h(x i,x j ). (1.1) 1 i<j n

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1 ESAIM: PS DOI: /ps/ ESAIM: Probability ad Statistics CRAMÉR TYPE MODERATE DEVIATIONS FOR STUDENTIZED U-STATISTICS,, Tze Leg Lai 1, Qi-Ma Shao 2 ad Qiyig Wag 3 Abstract. Let T be a Studetized U-statistic. It is proved that a Cramér type moderate deviatio P T x/1 Φx 1 holds uiformly i x [0,o 1/6 whe the kerel satisfies some regular coditios. Mathematics Subject Classificatio. 60F10, 60F05. Received September 20, Revised Jue 2, Itroductio ad mai results Let X, X 1,X 2,...,X be a sequece of idepedet ad idetically distributed i.i.d. radom variables, ad let hx 1,x 2 be a real-valued symmetric Borel measurable fuctio. Assume that θ = EhX 1,X 2. A ubiased estimator of θ is the Hoeffdig [7] U-statistic U = 2 1 1i<j hx i,x j. 1.1 The U-statistic elegatly ad usefully geeralizes the otio of a sample mea. Typical examples iclude i sample mea: hx 1,x 2 = 1 2 x 1 + x 2 ; ii sample variace: hx 1,x 2 = 1 2 x 1 x 2 2 ; iii Gii s mea differece: hx 1,x 2 = x 1 x 2 ; iv oe-sample Wilcoxo s statistic: hx 1,x 2 =1x 1 + x 2 0. The o-degeerate U-statistic shares may limitig properties with the sample mea. For example, if Eh 2 X 1, X 2 < ad σ 2 1 =VargX 1 > 0, where Keywords ad phrases. Moderate deviatio, u-statistic, studetized. gx =Ehx, X, 1.2 Research partially supported by NSF-MCS Research partially supported by Hog Kog RGC CERG ad Research partially supported by a Australia Research Coucil ARC discovery project. 1 Departmet of Statistics, Staford Uiversity Staford, CA CA , USA; lait@staford.edu 2 Departmet of Mathematics, Hog Kog Uiversity of Sciece ad Techology, Clear Water Bay, Kowloo Hog Kog, P.R. Chia; maqmshao@ust.hk 3 School of Mathematics ad Statistics, Uiversity of Sydey, NSW 2006, Australia; qiyig@maths.usyd.edu.au Article published by EDP Scieces c EDP Scieces, SMAI 2011

2 CRAMÉR TYPE MODERATE DEVIATIONS FOR STUDENTIZED U-STATISTICS 169 the the cetral limit theorem holds, i.e., sup P U θ x Φx 0, 1.3 x 2σ 1 where Φx is the stadard ormal distributio fuctio. A systematic presetatio of the theory of U-statistics was give i [10]. We refer the study o uiform Berry-Essee boud for U-statistics to Alberik ad Betkus [1, 2], Wag ad Weber [17] ad the refereces there. Oe ca also refer to Borovskich ad Weber [4,5] forlarge deviatios. However, sice σ 1 is typically ukow, it is ecessary to estimate σ 1 first ad the substitute it i 1.3. Therefore, what used i practice is actually the followig studetized U-statistic see, e.g., Arvese [3] where R 2 = T = U θ/r, 1.4 qi U 2 with q i = 1 1 hx i,x j. 1.5 Oe ca refer to Wag, Jig ad Zhao [16] o uiform Berry-Essee boud for studetized U-statistics. Also see Callaert ad Veraverbeke [6] adzhao[18]. We also refer to Vademaele ad Veraverbeke [14] ad Wag [15] for the Cramér type moderate deviatio. A special case of the studetized U-statics is the Studet t-statistic with hx 1,x 2 =x 1 + x 2 /2. Although the t-statistic has a close relatioship with the classical stadardized partial sum, it has bee foud that the t-statistic ejoys much better limitig properties. For example, Shao [11] proves that the large deviatio always holds for t-statistic without ay momet assumptio ad Shao [12] further shows that a Cramér type moderate deviatio is valid uder oly a fiite third momet. Jig, et al. [8] provedacramér type moderate deviatio result for idepedet radom variables uder a Lideberg type coditio. Jig, et al. [9] obtaied the saddlepoit approximatio without ay momet coditio. Thus, it is atural to ask whether similar results hold for the studetized U-statistics. The mai objective of this paper is to show that the studetized U-statistics share similar properties like the studet t-statistic does whe the kerel satisfies h 2 x 1,x 2 θ c 0 [σ g 2 x 1 θ + g 2 x 2 θ] 1.6 for some c 0 > 0. This coditio is satisfied by the typical examples of U-statistics listed at the begiig of this sectio. Theorem 1.1. Assume 0 <σ1 2 < ad that 1.6 holds for some c 0 > 0. The, for ay x with x ad x = o 1/2, If i additio E gx 1 3 <, the j=1 j i l P T x x 2 / P T x =1 Φx [ 1+o1 ] 1.8 holds uiformly i x [0,o 1/6. Assume θ =0. WriteS = j=1 gx jadv 2 = j=1 g2 X j. It is kow see Shao [11] that l P S /V x x 2 /2 1.9 for ay x with x ad x = o 1/2. It is also kow see Jig et al. [8] that if E gx 1 3 <, the [ P S /V x =1 Φx 1+O11 + x 3 1/2] 1.10

