A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS FOR A GENERAL CLASS OF STATISTICS

Statistica Siica 20 2010, 613-636 A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS FOR A GENERAL CLASS OF STATISTICS Big-Yi Jig ad Qiyig Wag Hog Kog Uiversity of Sciece ad Techology ad Uiversity of Sydey Abstract: This paper is cocered with Edgeworth expasios for a geeral class of statistics uder very weak coditios. Our approach uifies the treatmet for both stadardized ad studetized statistics that have bee traditioally studied separately uder usually differet coditios. These results are the applied to several special classes of well-kow statistics: U-statistics, L-statistics, ad fuctios of sample meas. Special attetio is paid to the studetized statistics. We establish Edgeworth expasios uder very weak or miimal momet coditios. Key words ad phrases: Edgeworth expasio, fuctio of sample meas, L-statistic, stadardizatio, studetizatio, U-statistic. 1. Itroductio Suppose we are iterested i the distributio of a statistic, T = T X 1,..., X, where X 1,..., X is a sequece of idepedet ad idetically distributed i.i.d. radom variables r.v. s. Typically, oe ca use the delta method to show that T coverges i distributio to a ormal distributio. The rates of covergece to ormality are usually of the order 1/2 ad ca be described by Berry-Essée bouds. To get a better approximatio tha asymptotic ormality, oe ca develop higher-order Edgeworth expasios uder appropriate coditios. The theory of Edgeworth expasios dates back a log way, the simplest case beig the Edgeworth expasio for the sample mea. Much effort has bee devoted to Edgeworth expasios for other classes of statistics, such as fuctios of meas, U-, L-statistics, ad others. O the other had, Edgeworth expasios for their studetized couterparts have also gaied much mometum, partly due to their usefuless i statistical iferece. It is worth poitig out that each of the methods for derivig Edgeworth expasios for the above-metioed statistics was tailored to the idividual structures of these statistics. A geeral uifyig approach is to cosider symmetric statistics, which iclude all the aforemetioed, as i Lai ad Wag 1993, Betkus, Götze ad va Zwet 1997, ad Putter ad va Zwet 1998 for istace.

614 BING-YI JING AND QIYING WANG A quick glace at the literature reveals that the momet coditios i Edgeworth expasios for the studetized statistics are typically stroger tha for the correspodig stadardized statistics; see Sectio 3 for more discussios. Oe otable exceptio is the case of the sample mea, where the third momet is eough for both the stadardized mea ad Studet s t-statistic, see Hall 1987 ad Betkus ad Götze 1996, for istace. This begs the questio whether the same pheomeo is also true for U-, L-statistics, ad other classes of statistics. I this paper we address this issue. We cosider Edgeworth expasios for a very geeral class of statistics, ad the apply them to some special cases of iterest. I particular, we cosider statistics of the form T /S, where T = 1/2 αx j + 3/2 S 2 1 = 1 + γx i, X j + V 2, 1 βx i, X j + V 1, with V 1 ad V 2 as remaider terms. Oe ca thik of T as the statistic of iterest ad S 2 as the ormalizig variable. We refer to T /S as the studetized statistic whe S is radom, ad as the stadardized statistic whe S = 1. There are several reasos to cosider such a class of statistics. First, a great may of commoly-see statistics belog to this class. These iclude stadardized or studetized U-, L-statistics, ad fuctio of sample meas. Secod, the approach take i this paper uifies the treatmet for both stadardized ad studetized statistics. For example, if γx, y = 0 ad V 2 is sufficietly small i the ormalizig variable S 2, the the studetized statistic T /S will reduce to the stadardized statistic T. Therefore, it is possible to derive asymptotic results for both cases uder the same set of coditios. Fially, the coditios required for our mai results are very weak, ad ofte miimal; see Sectio 3. Sectio 2 gives the mai results of the paper. They will be applied i Sectio 3 to several well-kow examples. Proofs ad techical details are give i Sectios 4 ad 5. Throughout, we use C to deote some positive costats, idepedet of, which may be differet at each occurrece. For a set B, let I B deote a idicator fuctio of B. The stadard ormal desity ad distributio fuctio are deoted by φx ad Φx, respectively. Fially, write, 1, <k 1 <k i j, i, i j i j k i,j,k=1 i j,j k,k i.

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 615 2. Mai Results Let X, X 1,..., X be a sequece of i.i.d. r.v.s. Let αx, βx, y, γx, y be some Borel measurable fuctios of x, y, ad z. Let V i V i X 1,..., X i = 1, 2 be fuctios of {X 1,..., X }. Let T = 1/2 βx i, X j + V 1, 2.1 αx j + 3/2 S 2 1 = 1 + γx i, X j + V 2. 2.2 1 The domiat term i T /S is 1/2 αx j, which coverges i distributio to a ormal distributio as uder weak coditios. We first give a Edgeworth expasio for T /S with remaider o 1/2 uder very weak coditios. Theorem 2.1. Assume the followig. a αx 1 is olattice; βx, y ad γx, y are symmetric i x, y. b EαX 1 = 0, Eα 2 X 1 = 1, E αx 1 3 <, E[βX 1, X 2 X 1 ] = 0, E βx 1, X 2 5/3 <, EγX 1, X 2 = 0, E γx 1, X 2 3/2 <. c P V j δ 1/2 = o 1/2, j = 1, 2, where δ > 0 ad δ 0. The we have sup x P T /S x E x = o 1/2, where E x = Φx Φ3 x 6 Eα 3 X 1 + 3EαX 1 αx 2 βx 1, X 2 xφ2 x 2 EαX 1γX 1, X 2. The ext corollary may be easier to use i some applicatios. Corollary 2.1. Assume the coditios of Theorem 2.1, except with S 2 replaced by S 2 = 1 + 3 i j k ηx i, X j, X k + V 2, where EηX 1, X 2, X 3 = 0, E ηx 1, X 2, X 3 3/2 <. The sup x P T /S E x = o 1/2, where E x = Φx Φ3 x 6 xφ2 x 2 Eα 3 X 1 + 3EαX 1 αx 2 βx 1, X 2 3 EαX i ηx 1, X 2, X 3. i=1

