 1 (Y k, E Y k, ) B (x) = E F (x) F (x) We have: F (x) F (x) =Ψ (x)+b (x) (8) Theorems 1, 2, 3 ad 4 are cosequeces of (8) ad the followg propostos. Proposto 1 (Potwse LDP ad MDP for (Ψ )). 1. Uder the assumptos (L1) ad (L2), the sequece (F (x) E (F (x))) satsfes a LDP wth speed () ad rate fucto I x. 2. Uder the assumptos (M1) (M4), the sequece (v Ψ (x)) satsfes a LDP wth speed (/v 2 ) ad rate fucto J x. Proposto 2 (Uform LDP ad MDP for (Ψ )). 1. Let (U1) (U3) hold. The for ay bouded subset U of R ad for all δ>0, lm 1 v 2 log P sup v Ψ (x) δ = g U (δ) 2. Let (U1) (U6) hold. The for ay subset U of R ad for all δ>0, g U (δ) lm f 1 v 2 log P sup v Ψ (x) δ lm sup 1 v sup 2 log P v Ψ (x) δ ξ ξ + d g U (δ)

7 Large ad moderate devato prcples 877 Proposto 3 (Potwse ad uform covergece rate of (B )). Let Assumptos (M1) (M3) hold. 1. If f s cotuous at x. We have If a 1/3, the If a>1/3, the B (x) =O ( h 2 ). ( ) B (x) =o 1 h. 2. If (U2) holds, the: If a 1/3, the sup B (x) = O ( h) 2. x R If a>1/3, the ( ) sup B (x) = o 1 h. x R Set x R; sce the assumptos of Theorems 1 guaratee that lm B (x) = 0, Theorem 1 s a straghtforward cosequece of the applcato of Part 1 (respectvely of Part 2) of Proposto 1. Moreover, uder the assumptos of Theorem 2, we have by applcato of Proposto 3, lm v B (x) = 0; Theorem 2 thus straghtfully follows from the applcato of Part 3 of Proposto 1. Faly, Theorem 3 ad 4 follows from Proposto 2 ad the secod part of Proposto 3. We ow state a prelmary lemma, whch wll be used the proof of Proposto 1. For ay u R, Set ( Λ,x (u) = 1 v exp 2 log E u ) Ψ (x) v Λ L x (u) = F (x)(ψ (u) u), Λ M u2 x (u) = 2 F (x) Lemma 1. Covergece of Λ,x 1. (Potwse covergece) If F s cotuous at x, the for all u R where Λ x (u) = lm Λ,x (u) =Λ x (u) (9) { Λ L x (u) whe v 1 Λ M x (u) whe v

8 878 Yousr Slaou 2. (Uform covergece) If F s uformly cotuous, the the covergece (9) holds uformly x U. Our proofs are ow orgazed as follows: Lemma 1 s proved Secto 3.1, Proposto 1 Secto 3.4 ad Proposto 2 Secto Proof of Lemma 1. Set u R, u = u/v ad a =. We have: Λ,x (u) = v2 log E exp (u a Ψ (x)) a ( ) = v2 log E exp u (Y k, E Y k, ) a = v2 a log E exp (u Y k, ) uv E Y 1, By Taylor expaso, there exsts c k, betwee 1 ad E exp (u Y k, ) such that log E exp (u Y k, ) = E exp (u Y k, ) 1 1 (E exp (u 2c 2 Y k, ) 1) 2 k, ad Λ,x ca be rewrte as Λ,x (u) = v2 a E exp (u Y k, ) 1 v2 2a 1 c 2 k, (E exp (u Y k, ) 1) 2 uv E Y 1, (10) Frst case: v. A Taylor s expaso mples the exstece of c k, betwee 0 ad u Y k, such that E exp (u Y k, ) 1 = u E Y k, u2 E Yk, u3 E Yk, 3 ec k, Therefore, Λ,x (u) = 1 2 u2 a v2 2a E Yk, 2 1 u + 6 u2 a 1 c 2 k, (E exp (u Y k, ) 1) 2 E Yk, 3 ec k, = 1 2 u2 F (x)+r,x (1) (u)+r(2),x (u) (11)

