Non-uniform Berry Esseen Bounds for Weighted U-Statistics and Generalized L-Statistics

Transcription

1 Coun Math Stat 0 :5 67 DOI 0.007/s Non-unifor Berry Esseen Bounds for Weighted U-Statistics and Generalized L-Statistics Haojun Hu Qi-Man Shao Received: 9 August 0 / Accepted: Septeber 0 / Published online: Noveber 0 School of Matheatical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 0 Abstract Weighted U-statistics and generalized L-statistics are coonly used in statistical inference and their asyptotic properties have been well developed. In this paper sharp non-unifor Berry Esseen bounds for weighted U-statistics and generalized L-statistic are established. Keywords Weighted U-statistics Generalized L-statistic L-statistic U-statistic Non-unifor Berry Esseen bound Noral approxiation Matheatics Subject Classification 00 Priary 60F05 Secondary 6E0 Introduction and Main Results Let {X i, i n} be a sequence of independent and identically distributed i.i.d. real-valued rando variables and let hx,...,x be a kernel of degree, that is, h is a real-valued easurable syetric function in arguents. The ain purpose of this paper is to establish non-unifor Berry Esseen bounds for weighted U-statistics and generalized L-statistics.. Weighted U-Statistics Let a : N R be a syetric function. A weighted U-statistic is given by U n ai,...,i hx i,...,x i.. i < <i n H. Hu Departent of Matheatics, Zhejiang University, Hangzhou, Zhejiang, China e-ail: huhaojun989@foxail.co Q.-M. Shao B Departent of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China e-ail: qshao@sta.cuhk.edu.hk

2 5 H. Hu, Q.-M. Shao Clearly, it reduces to the Hoeffding U-statistics when a. The weighted U-statistic was first introduced by Shapiro and Hubert 0] for and its asyptotic norality and non-norality were discussed in O Neil and Redner 8]. The results of O Neil and Redner 8] were extended by Major 7] for general. We refer to Rifi and Utzet 9] for the asyptotic behavior of the weighted U-statistics. To present a non-unifor Berry Esseen bound for the weighted U-statistic, assue EhX,...,X 0, Eh X,...,X <.. Put σ0 Eh X,...,X <, g x E hx,...,x X x ], σ Eg X., A l ai,...,i,l,.4 i < <i n, i j l,j {,..., } B k ai,...,i aj,...,j,.5 i,...,i,j,...,j H k and C k,l ai,...,i, laj,...,j,l,.6 i,...,i,j,...,j H k,l where H k denotes the set of i < <i n, j < <j n with k coon eleents between the two sets {i,...,i } and {j,...,j }, and H k,l denotes the set of i < <i n, j < <j n, i s l,j s l for each s with k coon eleents between the two sets {i,...,i } and {j,...,j }. The following theore gives a non-unifor Berry Esseen bound for U n. Theore. If EhX,...,X 0 and E g X p < for <p, then there exists a universal constant C such that P U n n j A z Φz j / σ 9σ k 0 kb k C σ0 z + σ nj A + k kb k / z + j p σ n j A j / + σ 0 n l A l k kc k,l / σ / n j A + E gx p n j A j p j σ p n j A..7 j p/ Reark. It is easy to see that when a, A l n n n, B k, n k k n n C k,l. k k

3 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 5 Result.7 recovers Theore. of Chen and Shao 4]. Reark. For letting al,l 0, we have n A l aj,l, B and j C,l i<j n n a j, l. j a i, j So the non-unifor Berry Esseen bound.7 is reduced to 9σ0 B C z + σ nj A + z + j p In particular, if / n B + ax i n A i A j 0, σ 0 B / σ n j A + E gx p n j A j p j / σ p n j A j p/ j then U n / n j A j / σ is asyptotically noral.. Generalized L-Statistics Let R n, R n,cn, be the order values of hx i,...,x i taken over -tuples {i,...,i : i < <i n}, where C n, n. A generalized L-statistic is defined by C n, T n c n,i R n,i,.8 i where i/cn, c n,i Jtdt.9 i /C n, and J is a weight function on 0, ]. The generalized L-statistic was first introduced by Serfling ]. It includes Hodges Lehann location estiator, U-statistics, tried U-statistics, Winsorized U-statistics as special cases. The asyptotic properties of the generalized L-statistics have been extensively studied. We refer to Serfling, ] Borovshikh and Weber ] for the asyptotic norality and for the general theory and applications, Cai ] for the oderate and large deviations, and Helers and Ruygaart 6], Helers, Jansseen and Serfling 5] for the unifor Berry Esseen bounds. Let Hy be the distribution function of the rando variable function hx,...,x and denote the epirical distribution function by n H n y I hx i,...,x i y, y R..0 i < <i n.