3 170 T.L. LAI, Q.-M. SHAO AND Q. WANG for 0 x O 1/6. The followig theorem shows that the studetized U-statistic T ca be approximated by the self-ormalized sum S /V uder the coditio 1.6. As a result, 1.7 ad1.8 follow from 1.9 ad 1.10, together with 1.11 below, respectively. Theorem 1.2. Assume that θ =0, 0 <σ1 2 = Eg2 X 1 < ad the kerel hx 1,x 2 satisfies the coditio 1.6. The there exists a costat η>0 depedig oly o σ1 2 ad c 0 such that, for all 4/ 1 ɛ < 1, 0 x /3 ad sufficietly large, P [ S /V 1 + ɛ x ] e η miɛ2, ɛ x} P T x P [ S /V 1 ɛ x ] e η miɛ2, ɛ x} This paper is orgaized as follows. I the ext sectio we will prove the mai theorems. A techical propositio will be postpoed to Sectio 3. We start with some prelimiaries. Write where R 2 = q2 i. Observe that 2. Proofs of theorems T = U /R, 2.1 q i U 2 = qi 2 2U q i + U 2 = q 2 i U 2. We have T = 1/ T 2 2 ad T x} = T x } [1 + 4x 2 1/ 2 2 ] 1/2 2.3 We ow establish a relatioship betwee T ad S /V. Todothis,furtherletψx 1,x 2 =hx 1,x 2 gx 1 gx 2, Δ = 1 ψx i,x j, W i = ψx i,x j, Λ 2 = W i 2. 1 It is easy to see that Also observe that j=1 j i 1i<j T j=1 j i hx i,x j = 2gX i +S + W i U /2 = S +Δ. 2.4 ad R 2 = j=1 j i 2 hx i,x j = 2 2 V 2 +Λ2 +3 4S gx i W i +2S W i.

4 CRAMÉR TYPE MODERATE DEVIATIONS FOR STUDENTIZED U-STATISTICS 171 Therefore, usig gx iw i V Λ, S by the Hőlder s iequality, we have W i S Λ ad Λ 2 i max W 2 1i R 2 = 4 1 V δ, 2.5 where δ [ Λ 2 V 2 Λ 2 V 2 + 3S2 V 2 + 4Λ V + 2 Λ +2 S ] Λ V + 3 S2 V 2 V By ad2.1 T S +Δ =, 2.7 d V 1 + δ 1/2 where d = / 1. Next propositio shows that Δ ad δ are egligible. Propositio 2.1. There exist costats δ 0 > 0 ad δ 1 > 0, depedig oly o σ1 2 ad c 0, such that for all y>0 P δ y exp δ 0 mi1,y,y 2 } 2.8 ad P Δ yv 2 +2exp δ 1 mi, y }. 2.9 The proof of Propositio 2.1 is postpoed to Sectio 3. We metio that the proof is based o expoetial iequalities for self-ormalized sums of martigaledifferece sequece Lems. 3.1 ad 3.4 ad for self-ormalized sums of idepedet radom variables Lems. 3.2 ad 3.3. These iequalities are iterestig i their ow rights. We are ow ready to prove our mai results. Proof of Theorem 1.2. Sice x 2 /9 ad0 ɛ < 1, it is easy to show that, for 0 x /3, τ 1 ɛ 1/2τ 4 1 ɛ /2, [ wheever is sufficietly large, where τ = 1 Propositio 2.1 that 1+ 4x ] 1/2. Hece it follows from 2.3, 2.7 ad P T x P [ S /V 1 ɛ x ] + P Δ /V x ɛ 1 + xτ 1 + δ 1/2} } Δ /V xɛ 1 + xτ P [ S /V 1 ɛ x ] + P P [ S /V 1 ɛ x ] + P + P } Δ /V xɛ /2 + P δ ɛ /4 P [ S /V 1 ɛ x ] e η miɛ2, ɛ x}, } δ ɛ /4 }