9 Large ad moderate devato prcples 879 wth R (1),x (u) = u2 R (2) R u 3,x (u) = v K (z) K ( z)f (x + zh ) F (x) dz a E Yk 3 e c k, v2 2a 1 c 2 k, (E exp (u Y k, ) 1) 2 Sce F s cotuous, we have lm F (x + zh ) F (x) = 0, ad thus, by the domated covergece theorem, (M1) mples that lm K (z) K ( z) F (x + zh ) F (x) dz =0, R R (1) t follows that lm,x (u) =0. Moreover, vew of (7), we have Y k, K, the c k, u Y k, u K (12) Notg that E Y k, 3 3 F R K (z) K2 (z) dz. Hece, t follows from (12), there exsts a postve costat c 1 such that, for large eough, u 3 1 v a E Yk 3 e c k, c 1 e u K u3 F v K (z) K 2 (z) dz (13) R whch goes to 0 as sce v. I the same way, there exsts a postve costat c 2 such that, for large eough, v 2 1 (E exp (u Y k, ) 1) 2 2a c 2 k, v2 2a (E exp (u Y k, ) 1) 2 ( u 2 c 2 2 h f 2 exp ( u K ) R (2) The combato of (13) ad (14) esures that lm obta from (11), lm Λ,x (u) =Λ M x (u). 2 K ( z) dz) (14) R,x (u) = 0. The, we

10 880 Yousr Slaou Secod case: (v ) 1. It follows from (10), that Λ,x (u) = 1 a ue Y 1, E exp (uy k, ) 1 1 2a 1 c 2 k, (E exp (uy k, ) 1) 2 Moreover, usg tegrato by parts, we get Λ,x (u) = uf (x) K (z) (exp (uk ( z)) 1) dz R,x (3) (u)+r,x (4) (u) (15) R wth R,x (3) (u) = 1 2a R,x (4) (u) = u R 1 c 2 k, (E exp (uy k, ) 1) 2 K (z) (exp (uk ( z)) 1) F (x + zh ) F (x) dz. R (3) It follows from (14), that lm,x (u) =0. Sce e t 1 t e t, we have R (4),x (u) u 2 e u K K (z) K( z) F (x + zh ) F (x) dz. R The, the domated covergece theorem esures that lm R,x (4) (u) =0. I the case F s uformly cotuous, set ε>0 ad let M>0such that 2 F K (z) K( z) dz ε/2. We eed to prove that for suffcetly z M large sup K (z) K( z) F (x + zh ) F (x) dz ε/2 x R z M whch s a straghtforward cosequece of the uform cotuty of F. The, t follows from (15), that lm Λ,x (u) = uf (x) K (z) (exp (uk ( z)) 1) dz R = F (x) (exp (u) 1 u) = Λ L x (u) ad thus Lemma 1 s proved.

11 Large ad moderate devato prcples Proof of Proposto 1 To prove Proposto 1, we apply Lemma 1 ad the followg result (see Puhalsk, 1994). Lemma 2. Let (Z ) be a sequece of real radom varables, (ν ) a postve sequece satsfyg lm ν =+, ad suppose that there exsts some covex o-egatve fucto Γ defed o R such that 1 u R, lm log E exp (uν Z ) = Γ (u). ν If the Legedre fucto Γ of Γ s a strctly covex fucto, the the sequece (Z ) satsfes a LDP of speed (ν ) ad good rate focto Γ. I our framework, whe v 1, we take Z = F (x) E (F (x)), ν = ad Γ = Λ L x. I ( ths case, ) the Legedre trasform of Γ = ΛL x s the rate fucto I x : t F (x) I 1+ t, sce ψ s strctly covex, the ts Cramer trasform F (x) I s a good rate fucto o R (see Dembo ad Zetou, 1998). Otherwse, whe, v, we take Z = v (F (x) E (F (x))), ν = /v 2 ad Γ = ΛM x ;Γ s the the quadratc rate fucto J x defed (3) ad thus Proposto 1 follows. 3.3 Proof of Proposto 2 I order to prove Proposto 2, we frst establsh some lemmas. Lemma 3. Let φ : R + R be the fucto defed for δ>0as { ( ) (ψ ) 1 1+ δ F φ (δ) = U, whe v 1, (L1) ad (L2) hold δ F U, whe v, (M1) (M4) hold 1. sup u R {uδ sup Λ x (u)} equals g U (δ) ad s acheved for u = φ (δ) > sup u R { uδ sup Λ x (u)} equals g U (δ) ad s acheved for u = φ ( δ) < 0. Proof of Lemma 3 oe beg smlar.. We just prove the frst part, the proof of the secod part Frst case v 1. Sce e t 1+t, for all t, we have ψ (u) u ad therefore, uδ sup Λ x (u) = uδ F U, (ψ (u) u) = ( ) δ F U, u 1+ F U, ψ (u)