4 54 H. Hu, Q.-M. Shao Let T be an L-functional TG 0 JtG t dt with G t inf{y : Gy t} for any distribution function G. Clearly, T n TH n and the generalized L-statistic T n is actually estiating the paraeter TH. Let Ax,...,x I hx,...,x y Hy J Hy dy,. Put g x E AX,...,X X x.. σ 0 Eh X,...,X, σ Var g X.. Then we have the following non-unifor Berry Esseen bound for T n. Theore. Assue that n/5 and there exists a constant c 0 such that Jt Js c0 t s for 0 s, t..4 If EhX,...,X 0 and E g X p < for <p, then there exists a universal constant C such that n P THn TH z Φz σ 6σ 0 J0 +c 0 + 6c0 nσ z + C J0 +c0 σ 0 + z + p n / + E g X p σ n p / σ p..5 Theore. recovers the unifor Berry Esseen bound of Helers, Janssen and Serfling 5] with a saller order in ters of. Proofs Our proof is based on the following non-unifor Berry Esseen inequality of Chen and Shao 4]: Proposition. Let ξ,...,ξ n be independent rando variables satisfying Eξ i 0 for i,...,n, and n i Eξ i. Let W n i ξ i and be a easurable function of {ξ i, i n}. If E ξ i p < for soe <p, then there exists an absolute constant C such that

5 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 55 PW + z Φz 9 + z + C z + p + n ξ i i + i n E ξ i, p. where i is any rando variable such that ξ i and W ξ i, i are independent, and ξ denotes Eξ /.. Proof of Theore. To apply Proposition., the first step is to rewrite n j A U n W +, j / σ where W is a su of independent rando variables and is a negligible ter. For k,let Then we can take and i h k x,...,x k E hx,...,x X x,...,x k x k, h k x,...,x k h k x,...,x k k g x i, i hx,...,x h x,...,x hx,...,x W i g x j. j n A j ξ i, ξ i n j A g X j, j / σ n j A j / σ i < <i n ai,...,i hx i,...,x i. Clearly, ξ,...,ξ n are independent rando variable with Eξ i 0 and ni Eξi. Set l n j A j / σ i < <i n, i j l,j {,...,} ai,...,i hx i,...,x i. Now by Proposition., Theore. follows by the next two leas. Lea. We have E σ 0 k kb k σ nj A.. j

6 56 H. Hu, Q.-M. Shao Lea. We have Proof of Lea. Observe that E nj A j σ E nj A j σ E l σ 0 k kc k,l σ nj A.. j i < <i n ai,...,i hx i,...,x i ai,...,i aj,...,j k i,...,i,j,...,j H k E h k X,...,X k nj A j σ B k E h k X,...,X k. k By Eq in ], we have E h k X,...,X k k σ 0 and hence. holds. Proof of Lea. Siilarly, we have n E l σ j A j { E i < <i n i < <i n, i j l,j {,...,} } ai,...,i hx i,...,x i ai,...,i, laj,...,j,l k i,...,i,j,...,j H k,l E h k+ X,...,X k,x n k C k,l E h k+ X,...,X k,x n. By Eq in ]again,wehave and hence. holds. E h k+ X,...,X k,x n k + σ 0

