1 Statistica Sinica 6006, EXACT CONVERGENCE RATE AND LEADING TERM IN THE CENTRAL LIMIT THEOREM FOR U-STATISTICS Qiying Wang and Neville C Weber The University of Sydney Abstract: The leading term in the normal approimation to the distribution of U- statistics of degree is derived. This result is applied to establish the eact rate of convergence in the Central Limit Theorem for U-statistics and to obtain the one-term Edgeworth epansion for the distribution function. Analogous results for more general U-type statistics are also considered. Key words and phrases: Berry-Esséen theorem, characterisation of rate of convergence, Edgeworth epansion, optimal moments, nonlattice condition, U-statistics, L-statistics.. Introduction and Main Results Let X, X,..., X n be a sequence of independent and identically distributed i.i.d. random variables. Let h, y be a real-valued Borel measurable function, symmetric in its arguments with EhX, X = 0. For n, a U-statistic of degree with kernel h, y is defined by n U n = hx i, X j. i<j n Write g = Eh, X and φ, y = h, y g gy. The statistic U n may be represented as U n = n n j= n gx j + i<j n φx i, X j := U n + U n. See, for eample, Lee 990, p.5. Throughout we assume that Eg X =. This assumption implies that nun / is a standard sum of non-degenerate iid random variables and its distribution may be approimated by a standard normal distribution Φ. Indeed, the classical result see Hall 98, p., for eample shows that sup P nu n Φ + n δn + n. 3
2 40 QIYING WANG AND NEVILLE C WEBER Here and below we define δ n = Eg X I gx n + n Eg 3 X I gx n +n Eg 4 X I gx n, and we say that two sequences of positive numbers a n } and b n } satisfy a n b n if 0 < lim inf n a n /b n lim sup n a n /b n <. Note that 3 yields concise results about the rate of convergence in the Central Limit Theorem. In past decades, there has been considerable interest in stating the accuracy of the normal approimation to the distribution of nu n / in a manner that is similar to 3. In increasing generality, the upper bound for the accuracy of the normal approimation has been established in a number of papers. We mention only Bickel 974, Chan and Wierman 977, Callaert and Janssen 978, Borovskikh 996, 00, Alberink and Bentkus 00, 00 and Wang 00. The result given by Borovskikh 00 also see Alberink and Bentkus 00, which is closest to the upper bound in 3, states that if E hx, X 5/3 <, then sup P nu n Φ [ Eg X I gx n + n E gx 3 I gx n A ] + On, 4 where A is an absolute positive constant. In contrast to rich results on the upper bound, there are only a few papers concerned with the lower bound for the accuracy of the normal approimation to the distribution of nu n /. Maesono 988, 99 obtained a lower bound of order On / under the condition Eh 4 X, X <. Only assuming the eistence of Eh X, X, Wang 99 derived a result for the distribution of nu n / that is similar to 3. In a slightly different problem Bentkus, Götze and Zitikis 994 proved that the best bound of order On / in 4 cannot be obtained under E hx, X 5/3 ɛ < for any ɛ > 0. In the present paper we give the leading term in a normal approimation to the distribution of nu n /. Using the leading term we derive the eact convergence rate in the Central Limit Theorem for U-statistics, up to terms of order On /, under E hx, X 5/3 <. As mentioned above, to get the terms of order On /, the latter moment condition is the best possible. We also show that, if in addition E gx 3 <, the leading term transforms into the conventional first term in an Edgeworth epansion of the distribution of U-statistics. Our main result is the following.