616 BING-YI JING AND QIYING WANG Remark 2.1. If γx, y = 0 i Theorem 2.1, we get sup x P T x E 0 x = o 1/2, where E 0 x = Φx 6 1 1/2 Φ 3 xeα 3 X 1 + 3EαX 1 αx 2 βx 1, X 2. The Edgeworth expasios for both types of statistics thus hold uder the same coditios if oe ca show that coditios for γx 1, X 2 i T /S are implied by those imposed o αx 1 ad βx 1, X 2. Such examples ca be see i Sectios 3.2 ad 3.3. Remark 2.2. The uderlyig distributio F of X i s is typically ukow i practice, so oe caot directly apply Theorem 2.1 ad Corollary 2.1 sice they ivolve ukow quatities such as Eα 3 X 1 ad EαX 1 αx 2 βx 1, X 2, etc. I such situatios, oe could use empirical Edgeworth expasios EEE Êx obtaied by estimatig the ukow quatities i E x by their empirical versios e.g., the jackkife. Remark 2.3. Sigh 1981 used a secod-order Edgeworth expasio for the mea of a sample from a olattice distributio to show, for the first time, that the bootstrap distributio approximates the true distributio of the stadardized statistic better tha the ormal approximatio uder a fiite third momet. This classical result has bee exteded by Blozelis ad Putter 2003 to Studet s t-statistic uder the same optimal coditios. It would be of great iterest to fid out whether similar results hold uder the coditios for the more geeral statistics cosidered i this paper. We hope to be able to report o this. 3. Some Importat Applicatios I this sectio, we apply the mai results i Sectio 2 to three well-kow examples: U- ad L-statistics ad fuctios of the sample mea, which results i secod-order Edgeworth expasios uder very weak, ofte optimal coditios. 3.1. U-statistics Let hx, y be a real-valued Borel measurable fuctio, symmetric i its argumets, with EhX 1, X 2 = θ. Defie a U-statistic of degree 2 with kerel hx, y by U = 2 1 hx i, X j. Let gx j = E hx i, X j θ X j, σ 2 g = V ar gx 1, ad R 2 = 4 1 2 2 i=1 1 1 2 hx i, X j U. j i

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 617 Note that 2 R 2 is the jackkife estimator of 4σg. 2 Let the distributios of the stadardized ad studetized U-statistics be, respectively, U θ U θ G 1 x = P x, ad G 2 x=p x. 2σ g R Asymptotic ormality of G 1 x ad G 2 x was established by Hoeffdig 1948 ad Arvese 1969, respectively, provided that Eh 2 X 1, X 2 < ad σ 2 g > 0. Berry-Essée bouds for G 1 x were studied by may authors, see Betkus, Götze ad Zitikis 1994 ad refereces therei. Also see Wag ad Weber 2006 for exact covergece rates ad leadig terms i the Cetral Limit Theorem. Berry-Essée bouds for G 2 x have bee give by may authors; see Wag, Jig ad Zhao 2000 for refereces. O the other had, Edgeworth expasios for U-statistics have also bee itesively studied i recet years. For G 1 x, uder the coditios that σ 2 g > 0, the d.f. of gx 1 is olattice, E gx 1 3 <, ad E hx 1, X 2 2+ɛ < for some ɛ > 0, Bickel Götze ad va Zwet 1986 showed that sup G 1 x E 0 x = o 1/2, 3.1 x where E 0 x = Φx φx 6 σg 3 x 2 1 { Eg 3 X 1 + 3EgX 1 gx 2 hx 1, X 2 }. Jig ad Wag 2003 weakeed the momet coditio E hx 1, X 2 2+ɛ < to E hx 1, X 2 5/3 <. O the other had, uder the coditios that σ 2 g > 0, the d.f. of gx 1 is olattice, ad E hx 1, X 2 4+ɛ < for some ɛ > 0, Helmers 1991 showed that where sup G 2 x E x = o 1/2, 3.2 x E x = Φx+ φx 6 { 2x 2 σg 3 +1Eg 3 X 1 +3x 2 +1EgX 1 gx 2 hx 1, X 2 }. Putter ad va Zwet 1998 weakeed the 4+ɛ-th momet coditio of Helmers 1991 to E hx 1, X 2 3+ɛ <. The followig result ca be obtaied from Corollary 2.1. Theorem 3.1. Suppose that the d.f. of gx 1 is olattice ad σ 2 g > 0. a If E gx 3 < ad E hx 1, X 2 5/3 <, the 3.1 holds.