7 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 57. Proof of Theore. As in the proof of Theore., the first step is to rewrite nt H n T H /σ as W +, where W is a su of independent rando variables and is negligible. To this end, let Ax,...,x and g x be defined as in. and. and put Ā x,...,x Ax,...,x g x j..4 Let ϕt t 0 Jsds. Observing that by the integration by parts see, e.g.,, p. 65] TH n TH ϕ Hn y ϕ Hy dy, we have where j n σ THn TH W +,.5 W n ξ i, ξ i g X i / nσ, i and n n Ā x i,...,x i σ i < <i n n + ϕ Hn y ϕ Hy H n y Hy J Hy ] dy..6 σ Clearly, ξ,...,ξ n are i.i.d. rando variable with Eξ i 0 and n i Eξi. Set n n l Ā x i,...,x i.7 σ where n σ H n,l y n i < <i n, i j l,j {,...,} ϕ Hn,l y ϕ Hy H n,l y Hy J Hy ] dy,.8 i < <i n, i j l,j {,...,} I hx i,...,x i y ] + n Hy..9 Now by Proposition., Theore. follows by the next two leas.

8 58 H. Hu, Q.-M. Shao Lea. We have Lea.4 We have E 4 J0 +c 0 + 6c 0 σ 0 nσ E l 6 J0 +8c 0 σ 0 n σ..0.. The proofs of the two leas are presented in the next two subsections... Proof of Lea. Let σ EA X,...,X. We first show that σ J0 + c0 σ 0.. Let ξ hx,...,x and η be an independent copy of ξ. Then σ E Iξ x Hx J Hx Iξ y Hy J Hy dxdy J0 + c 0 Hx y HxHy J Hx J Hy dxdy Hx y HxHy dxdy by the fact that.4 iplies Jt J0 +c 0 for 0 t. Also note that Hx y HxHy dxdy This proves.. Let E E x y x y ξ x<y<η Hx Hy dxdy Iξ x Iη y dxdy Iξ xiy < ηdx dy Eη ξ Iξ <η /Eξ η Eξ σ 0.. h x,...,x { I hx,...,x y Hy } { I hx +,...,x y Hy } dy.4

9 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 59 and σ Eh X,...,X. Then σ σ 0..5 In fact, by the Hölder inequality σ E I hx,...,x y Hy dy I hx+,...,x y Hy dy Hy Hy dy E hx,...,x Eh X,...,X σ0, as desired. Next we obtain a siilar result for Eh X,...,X,Y,...,Y, where Y,...Y has the sae distribution as that of X,...,X.Letξ hx,...,x and η hy,...,y. Then Eh X,...,X,Y,...,Y E Iξ x Hx Iη x Hx dx 0.5E Iξ x Hx + Iη x Hx dx E Iξ x Hx dx A direct calculation shows that for x y E Iξ x Hx Iξ y Hy E Iξ x Hx Iξ y Hy dxdy. Hx Hx Hy + HxH y Hx + H xh y Hy + H xh y Hx Hx Hy+ 5HxHy HxH y + H y Hx Hy+ HxHy HxH y Hx Hy + HxHy 4Hx Hy. Thus by., we have Eh X,...,X,Y,...,Y 4Eξ 4σ0..6 To estiate E, write A + B,

10 60 H. Hu, Q.-M. Shao where n n A Ā x i,...,x i, σ i < <i n n + B ϕ Hn y ϕ Hy H n y Hy J Hy ] dy. σ Clearly, Following the proof in, pp ], we have and hence E EA i < <i n n j E E A + B..7 j Ā x i,...,x i n j n σ nn + E A j X,...,X n n n n σ σ nn + σ n + σ J0 +c 0 σ0 nσ..8 To finish the proof of.0 by.7 and.8, it suffices to show that Recall that EB n σ E EB c 0 σ 0 nσ..9 ϕ Hn y ϕ Hy H n y Hy J Hy ] dy. By the definition of ϕt and the Lipschitz condition.4 ϕt ϕs t sjs t s Ju+ s Js du 0.5c 0 t s, we have 0.0

11 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 6 Write E ϕ Hn y ϕ Hy H n y Hy J Hy ] dy 0.5c0 E Hn y Hy dy.. Hn y Hy dy Qn + R n, where n Q n h X i,...,x i,x j...,x j n R n h X i,...,x i,x j...,x j, here and denote the su over all pairs of -tuples i,...,i, j,...,j C n, having all indices different and at least one ordered index equal, respectively, and h is defined in.4. We further write n Q n h n X k,...,x k, C n, where n n +h h n X k,...,x k X i,...,x i,x j...,x j and + denotes the su over all suands h X i,...,x i,x j...,x j with {i,...,i,j,...,j }{k,...,k }. Observe that E E ] h n X,...,X X,...,X k n n +h E E X i,...,x i,x j...,x j k] X n n 4 E E ] h X,...,X X,...,X k E E ]. h X,...,X X,...,X k By Eq in ], we have for k, E E h X,...,X X,...,X k ] k σ and by soe siple calculation, we have hence E E h X,...,X X ] 0,