3 EXACT CONVERGENCE RATE AND LEADING TERM 4 Theorem.. If E hx, X 5/3 <, then sup P nu n Φ L n + L n = oδ n + On, 5 where δ n is defined as in 3, [ L n = n EΦ gx } ] Φ n Φ, L n = Φ3 n E } gx gx φx, X I φx,x n 3. If in addition gx is nonlattice, then the right-hand side of 5 may be replaced by oδ n + n /. As is well-known, δ n 0, as n, and sup L n δ n, 6 see, for eample, Chapter of Hall 98. We show in Section 3 that sup L n = oδ n + On. 7 Together, 5 7 give concise results about the rate of convergence in the Central Limit Theorem for U-statistics. Indeed, if E hx, X 5/3 <, then sup P nu n Φ + n δ n + n. 8 Note that 8 refines 4 even for the upper bound. One application of 8 is to characterise the rate of convergence. The following theorem gives eamples. Generalizations of the eamples are readily derived, refer to Theorems.9 and.0 of Hall 98 for more details. Theorem.. Assume E hx, X 5/3 <. If 0 r < /, then n= n r sup P nu n if and only if E gx r+ <. If 0 < r < /, then sup P nu n if and only if Eg X I gx = o r. Φ < 9 Φ = On r 0
4 4 QIYING WANG AND NEVILLE C WEBER It is also interesting to note that the effect of L n in 5 on the rate of convergence appears only when δ n = On /. This can be easily seen from 7 and the following corollary, which provides the main result of Jing and Wang 003, about an Edgeworth epansion of the distribution of U-statistics under optimal conditions. Corollary. below also is an alternative to Theorem 5 of Borovskikh 998 with m =, where the Edgeworth epansion is obtained under lim sup t Ee itgx < instead of the condition that the distribution of gx is nonlattice. Borovskikh 998 also uses a weaker moment condition on the kernel h, y. Corollary.. Assume that E hx, X 5/3 <, E gx 3 <, and the distribution of gx is nonlattice. Then, as n, sup P nu n F n = on, where F n = Φ Φ 3 /6 n Eg 3 X + 3EgX gx hx, X }. The proof of all results will be given in Section 3. To conclude this section we mention that the rate of convergence in the Central Limit Theorem for U- statistics depends on the moment conditions for both hx, X and gx. If only E hx, X p <, where 4/3 < p < 5/3, the term of order On / in 8 has to be replaced by a term of lower order. This follows from Theorem. in the net section, which gives an etension of Theorem. to U-type statistics. Throughout the paper we denote constants by A, A, A,..., which may be different at each occurrence.. Etensions to U-type Statistics and L-statistics Let α and β, y be some real-valued Borel measurable functions of and y. Furthermore, let V n V n X,..., X n be real-valued functions of X,..., X n }. Define a U-type statistic by T n = n n αx j + n 3 βx i, X j + V n. j= i j In this section we derive the leading term in a normal approimation to the distribution of T n under mild conditions, which gives an etension of Theorem.. Theorem.. Assume that a EαX = 0 and Eα X = ; b E[βX, X X i ] = 0, i =,, and E βx, X p < for 4/3 < p 5/3;
5 EXACT CONVERGENCE RATE AND LEADING TERM 43 c P V n C 0 n / C n / for some constants C 0 > 0 and C > 0. Then, as n, sup P T n Φ L n + L n = oδ n + n 4 3p + On, 3 where δ n = Eα X I αx n + n Eα 3 X I αx n +n Eα 4 X I αx n [, L n = n EΦ αx ] } Φ n Φ, L n = Φ3 ]} n E αx αx [βx, X I 3 + βx, X I. β n β n 3 If the condition c is replaced by c P V n on / } on /, and in addition αx is nonlattice, then the right-hand side of 3 may be replaced by oδ n + n 4 3p/. Note that the U-type statistic T n defined by is quite general. We net consider an application to L-statistics. Let X,..., X n be i.i.d. real random variables with distribution function F. Define F n to be the empirical distribution, i.e., F n = n n j= IX i }, where I } is the indicator function. Let Jt be a real-valued function on [0, ] and T G = JG dg. The statistic T F n is called an L-statistic see Chapter 8 of Serfling 980. Write σ σ J, F = J F s J F t F mins, t} [ F mas, t}] dsdt, and define the distribution function of the standardized L-statistic T F n by H n = P nσ T F n T F. As is well-known, H n converges to Φ uniformly in provided E X <, σ > 0, and some smoothness conditions on Jt hold, see Serfling 980 and Helmers, Janssen and Serfling 990 for eample. The upper bounds for the rate of convergence to normality were investigated by Helmers 977 van Zwet 984, Helmers, Janssen and Serfling 990, Wang, Jing and Zhao 000 and Wang 00. As a consequence of Theorem., the following theorem derives the eact convergence rate two-sided bound in the Central Limit Theorem for L-statistics, up to terms of order On /, under mild conditions. Theorem.. Assume that