5 172 T.L. LAI, Q.-M. SHAO AND Q. WANG where η>0 is a costat depedig oly o σ1 2 ad c 0. This proves the upper boud of Similarly, for the lower boud of 1.11 P S /V 1 + ɛ x P T x+p Δ /V x 1 + ɛ xτ 1 + δ 1/2} P T x+p Δ /V x 1 + ɛ xτ 1 + ɛ /4 1/2} + P δ ɛ /4 } P T x+p Δ /V x 1 + ɛ x 1 + 1/ ɛ /8 + P δ ɛ /4 P T x+p Δ /V xɛ /2 + P δ ɛ /4 P T x e η miɛ2, ɛ x}. The proof of Theorem 1.2 is ow complete. ProofofTheorem1.1. Theorem 1.1 follows from adtheorem1.2 by a suitable choice of ɛ, together with some routie calculatios. Ideed, by the cetral limit theorem for T see, e.g., Thm.3.1,[16], the result 1.8 isobviouswhe0 x 1. I order to prove 1.8 forx [1,o 1/6, we choose ɛ = maxɛ x/, 1/8 }, where ɛ is a sequece of costats satisfyig ɛ ad ɛ x3 / 0forx [1,o 1/6. It is readily to see that ad hece uiformly i x [1,o 1/6, miɛ 2, ɛ x} ɛ x maxɛ x 2, 3/8 x}, e η miɛ2, ɛ x} = o [ 1 Φx ], 2.10 whe is sufficietly large, where we have used a well-kow fact: for x>0, 1 1 2π x 1 x 3 e x2 /2 1 Φx 1 1 /2 2π x e x2. The meaig of 2.10 is that for all sequece u with u = o 1/6 lim sup x [1,u ] O the other had, it follows from 1.10 that wherewehaveusedtheresult: P [ S /V 1 ɛ x ] e η miɛ2, ɛ x} /1 Φx = 0. 1 Φ [ 1 ɛ x ]}[ 1+O1x 3 / ] [ 1 Φx ] 1+ Φ[ 1 ɛ x ] Φx }[ 1+O1x 3 / ] 1 Φx = [ 1 Φx ][ 1+o1 ], 2.11 Φ [ 1 ɛ x ] Φx ɛ x e 1 ɛ2 x 2 /2 = o [ 1 Φx ], uiformly i [1,o 1/6, sice ɛ x 2 ɛ x 3 / = o1. By virtue of ad the upper boud of 1.11, we obtai P T x [ 1 Φx ] [1 + o1]. Similarly we have P T x [ 1 Φx ] [1 + o1]. This proves 1.8.