12 882 Yousr Slaou The fucto u uδ sup Λ x (u) has secod dervatve F U, ψ (u) < 0 ad thus t has a uque maxmum acheved for ) u 0 =(ψ ) (1+ 1 δ F U, Now, sce ψ s creasg ad sce ψ (0) = 1, we deduce that u 0 > 0. Secod case v. I ths case, we have uδ sup Λ x (u) = uδ u2 2 F U,. The fucto u uδ sup Λ x (u) has secod dervatve F U, < 0 ad thus t has a uque maxmum acheved for δ u 0 = > 0 F U, Lemma 4. I the case whe (v ) 1, let (L1) ad (L2) hold; I the case whe v, let (M1) (M4) hold. The for ay δ>0, v 2 lm log sup P v Ψ (x) δ = g U (δ) v 2 lm log sup P v Ψ (x) δ = g U ( δ) v 2 lm log sup P v Ψ (x) δ = g U ( δ) Proof of Lemma 4. The proof of Lemma 4 s smlar to the proof of Lemma 4 Mokkadem et al. (2006). Lemma 5. Let Assumptos (U1) (U3) hold ad assume that ether (v ) 1or (U4) holds. 1. If U s a bouded set, the for ay δ>0, we have v 2 lm log P sup v Ψ (x) g U (δ) 2. If U s a ubouded set, the, for ay b>0 ad δ>0, v 2 lm sup log P sup v Ψ (x) b g U (δ), x w where w = exp ( b v 2 ).

13 Large ad moderate devato prcples 883 Proof of Lemma 5. Set ρ 0,δ, let β deote ( the ) Hölder order of K, ad K H ts correspodg Hölder orm. Set w = exp b ad v 2 ( R = ρ 2 K H v h β We beg wth the proof of the secod part of Lemma 5. There exst N () pots of R, y () 1,y () 2,...,y () N () such that the ball {x R; x w } ca covered by the { } ( N () balls B () = x R; x y () R ad such that N 2w () 2 R ). Cosderg oly the N () balls that tersect {x U; x w }, we ca wrte For each {1,...,N()}, set x () P sup v Ψ (x) δ, x w ) 1 β {x U; x w } N() =1 B(). B () N() =1 U. We the have: P sup v Ψ (x) δ x B () N () max 1 N() P Now, for ay {1,...,N()} ad ay x B (), ( ) Ψ v Ψ (x) v x () + v ( ) ( x K Xk x () K h + v ( ) x E K Xk K h ( ( ) Ψ v x () v +2 K H ( ) v x () +2v K H h β R β ( ) Ψ v x () + ρ sup v Ψ (x) δ x B () ) X k ) X k h ( x () h x x () h ) β. Hece, we deduce that P sup v Ψ (x) δ, x w ( N () max P Ψ v 1 N() ( Ψ v N () sup P x () x () ) δ ρ ) δ ρ

14 884 Yousr Slaou Further, by defto of N () ad w, we have ad log N () log N () b v log 2 log R v 2 log R = 1 v 2 β log ρ log (2 K H) log v + β log h. The, vew of (U3), we have lm sup v 2 log N () b (16) The applcato of Lemma 4 the yelds v 2 lm sup log P sup v Ψ (x) δ, x w lm sup b g U (δ ρ). v 2 log N () g U (δ ρ) Sce the equalty holds for ay ρ 0,δ, part 2 of Lemma 5 thus follows from the cotuty of g U. Let us ow cosder part 1 of Lemma 5. Ths part s proved by followg the same steps as for part 2, except that the umber N () of balls coverg U s at most the teger part of (Δ/R ), where Δ deotes the dameter of U. Relato (16) the becomes lm sup v 2 log R 0 ad Lemma 5 s proved. Lemma 6. Let (U1) ), (M2) ad (U6) ) hold. Assume that ether (v ) 1or(U3) ad (U6) ) ( hold. ) Moreover assume that F s cotuous. For ay b>0 f we set w = exp b the, for ay ρ>0, we have, for large eough, v 2 sup, x w v h ( ) x K E Xk ρ Proof of Lemma 6. We have v ( ) x Xk E K = v K (z) F (x + zh ) dz. (17) R h