6 44 QIYING WANG AND NEVILLE C WEBER a Js Jt K s t, 0 < s < t <, for some K > 0; b EX < and σ > 0. Then, as n, sup H n Φ + n δ n + n, 4 where αx = σ JF tix t F t dt, and δ n is defined as in Theorem.. 3. Proofs Proof of Theorem.. The result is an immediate consequence of Theorem.. Proof of Theorem.. Without loss of generality we assume that β, y is symmetric. Otherwise it is enough to replace βx i, Y j by βx i, Y j + βx j, Y i. The proof is along the lines of Jing and Wang 003. Write βx i, X j = βx i, X j I βxi,x j n 3, α X j = E βxi, X j X j, α n X j = α X j I n α X j n, n Tn = n αx j + α X j + n 3 βxi, X j α X i α X j +V n. j= Noting EβX, X = 0, it is easily seen that α X j = E βx i, X j I βxi,x j n 3/ j X E βx i, X j I βxi,x j n 3/ X j, 5 i<j and, as in.4.5 of Jing and Wang 003, sup P T n P Tn np α X n + n P n 4 3p E βx, X p I βx,x n 3 = o n 4 3p βx, X n 3. 6 We further let, m 0 = [0 log n]+/b, where b > 0 is a constant to be chosen
7 EXACT CONVERGENCE RATE AND LEADING TERM 45 later, η j = αx j + α X j Eα X j, [ γ ij = βxi, X j α X i α X j + E βx ] i, X j, n S n = n η j, m = n 3 n,m = n k=m j= n j=m+ γ mj for m n, k if 0 < m < n, and n,m = 0 if m n. It follows immediately that Tn = S n + n,m0 + Ṽn + neα X n n / E βx, X, where Ṽn = V n + m 0 m= m. Note that for any fied m < k n and q, E n,m n,k q 8n 3q + k me γ q ; 7 see Theorem..3 in Koroljuk and Borovskich 994. It follows from 7 with q = p and E γ p < see below that for 4/3 p 5/3, P m 0 m= n m An p log +p n E γ p = o log n n 4 3p. 8 In terms of the condition c or c, 8 and the fact that neα X n n / E βx, X 3 ne βx, X I β n 3/ = on4 3p/, routine calculations show that, to prove 3, it suffices to prove I n := sup P S n + n,m0 Φ L n + L n = oδ n + n 4 3p + On 9 and, if in addition αx is nonlattice, then the right-hand side of 9 may be replaced by oδ n + n 4 3p/. We first establish five lemmas before the proof of 9. The proofs of these lemmas will be omitted. The details can be found in Wang and Weber 004, on which the present paper is based.
8 46 QIYING WANG AND NEVILLE C WEBER Lemma 3.. Write ˆαX = α X Eα X. We have E ˆαX λ E α X λ = o E γ q 6 βx, X q = o Eη = o δ n + n 4 3p n λ+ 3p, for λ, 0 E γ p 6 βx, X p <, n 3q p, for p < q, Lemma 3.. We have Eγ e itη +η n = t n E + o Eγ e itη +η n A min. 3 αx αx βx }, X δ n + n 4 3p n t + t 3, 4 }. 5 t n 4p p, n Eγ t + t 3 Net define, ft = Ee itη / n, gt = Ee itαx / n, and g n t = e t / + ngt + t /. Lemma 3.3. There eists a constant c 0 > 0 such that for all t c 0 n / and all sufficiently large n, ft e t 8n, gt e t 4n, 6 f n t e t A δ n + on 4 3p t + t 4 e t 6, 7 f n t g n t = o δ n + n 4 3p t + t 8 e t 6. 8 If in addition αx is nonlattice, then there eist constants b > 0 and ɛ n such that for c 0 t / n ɛ n, ft e b and gt e b. 9 Lemma 3.4. For any t c 0 n, where c0 is defined as in Lemma 3.3, E n,m0 e itsn + t e t n E = o δ n + n 4 3p αx αx [ βx, X + βx, X ]} t + t 6 e t To introduce the net lemma we first define some notation. As in.3 and
9 EXACT CONVERGENCE RATE AND LEADING TERM 47.6 of Jing and Wang 003, we have Z n t := Ee itsn+ n,m 0 Ee itsn ite n,m0 e itsn [ ] = Z n t + it Z n t + Z n t, where, l m,k = n 3/ n j=k+ γ mj, jm is the largest integer such that mjm < n and Z n t = n Z n n t = m=m 0 Ee itsn+ n,m+ e it m it m, jm m=m 0 j= Z n n t = jm m=m 0 j= t c 0 n El m,jm e itsn e it n,jm+ e it n,j+m+, E l m,jm l m,j+m e its n e it n,j+m+. Lemma 3.5. For 4/3 < p 5/3, we have t c 0 n t Z nt dt = o δ n + n 4 3p, 3 t + Z n t dt = o δ n + n 4 3p, 3 Z n where c 0 is defined as in Lemma 3.3. We are now ready to prove 9. We continue to use the notation defined in Lemmas Furthermore write ϕ n t = t B n e t /, where B n = ]} n E αx αx [βx, X I + βx β n 3, X I. β n 3 Using Lemmas we have J n := Ee itsn+ n,m 0 g n t itϕ n t dt t c 0 n t t c 0 n t Z nt dt + t c 0 n t f n t g n t dt + E n,m0 e itsn ϕ n t dt t c 0 n = oδ n + n 4 3p. 33