6 CRAMÉR TYPE MODERATE DEVIATIONS FOR STUDENTIZED U-STATISTICS 173 I a similar matter, by choosig ɛ =max 1/8,ɛ } where ɛ is a sequece of costats such that ɛ 0 so slowly that ɛ 2 /x2,wehave ad therefore which together with 1.11 ad1.9 proves1.7. The proof of Theorem 1.2 is ow complete. = o1e 0.5η miɛ2, ɛ x, x 2 = omiɛ 2, ɛ x e η miɛ2, ɛ x = o1e x2 /4, 3. Proof of propositio 2.1 I this sectio, we give the proof of Propositio 2.1. Lemma 3.1 is iterestig i itself as it provides a expoetial boud for martigale differece uder fiite momet coditios. Lemma 3.1. Let ξ i, F i,i 1} be a sequece of martigale differece with Eξi 2 < ad put d2 i = Eξ2 i F i 1. The P ξ i x 2exp x 2 /8 3.1 ξ2 i +2d2 i +3Eξ2 i 1/2 for all x>0. Proof. We first show that e x x2 1+x1 x 1/2} 3.2 for all x R. It is easy to see that 3.2 holdsforx< 1/2. For x 1/2 letfx =x x 2 l1 + x. Observe that f x = 1 2x 1 x1 + 2x = 1+x 1+x > 0 for 1/2 <x<0, =0 forx =0, < 0 for x>0. Therefore f achieves maximum at x =0,thatis,fx f0 = 0 for x> 1/2. This proves 3.2. It follows from 3.2 thatfort R E exptξ i t 2 ξi 2 +2d 2 i F i 1 = e 2t2 d 2 i E exptξ i t 2 ξi 2 F i 1 e 2t2 d 2 i 1+Etξ i 1 tξi 1/2} F i 1 = e 2t2 d 2 i 1 Etξ i 1 tξi< 1/2} F i 1 e 2t2 d 2 i 1+2Etξ i 2 F i 1 1. This shows that exp t i j=1 ξ j t 2 } i j=1 ξ2 j +2d2 j, F i,i 1 is a super-martigale ad hece E exp t ξ j t 2 j=1 j=1 ξj 2 +2d 2 j

7 174 T.L. LAI, Q.-M. SHAO AND Q. WANG By 3.3 adtheorem2.1of, E exp a ξ i 2 ξ2 i +2d2 i +3Eξ2 i 1/2 for all a>0. Lettig a = x/2 2 together with Markov s iequality yields P ξ i x ξ2 i +2d2 i +3Eξ2 i 1/2 2expa e ax/ 2 a E exp ξ i 2 ξ2 i +2d2 i +3Eξ2 i 1/2 2exp ax/ 2+a 2 = 2exp x 2 /8. This proves 3.1. Lemma 3.2. Let ξ i,i 1} be idepedet radom variables with zero meas ad fiite variaces. Put S = ξ i, V 2 = ξi 2, B 2 = Eξi 2. The P S xv 2 +5B2 1/2 2exp x 2 /8 for x>0 3.5 ad ES 2 I S xv +4B 23B 2 e x2 /4 3.6 Proof. Result 3.5 follows from 3.1 directly because Eξi 2 F i 1 =Eξi 2 by idepedece of radom variables. Whe 0 <x<3, we have 23e x2 /4 1adES 2I S xv +4B ES 2 = B2 ad hece 3.6 holds. Whe x>3, let η i, 1 i } be a idepedet copy of ξ i, 1 i }. Set By the Chebyshev iequality, Notig that P S 2B, V 2 S = η i,v 2 = ηi 2. 4B2 1 P S > 2B P V 2 > 4B2 1 1/4 1/4 =1/2. S x4b + V, S 2B,V 2 4B2 } S S x4b + ξ i η i 2 1/2 V S S x2b + ξ i η i 2 1/2 2B, S 2B } 1/2 S S x ξ i η i 2, S 2B }, 2B, S 2B,V 2 4B 2 }

8 CRAMÉR TYPE MODERATE DEVIATIONS FOR STUDENTIZED U-STATISTICS 175 we have ESI S 2 xv +4B = ES2 I S xv +4B I S 2B,V 2 4B2 P S 2B,V 2 4B2 1/2 2ESI 2 S S x ξ i η i 2, S 2B 1/2 4ES S 2 I S S x ξ i η i 2, S 2B +4ES 2 I 1/2 S S x ξ i η i 2, S 2B [by the fact that S 2 2S S 2 +2S 2 1/2 4ES S 2 I S S x ξ i η i 2 1/2 +16BP 2 S S x ξ i η i ] Let ε i, 1 i } be a Rademacher sequece idepedet of ξ i, 1 i } ad η i, 1 i }. Notig that ξ i η i, 1 i } is a sequece of idepedet symmetric radom variables, ε i ξ i η i, 1 i } ad ξ i η i, 1 i } have the same joit distributio. It is kow that P a i ε i x a 2 i 1/2 2e x2 /2 3.8 for ay real umbers a i }. Hece with Y = a iε i / 1/2 a2 i 1 2 a 2 i E a i ε i I a i ε i x a 2 i 1/2 = EY 2 IY x = x 2 P Y x+2 2x 2 e x2 /2 +4 x x tp Y tdt te t2 /2 dt = 22+x 2 e x2 /2 2.4e x2 /4 3.9 for x>3. Thus by 3.8 ad3.9 forx>3 1/2 P S S x ξ i η i 2 2e x2 /2 0.22e x2 /4 3.10