618 BING-YI JING AND QIYING WANG b If E hx 1, X 2 3 <, the 3.2 holds. Remark 3.1. The momet coditios E gx 1 3 < ad E hx 1, X 2 5/3 < are optimal for Edgeworth expasio of stadardized U-statistics; see Jig ad Wag 2003. The coditios for Edgeworth expasio of studetized U-statistics are suboptimal i the followig sese: the coditio E hx 1, X 2 3 < caot be weakeed to E hx 1, X 2 3 δ < for ay δ > 0, but it might be weakeed to E gx 1 3 <, E hx 1, X 2 5/3 <, ad E hx 1, X 2 hx 1, X 3 3/2 <. Remark 3.2. If hx, y = x + y/2, the U θ/r reduces to Studet s t-statistic. From Theorem 3.1, the momet coditio for Edgeworth expasios of error size o 1/2 for Studet s t-statistic is E X 1 3 <, which is optimal; see Hall 1987. Proof of Theorem 3.1. Part a ca be proved by applyig Theorem 2.1 directly. We prove part b. Similar to A 3 i Callaert ad Veraverbeke 1981 also see Serflig 1980, we may write U θ R = U θ/2σ g R /2σ g where T ad S 2 are defied as i Corollary 2.1, with αx j = σ 1 g gx j, βx i, X j = σ 1 g [hx i, X j θ gx i gx j ], ηx i, X j, X k = σ 2 g [hx i, X j θ] [hx i, X k θ] 1, V 1 = 3/2 1 1 βx i, X j, 2σ 2 g V 2 = 1 2 2 hx i, X j θ 2 + 22 + 4 1 3 1 2 2 i j k T S, 1σ 2 g 2 2 U θ 2 ηx i, X j, X k + 2 2. By the properties of coditioal expectatio, it ca easily show that EαX i αx 2 βx i, X j = σg 3 EgX 1 gx 2 hx 1, X 2, if 1 i j 2, EαX i ηx 1, X 2, X 3 = σg 3 Eg 3 X 1, if i = 1, = σg 3 EgX 1 gx 2 hx 1, X 2, if i = 2, 3. I view of these estimates ad the relatios Φ 2 x = xφx ad Φ 3 x = x 2 1φx, oe ca apply Corollary 2.1 to obtai E x = Φx+ φx 6 { 2x 2 σg 3 +1Eg 3 X 1 +3x 2 + 1EgX 1 gx 2 hx 1, X 2 }.

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 619 O the other had, coditio a of Corollary 2.1 ca be easily checked. The Theorem 3.1 follows from Corollary 2.1 if we ca show P V j 1/2 log 1 = o 1/2 for j = 1, 2, but this follows from Lemma 4.1 below. 3.2. L-statistics Let X 1,..., X be i.i.d. r.v.s with distributio fuctio F. Deote the empirical distributio by F x = 1 I X i x. Let Jt be a real-valued fuctio o [0, 1] ad T G = xjgx dgx. The T F is called a L-statistic; see Serflig 1980. Write s t = mi{s, t}, s t = max{s, t}, ad σ 2 σ 2 J, F = J F s J F t F s t [1 F s t] dsdt, A atural estimate of σ 2 is σ 2 σ 2 J, F. Let the distributios of the stadardized ad studetized L-statistic T F be, respectively, T F T F L 1 x = P x, ad σ T F T F L 2 x = P x. σ Asymptotic ormality of L 1 x ad L 2 x holds if E X 1 2 <, σ 2 > 0, ad some smoothess coditios o Jt; see Serflig 1980. Berry-Essée bouds were give for them by Helmers 1977, va Zwet 1984, Helmers, Jasse ad Serflig 1990, Helmers 1982, ad Wag, Jig ad Zhao 2000. Also see Wag ad Weber for exact covergece rates ad leadig terms i the Cetral Limit Theorem. O the other had, Edgeworth expasios for L 1 x were give by Helmers 1982, Lai ad Wag 1993, amog others. Edgeworth expasios for L 2 x were studied by Putter ad va Zwet 1998 uder some smoothess coditios o Jt ad the momet coditio E X 1 3+ɛ < for some ɛ > 0. The ext theorem shows that this momet coditio ca be weakeed. Theorem 3.2. Assume the followig. a J t is bouded o t [0, 1]. b σ 2 > 0 ad [F t1 F t] 1/3 dt <. c the d.f. of Y = JF s I X1 s F s ds is olattice. The we have sup L 1 x Ẽ0x = o 1/2 ad sup L 2 x Ẽx = o 1/2, 3.3 x x

620 BING-YI JING AND QIYING WANG where Ẽ0x = Φx + 6 1 σ 3 1/2 3a 1 Φ 1 x + a 3 Φ 3 x ad 1 Ẽ x = Φx + 6σ 3 3a 1 Φ 1 x + 3a 2 xφ 2 x + a 3 Φ 3 x. Here, J 0 t = JF t, a 1 = σ 2 J 0xF x[1 F x]dx, a 2 = J 0 yj 0 z [ J 0 xk 1 x, y, z + J 0xK 2 x, y, z ] dxdydz, a 3 = J 0 xj 0 y [ J 0 zk 3 x, y, z + 3J 0zK 4 x, zk 4 y, z ] dxdydz, K 1 x, y, z = [F x y z F x yf z] [1 F x y] +F x yf x y [F z 1], K 2 x, y, z = F x y [1 F x y] [F y z F yf z], K 3 x, y, z = F x y z F xf y z F yf x z F zf x y + 2F xf yf z, K 4 x, y = F x y F xf y. Remark 3.3. Note that [F t1 F t] 1/3 dt < is weaker tha E X 1 3+ɛ < for every ɛ > 0. To see this, first apply Markov s iequality to get F t{1 F t} E X 1 3+ɛ / t 3+ɛ. The, {F t[1 F t]} 1/3 dt 1dt + {F t[1 F t]} 1/3 dt t 1 t >1 2 + E X 1 3+ɛ 1/3 t 1 ɛ/3 dt <. Proof of Theorem 3.2. We oly prove the secod relatio i 3.3, the first ca be doe similarly. Write t >1 J t = JF t, Zs, t, F = F s t1 F s t, ξx i, X j = σ 2 J 0 sj 0 t I Xi s ti Xj >s t Zs, t, F dsdt, ϕx i, X j, X k = σ 2 J 0sJ 0 t [ I Xi t F t ] I Xj s ti Xk >s t dsdt. From Lemma B of Serflig 1980, p.265, we have T F T F = [K 1 F x K 1 F x]dx, where K 1 t = t 0 Judu. We ow ca write T F T F T F T F /σ = T + 1/2 EβX 1, X 1, σ σ/σ S