15 Large ad moderate devato prcples 885 Set ρ>0. I the case (v ) 1, we set M such that F K (z) dz ρ/2; z >M t follows that v ( ) x K E Xk h ρ 2 + F (x) K (z) dz + K (z) F (x + zh ) F (x) dz. z M z >M Lemma 6 the follows from the fact that F fulflls (U6) ). As matter of fact, ths codtos mples that lm x, F (x) = 0 ad that the thrd term the rght-had-sde of the prevous equalty goes to 0 as (by the domated covergece). Let us ow assume that lm v = ; relato (17) ca be rewrtte as v E K ( ) x Xk h = v K (z) F (x + zh ) dz z w /2 +v K (z) F (x + zh ) dz. z w /2 Frst, sce x w ad z w /2, we have x + zh w (1 h /2) w /2 for large eough. Moreover, vew of assumptos (U3), for all ξ>0, lm v w ξ { = lm exp bξ ( 1 1 v 2 bξ v 2 log v )} =0. (18) Set M f = sup x R x η F (x). Assumpto (U6) ) ad equato (18) mple that, for suffcetly large, sup v K (z) F (x + zh ) dz x w z w /2 M f sup v K (z) x + zh η dz x w z w /2 2 η v M f w η K (z) dz R ρ 2.

16 886 Yousr Slaou Moreover, vew of (U3), (U6) ) ad (18), for suffcetly large, sup v K (z) F (x + zh ) dz x w z >w /2 2 ζ v M f z ζ K (z) dz w ζ z >w /2 ρ 2. Ths cocludes the proof of Lemma 6. Sce K s a bouded fucto that vashes at fty, we have lm x Ψ (x) = 0 for every 1. Moreover, sce K s assumed to be cotuous, Ψ s cotuous, ad ths esures the exstece of a radom varable s such that Ψ (s ) = sup Ψ (x). Lemma 7. Let Assumptos (U1) (U3), (U4) ) ( ad (U5) ) hold. Suppose ether (v ) 1or (H6) hold. For ay b>0, set w = exp b ; for ay δ>0, we have v 2 lm sup v 2 log P s w ad Ψ (s ) δ bβ (19) Proof of Lemma 7. We frst ote that s U ad therefore s w ad v Ψ (s ) δ s w ad v ( ) s X k K h + v E ( ) s X k K h δ s w ad v ( ) K s X k δ h v ( ) sup E K s X k x w, Set ρ 0,δ; the applcato of Lemma 6 esures that, for large eough, h s w ad v Ψ (s ) δ s w ad v ( ) s X k K δ ρ. h

17 Large ad moderate devato prcples 887 Set κ = sup x R x γ K (x) (see Assumpto (U5)). We obta, for suffcetly large, s w ad v Ψ (s ) δ ( ) s X k s w ad k {1,...,} such that v K δ ρ s w ad k {1,...,} such that κh γ v 1 s X k γ (δ ρ) κv h γ 1 γ s w ad k {1,...,} such that s X k δ ρ κv h γ 1 γ s w ad k {1,...,} such that X k s δ ρ s w ad k {1,...,} such that X k w (1 u,k ) wth ( ) u,k = w 1 v 1 1 γ κ γ h. δ ρ Moreover, we ca wrte u,k as ( u,k = exp b 1 v2 log v v 2 1 bγ v2 log (h ) h ) ( 1 κ ) 1 γ b δ ρ ad assumpto (U3) esure that lm u,k = 0, t the follows that 1 u,k > 0 for suffcetly large; therefore we ca deduce that (see Assumpto (U4) )): P s w ad v Ψ (s ) δ Cosequetly, =1 =1 P X k β w β (1 u,k) β E ( X k β) w β E ( X 1 β) w β (1 u,k) β max (1 u,k) β. 1 k v 2 log P s w ad v Ψ (s ) δ v2 log + log E ( X 1 β) bβv 2 β log max (1 u,k) 1 k ad, thaks to assumptos (U3), t follows that lm sup v 2 log P s w ad v Ψ (s ) δ bβ, whch cocludes the proof of Lemma 7.,