9 176 T.L. LAI, Q.-M. SHAO AND Q. WANG ad 2 ES S 2 I S S x ξ i η i 1/2I S 2 2B =E ε i ξ i η i I ε i ξ i η i x ξ i η i 2 1/2 2.4e x2 /4 E ξ i η i 2 = 4.8Be 2 x2 / This proves 3.6 by3.7, 3.10 ad3.11. I the followig two lemmas we cotiue to use the otatios give i Sectio 2. Lemma 3.3. Assume σ1 2 =1. The for all y>0, P S yv + 5 2e y2 / ad where η 0 =1/32a 2 0 ad a 0 satisfied P V 2 /2 e η EgX 1 2 I gx 1 a 0 }1/ Proof. Recall EgX 1 =0adEgX 1 2 = is a special case of 3.5. We ext prove Let Y k = gx k I gxk a 0. Sicee x 1 x + x 2 /2forx>0, we have with t =1/4a 2 0 P V 2 /2 P k=1 Y 2 k /2 e t/2 Ee t k=1 Y 2 k =e t/2 Ee ty 2 1 e t/2 1 tey1 2 + t 2 EY1 4 /2 e t/2 1 3/4t + t 2 a 2 0/2 exp t/4 t 2 a 2 0 /2 =exp 32a 2 0, as desired. Lemma 3.4. Assume σ1 2 =1.The,forally 0, P [ Λ 2 a 0 y 2 7 V ] 2 e y2 / where a 0 =2c 0 +4,ad P where a 2 1 = 46c i<j ψx i,x j a 1 y 2 V /2 2 +2e y2 /8, 3.16

10 CRAMÉR TYPE MODERATE DEVIATIONS FOR STUDENTIZED U-STATISTICS 177 Proof. First prove Note that, give X i, W i follows from 3.5 that P W i is a sum of i.i.d. radom variables with zero meas. It y [ V i2 +5 1τ 2 X i ] } 1/2 2e y2 / where V i2 = j=1 ψ 2 X i,x j adτ 2 x =Eψ 2 X 1,X j X j = x. Note that ψ 2 x 1,x 2 2c 0 +4[1+ j i g 2 x 1 +g 2 x 2 ]. We have V i2 +5 1τ 2 X i 2c 0 +4 This, together with 3.17 ad the fact that yields that [ 11 +6g 2 X i + [ 11 +6g 2 X i + ] g 2 X i. ] g 2 X i = 7 V 2 +11, [ ] P Λ 2 a 0 y 2 7 V P W i 2 e y2 /8, y [ V i2 +5 1τ 2 X i ] } 1/2 as required. We ext prove Let F j = σx i,i j adrewrite 1i<j ψx i,x j = Y j, where Y j = j 1 ψx i,x j. The Y j, F j,j 2} is a martigale differece sequece. By 3.1, we have P Y j y [ Y 2 j +3EYj 2 +2EY j 2 F j 1 ] 1/2 2e y 2 / Note that EYj 2 prove j 1Eh 2 X 1,X 2 3j 1 by 1.6 adeg 2 X 1 =1. Theresult3.16 follows if we ] P [T1 2 a 2 y 2 V e y2 /8, 3.19 where T1 2 = Y j 2 ad a 2 = 14c 0 +4,ad where T 2 2 = EY 2 j F j 1 ada 3 = 16c ] P [T2 2 a 3 y 2 V e y2 /4, 3.20