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 621 where T ad S 2 are defied as i Corollary 2.1, with [ ] αx j = σ 1 J 0 t I Xj t F t dt, βx i, X j = σ 1 J 0t [ I Xi t F t ] [ ] I Xj t F t dt, ηx i, X j, X k = ξx i, X j + ϕx i, X j, X k, V 1 1 3/2 [βx j, X j EβX 1, X 1 ] +C 1/2 V 2 = Q 1 + Q 2 + Q 3, F t F t 3 dt, where the Q i s are Q 1 = 2σ 2 [J s J 0 s J 0sF s F s ] J 0 tzs, t, F dsdt, Q 2 = σ 2 [J s J 0 s] [J t J 0 t] Zs, t, F dsdt, Q 3 = 3 [ξx j, X k + ϕx j, X j, X k + ϕx k, X j, X k ] j k 1 σ 2 F s t [1 F s t] dsdt. Coditios a ad b i Corollary 2.1 ca be easily checked. Let us check coditio c. It suffices to show that P V 1 1/2 log 1 = o 1/2, P Q j 1/2 log 1 = o 1/2, for j = 1, 2, 3, 3.4 For illustratio, we oly prove 3.4 for j = 1. Others ca be show similarly. Notig that Zs, t, F [F s1 F s] 1/2 [F t1 F t] 1/2, we have Q 1 σ 2 sup J 0 xj 0 y F s F s 2 Zs, t, F dsdt x,y Cσ 2 F s F s 2 ds F 1/2 t1 F t 1/2 dt =: Cσ 2 Q 6 Q 7, say. 3.5 Usig the iequality E F t F t k C k/2 F t1 F t, we get EQ 3 6 = E { F s F s 2 F t F t 2 F v F v 2} dsdtdv

622 BING-YI JING AND QIYING WANG E F s F s 6 3 1/3 ds C 3 F 1/3 s1 F s 1/3 ds 3. Similarly, we ca show EQ 3 7 F 1/3 s1 F s 1/3 ds 3. Therefore, P Q 1 1/2 log 1 3/2 1/2 log E Q1 3/2 3/2 EQ 3 1/2 6 EQ 3 1/2 7 C 3/2 σ 3 1/2 log C 3/4 log 3/2 = o 1/2. 3 [F t1 F t] 1/3 dt Fially, we ca apply Corollary 2.1 to get sup x P T /S x E x =o 1/2, From this, ad usig the similar method as i proof of Theorem 2.1, we ca get sup x P T + 1/2 EβX 1, X 1 x 1/2 φxeβx 1, X 1 E x S = o 1/2. This reduces to 3.3 after some tedious but routie calculatios. 3.3. Fuctios of the sample mea Let X 1,..., X be i.i.d. r.v.s with EX 1 = µ ad V arx 1 = σ 2 < ; let f be differetiable i a eighborhood of µ with f µ 0. The asymptotic variace of fx is σf 2 = f µ 2 σ 2. Write X = 1 i=1 X i ad σ 2 = 1 i=1 X i X 2. A simple estimator of σf 2 is f X 2 σ, ad the jackkife variace estimator of σf 2 is σ 2 f = 1 fx j 2 j 1 fx, where X = X i X j. 1 i=1 Write the distributios of the stadardized ad studetized fx, respectively, as fx fµ fx fµ H 1 x=p x, H 2 x=p x. σ f σ f Asymptotic properties of H 1 x have bee well studied see Bhattacharya ad Ghosh 1978 for istace. O the other had, Miller 1964 showed that σ 2 f is a cosistet estimator of σ2 f ad hece proved that H 2x is asymptotically

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 623 ormal. Applyig Bai ad Rao 1991, Edgeworth expasios for H 1 x with error size o 1/2 hold uder the miimal momet coditio E X 1 3 <. Putter ad va Zwet 1998 gave Edgeworth expasios for H 2 x uder E X 1 3+ɛ < for some ɛ > 0. The ext theorem gives the optimal momet coditio. Theorem 3.3. Assume that f 3 x is bouded i a eighborhood of µ ad f µ 0, E X 1 3 <, ad the d.f. of X 1 is olattice. The we have sup H 1 x Ẽ0x = o 1/2, sup H 2 x Ẽx = o 1/2, 3.6 x x where ρ = EX 1 µ 3 /σ 3, b = 2 1 σf u/f u, ad Ẽ 0 x = Φx + φx 6 1 x 2 ρ + 62 x 2 b, Ẽ x = Φx + φx 6 2x 2 + 1ρ + 6b. Proof. We oly prove the secod relatio i 3.6, the first ca be doe similarly. Applyig Taylor s expasio to fx fµ ad σ f 2, we get fx fµ σ f = T + 1/2 b S, where T ad S 2 are defied i 2.1 ad 2.2, with αx j = X j µ, βx i, X j = 2bαX i αx j, σ γx i, X j = α 2 X i + α 2 X j 2 + 2bσ 1 αx i αx j [αx i + αx j ], V 1 C Xj 3/2 µ 2 σ 2 + C X µ 3, V 2 C 4 X µ k + C Xj 2 µ 2 + X j µ 3 + C 3 X j µ 4 k=2 + C X µ2 X j µ 2 + C 3 γx i, X j. Coditios a ad b i Theorem 2.1 ca be easily checked. Coditio c ca be verified by applyig Lemmas 4.1 4.2 i the Appedix. Applyig Theorem 2.1, we get sup x P T x S i j E x = o 1/2, 3.7