10 48 QIYING WANG AND NEVILLE C WEBER Note that eit d Φ + L n L n = g n t + itϕ n t. It follows from Esseen s smoothing lemma and 33 that I n Ee itsn+ n,m 0 g n t itϕ n t dt + t c 0 n t = oδ n + n 4 3p + On. 34 This proves the first part of 9. If αx is nonlatice, it follows from the fact that n,m0 only depends on X m0 +,..., X n, and 9, that for any ɛ n, J n := Ee itsn+ n,m 0 g n t itϕ n t dt c 0 n t ɛn n t c 0 n t ɛn n t ft m 0 dt + c 0 n t ɛn n t g nt + itϕ n t dt = oδ n + n 4 3p. 35 Using 33, 35 and Esséen s smoothing lemma again, we obtain for 4/3 p 5/3, I n Ee itsn+ n,m 0 g n t itϕ n t dt + A t ɛ n n t ɛ n n J n + J n + on = oδ n + n 4 3p. This implies the second part of 9 and hence the proof of 9. The proof of Theorem. is now complete. Proof of Theorem.. Write η j t = IX j t} F t, αx j = σ JF tη j tdt, βx i, X j = Kσ As in 9 of Wang 00, we have n n αx j n 3 βx i, X j V n j= i j A n η i tη j tdt. n T Fn T F σ n n αx j + n 3 βx i, X j + V n, 36 j= i j
11 EXACT CONVERGENCE RATE AND LEADING TERM 49 where V n = n 3/ n j= ZX j with ZX j = Kσ ηj tdt. It is readily seen that EαX = 0, Eα X =, EβX i, X j X i = 0, i j, and similar to the proof of Lemma A in Serfling 980, p.88, αx j + βx i, X j + ZX j Aσ X j + E X. 37 In terms of these facts, 4 follows easily from Theorem.. We omit the details. Proof of Corollary.. Equation follows easily from Theorem., the classical result sup L n + Φ3 6 n Eg3 X = on, and by Hölder s inequality, that L n Φ3 n E gx gx φx, X } A n E gx gx φx, X I φx,x n 3 sup A n E gx E φx, X I φx,x n 3 } = on. Proof of 7. It suffices to show that E gx gx φx, X I φx,x n } 3 n = oδ n + On. 38 Write φx, X = φx, X I φx,x n 3/. It is readily seen that E gx gx φx }, X = E gx I gx n gx φx }, X +E gx I gx < n gx I gx φx } n, X +E gx I gx < ngx I gx < φx } n, X := I 6n + I 7n + I 8n. 39 By Hölder s inequality, and similar to, we have I 6n + I 7n EgX I n gx n E φx, X n [ ] δ n on = o δ n + n. 40
12 40 QIYING WANG AND NEVILLE C WEBER Similarly, by noting E φx, X 5/3 <, I 8n / n n E gx 5 4 I gx < 5 n E φx, X An E gx 5 4 I gx < 5 n. 4 In terms of 4, if E gx 5/ I gx < n <, then I 8n n = On. 4 We show that if E gx 5/ I gx < n =, then ne gx 5 I gx < n A E gx 4 I gx < n, 43 and hence it follows from 4, that I 8n / n = o n E gx 5 I gx < n = o n E gx 4 I gx < n = oδ n. 44 Combining 39 40, 4 and 44, we obtain the proof of 38. We net prove 43. Write l τ = E gx τ I gx <. Note that l 4 is a non-decreasing function and l 4 A. It follows from Proposition.. of Bingham, Goldie and Teugels 987 that lim sup l 4 /l 4 <. Now 43 follows easily from question 34 on page 89 of Feller 97 also see Feller 969 or Theorem.6.6 of Bingham, Goldie and Teugels 987. The proof of 7 is now complete. Acknowledgements The authors would like to thank two referees and an associate editor for their detailed reading of this paper and valuable comments. The research is partially supported by ARC discovery grant DP0457 and Sesqui NSSS grant 004. References Alberink, I. B. and Bentkus, V. 00. Berry-Esseen bounds for von-mises and U-statistics. Lithuanian Math. J. 4, -6. Alberink, I. B. and Bentkus, V. 00. Lyapunov type bounds for U-statistics. Theory Probab. Appl. 46, Bentkus, V., Götze, F. and Zitikis, M Lower estimates of the convergence rate for U-statistics. Ann. Probab., Bickel, P. J Edgeworth epansions in nonparametric statistics. Ann. Statist., -0.
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14 4 QIYING WANG AND NEVILLE C WEBER School of Mathematics and Statistics, The University of Sydney, NSW 006, Australia. School of Mathematics and Statistics, The University of Sydney, NSW 006, Australia. Received April 004; accepted October 004