11 178 T.L. LAI, Q.-M. SHAO AND Q. WANG We oly prove The proof of 3.19 is similar to We omit the details. Without loss of geerality, assume y 1. Otherwise 3.20 isobvious. WriteV j = V ψ,j +4j 1 1/2 τx j, where Vψ,j 2 = j 1 ψ2 X i,x j. Observe that P T2 2 2y 2[ 4 y 2[ 4 y 2[ 4 τ 2 X j Eτ 2 X 1 ]} P E [ Yj 2 I Y j yv j F ] j 1 τ 2 X j Eτ 2 X 1 ]} + P E [ Yj 2 I Y j >yv j F ] j 1 τ 2 X j Eτ 2 X 1 ]} := J 1 + J Note that J 1 P y 2 E [ V j 2 ] F j 1 y 2 4 τ 2 X j Eτ 2 X 1 j 1 = P 2τ 2 X i +32 j 1Eτ 2 X 1 4 τ 2 X j Eτ 2 X 1 = ad that recall y 1 J 2 = 1 64y 2 2 Eτ 2 X y 2 2 Eτ 2 X y 2 2 Eτ 2 X 1 E [ Y 2 j I Y j >yv j ] E E [ Yj 2 I Y j >yv j ] } X j E [ jτ 2 X 1 ] e y2 /4 by 3.6 e y2 / The result 3.20 ow follows from ad the fact that 4 τ 2 X j Eτ 2 X 1 8c V 2, as τ 2 x 2c 0 +4[2+gx]. This also completes the proof of Lemma 3.4. We are ow ready to prove Propositio 2.1. Without loss of geerality, assume σ1 2 =1. Otherwise,cosider h/σ 1 i the place of h. Weolyprove2.8. The proof of 2.9 is give i a similar maer except we use 3.16 i the place of 3.15.

12 CRAMÉR TYPE MODERATE DEVIATIONS FOR STUDENTIZED U-STATISTICS 179 By 3.12 ad3.13, for ay x>0 P S 5xV P V 2 /2 + P [ S xv + 5 ] 2e x2 /8 +e η0. By 3.15 ad3.13, for ay x>0, P Λ 7a 0 +22x V [ ] P V 2 /2 + P Λ 2 a 0 x 2 7 V These facts imply that, for ay y>0, 2 e x2 /8 +e η0. P δ y 2 P S y 2 V /3 + 2P Λ y 2 V /4 + P Λ y 2 V / e δ 0 y +2 2e δ 0 y2 +5e η exp δ 0 mi1,y,y 2 }, where δ 0, δ 0 ad δ 0 are costats depedig oly o σ 2 1 ad c 0. This proves 2.8 ad hece completes the proofofpropositio2.1. Ackowledgemets. The authors thak the referee for his/her valuable commets. Refereces [1] I.B. Alberik ad V. Betkus, Berry-Essee bouds for vo-mises ad U-statistics. Lith. Math. J [2] I.B. Alberik ad V. Betkus, Lyapuov type bouds for U-statistics. Theory Probab. Appl [3] J.N. Arvese, Jackkifig U-statistics. A. Math. Statist [4] Y.V. Borovskikh ad N.C. Weber, Large deviatios of U-statistics I. Lietuvos Matematikos Rikiys [5] Y.V. Borovskikh ad N.C. Weber, Large deviatios of U-statistics I. Lietuvos Matematikos Rikiys [6] H. Callaert ad N. Veraverbeke, The order of the ormal approximatio for a studetized U-statistics. A. Statist [7] W. Hoeffdig, A class of statistics with asymptotically ormal distributio. A. Math. Statist [8] B.-Y. Jig, Q.M. Shao ad Q. Wag, Self-ormalized Cramér-type large deviatio for idepedet radom variables. A. Probab [9] B.-Y. Jig, Q.M. Shao, W. Zhou, Saddlepoit approximatio for Studet s t-statistic with o momet coditios. A. Statist [10] V.S. Koroljuk ad V. Yu. Borovskich, Theory of U-statistics. Kluwer Academic Publishers, Dordrecht [11] Q.M. Shao, Self-ormalized large deviatios. A. Probab [12] Q.M. Shao, Cramér-type large deviatio for Studet s t statistic. J. Theorect. Probab [13] V.H. De La Pea, M.J. Klass ad T.L. Lai, Self-ormalized processes: expoetial iequalities, momet boud ad iterated logarithm laws. A. Probab [14] M. Vardemaele ad N. Veraverbeke, Cramer type large deviatios for studetized U-statistics. Metrika [15] Q. Wag, Berstei type iequalities for degeerate U-statistics with applicatios. A. Math. Ser. B [16] Q. Wag, B.-Y. Jig ad L. Zhao, The Berry-Essée boud for studetized statistics. A. Probab [17] Q. Wag ad N.C. Weber, Exact covergece rate ad leadig term i the cetral limit theorem for U-statistics. Statist. Siica [18] L. Zhao, The rate of the ormal approximatio for a studetized U-statistic. Sciece Exploratio