624 BING-YI JING AND QIYING WANG where E x is give i Theorem 2.1. It follows from 3.7, ad a similar method to the oe used i the proof of Theorem 2.1, that sup P T + b/ x b Φ 1 x E x x = o 1/2. S Theorem 3.3 the follows from Φ 2 x = xφx, Φ 3 x = x 2 1φx, ad Eα 3 X 1 = ρ, EαX 1 αx 2 βx 1, X 2 = 2b, EαX 1 γx 2, X 1 = ρ + 2b. 4. Proof of Theorem 2.1 4.1. Some useful lemmas Lemma 4.1. Let gx 1,..., x k be symmetric i its argumets. Assume that 1 U g = gx i1,..., X ik k 1 i 1 < <i k is a degeerate U-statistic of order m, i.e., Egx 1,..., x m, X m+1,..., X k =0. a If E gx 1,..., X k p <, E U g p C m+11 p, for 1 p 2, E U g p C m+1p/2, for p 2. b If E gx 1,..., X k 2 <, the P U g C 1/2 = o 1/2 for m 1. c If E gx 1,..., X k 3/2 <, the P U g C 1/2 log 1 = o 1/2, for m = 0, P U g C 3/10 = o 1/2, for m = 1, P U g C 1/2 log 1 = o 1/2, for m = 2. d If E gx 1,..., X k 3 <, the for all m 0 ad k > 0, P U g C 1/4 log k = o 1/2. The proof of a ca be foud i Theorems 2.1.3 ad 2.1.4 of Koroljuk ad Borovskich 1994. Others ca be show by a of the preset lemma ad Markov s iequality. Lemma 4.2. a If E X 1 <, there exists a δ 0 such that P X δ = o 1/2.

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 625 b If EX 1 = 0 ad E X 1 3/2 <, there exists a δ 0 such that P X δ = o 1/2. c If E X 1 3/4 <, the P X C = o 1/2. d If EX 1 = 0, EX1 2 = 1, ad E X 1 3 <, the sup x 2 10 log P X x 1/2 /3 = o 1/2. { Proof. To prove a, let δ 3 = max E X 1 I X1 1/4, 2 1/2 E X 1 } 3. Sice E X 1 <, we have δ 0 ad P X δ P X 1 3/2 1 +P 3/2 X j I Xj 3/2 EX ji Xj 3/2 δ E X 1 1/2 1/2 E X 1 I X1 3/2 + 8 2 δ 2 o 1/2 + EX 2 1I X1 1/4 + EX2 1I 1/4 < X 1 3/2 8 E X 1 2 EX2 1I X1 1/4 + 8δ 1/2 = o 1/2. This proves a. Similarly, we ca prove b ad c. We prove d ext. From Chapter V of Petrov 1975, sup x 1+ x 3 P X x Φx C 1/2 E X 1 3. From this ad the iequality 1 Φx 1 2π e x2 /2 2, for x 1, 4.1 1 + x 3 we have P X x /3 1 Φ x /3 + C 1/2 log 1 = o 1/2 for 3 ad x 2 10 log. Similarly, P X x /3 = o 1/2 for 3 ad x 2 10 log. We have proved d. The proof of the ext lemma ca be foud i Jig ad Wag 2003. Lemma 4.3. Let V x ad W x, y be real Borel-measurable fuctios ad W x, y be symmetric i its argumets. Assume the followig. a the d.f. of V X 1 is olattice for sufficietly large. b EV X 1 = 0, EV 2 X 1 = 1, sup 1 E V X 1 3 <. c E[W X 1, X 2 X 1 ] = 0, sup 2 E W X 1, X 2 5/3 <. The, sup x 1 P V X j + 1 3/2 W X i, X j x E x = o 1/2,

626 BING-YI JING AND QIYING WANG where L x = { EΦ x 1/2 V X 1 Φx } Φ 2 x/2 ad E x = Φx + L x Φ3 x 2 EV X 1 V X 2 W X 1, X 2. The ext lemma may be of idepedet iterest. Lemma 4.4. Let ξ j x, ϕ j x, y, ad V V X 1,..., X be real Borel measurable fuctios i their argumets, ad ϕ j x, y = ϕ j y, x. Assume the followig. a The d.f. of ξ 1 X 1 is olattice ad Eξ 1 X 1 = 0, Eξ 1 X 1 2 = 1, E ξ 1 X 1 3 <. b Eξ 2 X 1 = 0, E ξ 2 X 1 3/2 <, E ξ 3 X 1 3/4 <. c Eϕ j X 1, X 2 X 1 = 0, j = 1, 2; E ϕ 1 X 1, X 2 5/3 <, E ϕ 2 X 1, X 2 3/2 <. d P V o 1/2 = o 1/2. Let ς j = ξ 2 X j + 1 ξ 3 X j ad ψ ij x = ϕ 1 X i, X j + x 1/2 ϕ 2 X i, X j, K x = 1 ξ 1 X j + x ς j + 1 3/2 ψ ij x, E K x = Φx 6 1 1/2 Φ 3 x Eξ 3 1X 1 + 3Eξ 1 X 1 ξ 1 X 2 ϕ 1 X 1, X 2 + 1/2 xφ 2 xeξ 1 X 1 ξ 2 X 1. The, sup x P K x x1 + V E K x = o 1/2 as,. Proof. Without loss of geerality, we assume that For, if ot, we ca defie ϕ 2 X i, X j 4 2 for all i, j. 4.2 ϕ 3 X i, X j = ϕ 2 X i, X j I ϕ2 2 Eϕ 2 X i, X j I ϕ2 2, ϕ 4 X i, X j = ϕ 3 X i, X j Eϕ 3 X i, X j X i Eϕ 3 X i, X j X j, ψ ij x = ϕ 1 X i, X j + x ϕ 4 X i, X j. The we have 1 3/2 ψ ij x = 1 ψ 3/2 ij x + xr, say.