18 888 Yousr Slaou 3.4 Proof of Proposto 2 Let us at frst ote that the lower boud v 2 lm f log P sup v Ψ (x) δ g U (δ) (20) follows from the applcato of Proposto 1 at a pot x 0 U such that F (x 0 )= F U,. I the case U s bouded, Proposto 2 s thus a straghtforward cosequece of (20) ad the frst part of Lemma 5. Let us ow cosder the case U s ubouded. Set δ>0 ad, for ay b>0 set w = exp P sup P v Ψ (x) δ sup v Ψ (x) δ, x w t follows from Lemmas 5 ad 7 that v 2 lm sup log P sup v Ψ (x) δ ad cosequetly lm sup v 2 log P sup v Ψ (x) δ ( b v 2 ). Sce, by defto of s, + P s w ad v Ψ (x) δ, max { bβ; b g U (δ)} f max { bβ; b g U (δ)}. b>0 Sce the fmum the rght-had-sde of the prevous boud s acheved for b = g U (δ) / (β + 1) ad equals β g U (δ) / (β + 1), we obta the upper boud v 2 lm sup log P sup v Ψ (x) δ β β +1 g U (δ) whch cocludes the proof of Proposto Proof of Proposto 3 It follows from (1), that E F (x) = ( ) x y K f (y) dy R h = K (z) F (x + zh ) dz = R F (x)+ 1 2 h2 F (2) (x) z 2 K (z) dz + η (x) (21) R

19 Large ad moderate devato prcples 889 wth η (x) = R F (x + zh ) F (x) zh F (x) 1 2 z2 h 2 F (2) (x) K (z) dz Sce F s cotuous, we have lm F (x + zh ) F (x) zh F (x) 1 2 z2 h 2 F (2) (x) = 0, ad thus by the domated covergece theorem, we have lm η (x) = 0, ad thus Part 1 of Proposto 3 s completed. Sce sup x R F (2) (x) < +, Part 2 follows. Refereces A. Azzal, A ote o the estmato of a dstrbuto fucto ad quatles by a kerel method, Bometrka, 68 (1981), R. Bojac ad E. Seeta, A ufed theory of regularly varyg sequeces, Math. Z, 134 (1973), A. Dembo ad O. Zetou, Large devatos techques ad applcatos. Sprger, Applcato of mathematcs, New-York, 1998 M. Falk, Relatve effcecy ad defcecy of kerel type estmator of smooth dstrbuto fuctos, Stat. Neerl, 37 (1983), J. Galambos ad E. Seeta, Regularly varyg sequeces, Amer. Math. Soc, 41 (1973), F. Gao, Moderate devatos ad large devatos for kerel desty estmators, J. Theoret. Probab, 16 (2003), P.D. Hll, Kerel estmato of a dstrbuto fucto, Commu. Stat-Theor. M, 14 (1985), D. Loua, Large devatos lmt theorems for the kerel desty estmator, Scad. J. Statst, 25 (1998), A. Mokkadem & M. Pelleter ad J. Worms, Large ad moderate devatos prcples for kerel estmato of a multvarate desty ad ts partal dervatves, Austral. J. Statst, 4 (2005), A. Mokkadem, M. Pelleter ad B. Tham, Large ad moderate devatos prcples for recursve kerel estmato of a multvarate desty ad ts partal dervatves, Serdca Math. J, 32 (2006), E.A. Nadaraya, Some ew estmates for dstrbuto fuctos. Theory Prob. Appl, 9 (1964),

20 890 Yousr Slaou E. Parze, O estmate of a probablty desty ad mode, A. Math. Statst, 33 (1962), A.A. Puhalsk, The method of stochastc expoetals for large devatos, Stochastc Process. Appl, 54 (1994), R.D. Ress, Noparametrc estmato of smooth dstrbuto fucto, Scad. J. Statst, 8 (1981), M. Roseblatt, Remarks o some oparametrc estmate of a desty fucto, A. Math. Statst, 27 (1956), J. Worms, Moderate ad large devatos of some depedet varables, Part II: Some kerel estmators, Math. Methods Statst, 10 (2001), Receved: May 1, 2014