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 627 Write δ 2 = E ϕ 2 X 1, X 2 3/2 I ϕ2 2. Sice E [ϕ 2 X 1, X 2 X 1 ] = 0, we have P R δ 1 = P 2 [ϕ 2 X i, X j ϕ 4 X i, X j ] δ 4 δ 1 E ϕ 2 X 1, X 2 I ϕ2 2 4 δ 1 1 E ϕ 2 X 1, X 2 3/2 I ϕ2 2 4δ 1/2. It is easy to show that δ 0 ad that ϕ 4 x, y is a symmetric fuctio satisfyig Eϕ 4 X 1, X 2 X 1 = 0, E ϕ 4 X 1, X 2 3/2 <, ad ϕ 4 X i, X j 4 2. Therefore, if 4.2 does ot hold, we ca replace ϕ 2 X i, X j by ϕ 4 X i, X j, ad V by V R. For simplicity, take V = 0. Clearly, this will ot affect the result sice V oly makes cotributio of size o 1/2 to the Edgeworth expasio. Write ξ2 X j = ξ 2 X j I ξ2 X j /1+x 2, ξ3 X j = ξ 3 X j I ξ3 X j 2 /1+x 2, ςj = ξ2 X j + ξ3 X j/, ad The we have K x = 1 ξ 1 X j + x ςj + 1 3/2 ψ ij x. sup P K x x E K x x sup x 2 10 log P K x x E K x + sup x 2 10 log P Kx x E K x + sup x 2 10 log P Kx x P K x x =: P 1 + P 2 + P 3, say. It suffices to show that P i = o 1/2 for i = 1, 2, 3. i First, we prove P 1 = o 1/2. Write P 1 = P + 1 P + 1 = sup x 10 log 1/2 P K x x E K x + P 1, where sup P K x x + sup 1 E K x x 10 log 1/2 x 10 log 1/2 1 sup P x 10 log 1/2 + sup P x 1 ξ 1 X j x 3 1 3/2 ψ ij x x 3 1 + P ς j 1 3 + sup 1 E K x x 10 log 1/2

628 BING-YI JING AND QIYING WANG = o 1/2, where the last equality follows from Lemmas 4.1 4.2 ad the iequality 4.1. Similarly, we ca show that P 1 = o 1/2. This proves P 1 = o 1/2. ii We show that P 3 = o 1/2 ext. Clearly P 3 sup P Kx x P K x x x 2 1 + sup P K x x, ς j ςj, for some j 1 x 2 10 log + sup 1 x 2 10 log P Kx x, ς j ςj, for some j =: Ω 0 + Ω 1 + Ω 2, say. First cosider Ω 0. It is easy to see that P ς j ςj P ξ 2 X 1 1 + x 2 + P ξ 3 X 1 2 1 + x 2 1 + x 3/2 + x 3 3/2 E ξ 2 X 1 3/2 I ξ2 X 1 /1+x 2 + E ξ 3 X 1 3/4 I ξ3 X 1 2 /1+x 2.4.3 It follows from 4.3 that Ω 0 sup x 2 1 P ς j ς j = o 1/2. Next we ivestigate Ω 1. Without loss of geerality, we assume that x 1. The i view of 4.3 ad idepedece of X k, we obtai 1 sup P 1 x 2 10 log sup 1 x 2 10 log ξ 1 X k x 3, ς j ςj, k=1 P 1 1 P ξ 1 X j 1 6 = o 1/2 + = o 1/2, sup for some j ξ 1 X k x 3, ς j ςj k=1 + 1 x 2 10 log sup 1 x 2 10 log 1 P 1 P ξ 1 X k x 6 k=1 k j ξ 1 X k x 6, ς j ςj k=1 k j P ς j ςj where we used the iequality: P 1/2 k=1 ξ 1 X k x/6 Cx 3 for all k j

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 629 1 j. From this ad Lemmas 4.1-4.2, we get 1 Ω 1 sup P x 2 1 3/2 1 + sup P 1 x 2 10 log ψ ij x x 3 1 + P ς j 1 3 ξ 1 X k x/3, ς j ςj, for some j k=1 = o 1/2. 4.4 Similarly, we ca show Ω 2 = o 1/2. Thus, we have show P 3 = o 1/2. iii Fially, we prove P 2 = o 1/2. Write ad defie Y j x = ξ 1 X j + x ς j Eς j, σx 2 = EY1x, 2 θ x = x σ x 1 Eς 1. { L y = EΦ y Y } 1x Φy 1 σ x 2 Φ2 y, K x = 1 1 σ x Y jx + 1 1 3/2 σ x ψ ijx, Φ3 y E y = Φy + L y 2 σx 3 EY 1xY 2 xψ 12 x, E y = Φy Φ3 y 6 σ 3 x The we have P 2 sup sup P K x y E y + x 2 10 log y + sup Eθ x E K x x 2 10 log =: I 1 + I 2 + I 3, say. Thus, P 2 = o 1/2 follows if we ca show EY 3 1 x + 3EY 1 xy 2 xψ 12 x. sup x 2 10 log sup E y Ey y I j = o 1/2, for j = 1, 2, 3. 4.5 Uder coditio b, we obtai that for all x 2 10 log, 1 + x Eξ2X 1 E ξ 2 X 1 I ξ2 X 1 /1+x 2 = o, 4.6

630 BING-YI JING AND QIYING WANG E ξ2x 1 α E ξ 2 X 1 α I ξ2 X 1 + Similarly, we have α 3/2 E ξ 2 X 1 3/2 I ξ2x1 1+x 2 α 3/2 = o 1 + x 2, for α > 3 2. 4.7 E ξ3x 1 α 2 α 3/4 = o 1 + x 2, for α > 3 4. 4.8 Recallig ςj = ξ 2 X j + 1 ξ3 X j, it follows from 4.6 4.8 that for x 2 10 log, Eς1 = Eξ 2X 1 + 1 Eξ 3X 1 1 + x = o, 4.9 E ς1 E ξ2x 1 + 1 E ξ 3X 1 = O1, 4.10 Eξ 2X 1 2 + Eξ 3 X 1 2 1 + x 2 = o,4.11 x 3 E ξ 2X 1 3 + 1 3 E ξ 3X 1 3 = o1. 4.12 x 2 Eς 1 2 2x2 x 3 E ς 1 3 6 By usig 4.9 4.12, together with Hölder s iequality, we get that if x 2 10 log, the σx 2 = 1 + 2x 1 + x Eξ 1 X 1 ξ 2 X 1 + o, 4.13 σ 1 x = 1 x 1 + x Eξ 1 X 1 ξ 2 X 1 + o, 4.14 EY 3 1x = Eξ 1 X 1 3 + o1, E Y 1 x 3 = O1, 4.15 EY 1 xy 2 xψ 12 x = Eξ 1 X 1 ξ 1 X 2 ϕ 1 X 1, X 2 + o1. 4.16 We oly check 4.16 below, others ca be checked similarly. Let µ j x = x ς 1/2 j Eς j. It follows from 4.12 that E µ 1 x 3 = o1. The, EY 1 xy 2 xψ 12 x = Eξ 1 X 1 ξ 1 X 2 ϕ 1 X 1, X 2 + B 1 + B 2 where, by otig idepedece of X k ad 4.15, B 1 E {ξ 1 X 1 µ 2 x + µ 1 xy 2 x} ϕ 1 X 1, X 2 3 E µ 1 x 3 1/3 E ξ1 X 1 3 + E Y 1 x 3 1/3 E ϕ 1 X 1, X 2 3/2 2/3

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 631 = o1, B 2 E Y 1 xy 2 xϕ 2 X 1, X 2 E Y 1 x 3 2/3 E ϕ 2 X 1, X 2 3/2 2/3 = O1. This proves 4.16. We tur back to the proof of 4.5. To show I 3 = o 1/2, it suffices to show sup x 2 10 log Φθ x Φx xφ2 x Eξ 1 X 1 ξ 1 X 2 = o 1/2, 4.17 sup Φ 3 θ x Φ 3 x = O 1/2, 4.18 x 2 10 log sup x 2 10 log sup x 2 10 log EY 3 1 x σ 3 x Eξ 1 X 1 3 = o1, 4.19 EY 1 xy 2 xψ 12 x σ 3 x Eξ 1 X 1 ξ 1 X 2 ϕ 1 X 1, X 2 = o1. 4.20 Clearly, 4.18 4.20 follow easily from 4.13 4.16. Now let us check 4.17. Usig 4.9 ad 4.14, we have that, for all x 2 10 log, θ x = x σ x 1 Eς 1 = x 1 x 1 + x 2 Eξ 1 X 1 ξ 1 X 2 + o. Hece, for sufficietly large, we have x/2 θ x 3x/2. From these ad Taylor s expasio, there exists 1/2 δ 3/2 such that for all x 2 10 log, Φθ x = Φx + θ x xφx + θ x x 2 Φ 2 δx 2 = Φx x2 φx Eξ 1 X 1 ξ 2 X 1 + o 1/2 fxφ x 2, where fx is a polyomial i x. Sice Φ 2 x = xφx, 4.17 is show. Thus, I 3 = o 1/2. Next we show that I 2 = o 1/2. Note that sup E y Ey y = sup y {EΦ y 6 σx 3 Φ3 y Y } 1x Φy 1 σ x 2 Φ2 y + EY 1 3 x C σ 3 x E Y 1x 3 I Y1 x σ x + C σx 4 E Y 1x 4 I Y1 x σ x,

632 BING-YI JING AND QIYING WANG where the last iequality follows from Theorem 3.2 of Hall 1982. It follows from 4.13 that for sufficietly large ad all x 2 10 log, 1 2 < σ x < 3 2. 4.21 It follows from 4.9 that for sufficietly large ad all x 2 10 log, Y 1 x = ξ 1X 1 + x ς 1 Eς1 1 + ξ 1 X 1 + ξ 2 X 1 1/2 + ξ 3 X 1 1/4 =: κx 1, say.4.22 Notig 4.21 ad Eκ 3 X 1 <, we get for sufficietly large, I 2 = sup sup E y Ey x 2 10 log y C 1/2 sup E Y 1 x 3 I Y1 x 1/4 + 1 E Y 1 x 4 I x 2 10 log Y1 x 1/4 C 1/2 Eκ 3 X 1 I κx1 1/4 + 1 1/4 Eκ3 X 1 =o 1/2. Fially we use Lemma 4.3 to show that I 1 = o 1/2 by takig V X j = Y j x/σ x, W X i, X j = ψ ij x/σ x. First, we check coditio a of Lemma 4.3. By Theorem 1.3 of Petrov 1995, a d.f. with the characteristic fuctio ft is a olattice d.f. if ad oly if for every fixed umber s 0 0, we have fs 0 < 1 or, equivaletly, if ad oly if sup δ t t0 ft < 1 for ay t 0 > 0 ad δ > 0. Hece, to show that V X j is olattice uiformly for x 2 10 log for sufficietly large, we oly eed to prove b =: sup x 2 10 log sup δ t t0 Ee itv X j < 1 for sufficietly large ad each δ > 0. Notig that sup δ t t0 Ee itξ 1X 1 < d < 1, ad from 4.10 ad 4.14, we have b sup sup Ee itv X j Ee itξ 1X 1 + sup Ee itξ 1X 1 x 2 10 log δ t t 0 t 0 t 0 sup x 2 10 log sup x 2 10 log E V X j ξ 1 X 1 + d δ t t 0 σ x 1 1 + x E ς 1 Eς1 + d = d + o1 < 1, for sufficietly large. Coditio a of Lemma 4.3 also holds here. To check coditio b of Lemma 4.3, it is easy to see that EV X 1 = 0 ad EV 2 X 1 = 1. For x 2 10 log, from 4.21 4.22, we have sup E V X 1 3 <.

A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS 633 To check coditio c of Lemma 4.3, it is easy to see that E[W X 1, X 2 X 1 ] = 0. For x 2 10 log, by Mikowski s iequality ad the assumptios, we get E ψ ij x 5/3 CE ϕ 1 X i, X j 5/3 + Cx5/3 5/6 E ϕ 2X i, X j 5/3 I{ ϕ 2 X i, X j 4 2 } CE ϕ 1 X i, X j 5/3 + Cx5/3 1/2 E ϕ 2X i, X j 3/2 C, which, together with 4.21, yields sup 2 E W X i, X j 5/3 <. Hece, applyig Lemma 4.3, we have that, for all sufficietly large, I 1 = sup sup P K x y E y = o 1/2. x 2 10 log y The, P 2 = o 1/2. The proof of Lemma 4.4 is complete. 4.2. Proof of Theorem 2.1 Without loss of geerality, we assume V j = 0. It will be clear that this assumptio does ot affect the proof of the mai results sice their cotributios to the Edgeworth expasio is oly of size o 1/2. Let γ 1 x = Eγ x, X 1 ad γ 2 x, y = γx, y γ 1 x γ 1 y. It is easy to show that S 2 = 1 + 1 1 γ 1 X j + γ 2 X i, X j =: 1 + Z + R, say. 4.23 1 Notig that 1 + u/2 u 2 /6 1 + u 1/2 1 + u/2 + u 2 /6 for u 1/9, if Z + R 1/9, we have 1 + 1 2 Z + R 1 3 Z 2 + R 2 S 1 + 1 2 Z + R + 1 Z 2 3 + R 2.4.24 Put s = Z /2 + 1/2R + sz. 2 The from 4.23 ad 4.24, we have T T P x P x, Z + R 1 + P Z + R 1 S S 9 9 { P T x 1 + 1 3 + R } 2 + R2 + P Z + R 1 3 9 P T x {1 + 13 } + 3/5 +P Z + R 1 9 + P R 2 + R2 3 3/5. 4.25 Similarly, we get P T /S x P T x {1 + 13 } 3/5

634 BING-YI JING AND QIYING WANG P Z + R 1 9 P R 2 R2 3 3/5. 4.26 Usig Jese s iequality, we ca easily see that E γ 1 X 1 3/2 < ad E γ 2 X 1, X 2 3/2 <. Sice 2R is a degeerate U-statistic of order 1, it follows from Lemmas 4.1 4.2 that 1 P 2 R ± 1 3 R2 3/5 P R 1 + P R 3/10 = o 1/2, P Z + R 1 9 P Z 1 + P R 1 = o 1/2. 18 18 I view of these iequalities, 4.25 ad 4.26, Theorem 2.1 follows if sup P T x 1 + s + A E x = o 1/2, 4.27 x where A 3/5 ad s 1/3. To prove 4.27, let 1 ς j x = 2 γ 1X j + s γ2 1X j, ψ ij x = βx i, X j A elemetary calculatio shows that x 2sγ 1 X i γ 1 X j + 1 2 γ 2X i, X j. P T x 1 + s + A 1 = P αx j + x ς j + 1 3/2 ψ ij x x 1 + A. It is easy to check that coditios of Lemma 4.4 are satisfied with s 1/3, V = 3/5, ad with ξ 1 X j = αx j, ξ 2 X j = γ 1 X j /2, ξ 3 X j = sγ1 2X j, ad ϕ 1 X i, X j = βx i, X j, ϕ 2 X i, X j = 2sγ 1 X i γ 1 X j + 1 2 γ 2X i, X j. So 4.27 follows from Lemma 4.4 ad the relatio EαX 1 γ 1 X 1 = EαX 1 γx 1, X 2. Ackowledgemets Jig s research is partially supported by Hog Kog RGC grats HKUST6011 /07P ad HKUST6015/08P. Wag s research is partially supported by a Australia Research Coucil discovery project. We thak the Editors ad three referees for very useful commets ad costructive criticisms which have greatly improved the paper.

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636 BING-YI JING AND QIYING WANG Serflig, R. J. 1980. Approximatio Theorems of Mathematical Statistics. Joh Wiley, New York. Sigh, K. 1981. O the asymptotic accuracy of Efro s bootstrap. A. Statist. 9, 1187-1195. Wag, Q., Jig, B.-Y. ad Zhao, L. 2000. The Berry-Essée boud for studetized statistics. A. Probab. 28, 511-535. Wag, Q. ad Weber, N. C. 2006. Exact covergece rate ad leadig term i the cetral limit theorem for U-statistics, Statist. Siica 16, 1409-1422. va Zwet, W. R. 1984. A Berry-Essée Boud for symmetric statistics. Z. Wahrsch. Verw. Gebiete 66, 425-440. Departmet of Mathematics, Hog Kog Uiversity of Sciece ad Techology, Clear Water Bay, Kowloo, Hog Kog. E-mail: majig@ust.hk School of Mathematics ad Statistics, Uiversity of Sydey, NSW, Australia. E-mail: qiyig@maths.usyd.edu.au Received December 2007; accepted December 2008