12 6 H. Hu, Q.-M. Shao EQ n n n k k k k nn + σ n k n k E E h n X,...,X X,...,X k ] k σ 8 σ0 n.. Next we give an upper bound of ERn. Notice that the nuber of ters in equals n n n Card n n n n n + + n + n n + n. n Hence, by.6 and the Minkowski inequality Thus, by. and. n + ER n 64 σ 0 n.. EB 0.5nc 0 σ E Q n + R n 0.5nc 0 σ 8 σ0 n + 64 σ0 n This proves.9... Proof of Lea.4 c 0 σ 0 nσ. Let l C + D, where n n C σ n D σ i < <i n, i j l,j {,...,} Ā x i,...,x i, ϕ Hn,l y ϕ Hy H n,l y Hy J Hy ] dy.

13 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 6 Clearly, we have E l EA C + EB D. Note that n n EA C E σ Ā x i,...,x i. i < <i n Following the proof in, p. 87] and siilarly to.8 wehave EA C σ nn + σ J0 +c 0 σ 0 nn + σ 4 J0 +c 0 σ0 n σ To coplete the proof of., it suffices to show that Let L L l Siilarly to., we have σ n E EB D + c 0 E EB D 0 c 0 σ 0 n σ. i < <i n, i j l,j {,...,}..4 ϕ Hn y ϕ Hy H n y Hy J Hy ] dy, ϕ Hn,l y ϕ Hy H n,l y Hy J Hy ] dy. ϕ Hn y ϕ H n,l y H n y H n,l y J H n,l y ] dy Hn y H n,l y J H n,l y J Hy ] dy Hn y H n,l y dy H n y H n,l y H n,l y Hy dy

14 64 H. Hu, Q.-M. Shao where 0.5c0 E Hn y H n,l y dy + c0 E Hn y H n,l y Hn,l y Hy dy 0.5c 0 L + c 0 L l,.5 L E L l E We first consider L. By.0 and.9, Hn y H n,l y dy n Hn y H n,l y dy, Hn y H n,l y Hn,l y Hy dy. i < <i n I hxi,...,x i,x n y ] Hy dy n h X,...,X i i,x j,...,x j,x n, where denotes the su over all pairs of -tuples i,...,i, j,..., j C n, and h x,...,x,x,...,x,x n I hx,...,x,x n y ] Hy I hx,...,x,x n y ] Hy dy. A direct calculation gives Card n. So, by.6 and the Minkowski inequality, we obtain L 44 σ0 n 4..6 As to L l, note that Hn y H n,l y Hn,l y Hy dy n I hxi,...,x i,x n y ] Hy i < <i n I hxj,...,x j y ] Hy dy j < <j n

15 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 65 n I hxi,...,x i,x n y ] Hy i < <i n I hxj,...,x j y ] Hy dy. j < <j n Thus, by the Minkowski inequality again n 4 n L l E I hx,...,x,x n y ] Hy I hxj,...,x j y ] Hy dy where j < <j n L l, + L l,,.7 L l, n E I hx n,...,x,x n y ] Hy I hxj,...,x j y ] Hy dy, L l, n E I n hx,...,x,x n y ] Hy I hxj,...,x j y ] Hy dy, 4 denotes the su over j,...,j C n, {,..., } and 4 denotes the su over all pairs i,...,i,n C n, and {j,...,j } {,..., }. Observing that n n Card n n we have L l, 4 n n 4 n E I hx,...,x,x n y ] Hy 4 I hx j,...,x j y ] Hy dy 4 n 4 n n σ 0 4 n 4 σ0 84 σ0 n..8 In the second inequality above, we used.6. To estiate L l,, note that

16 66 H. Hu, Q.-M. Shao E I hx,...,x,x n y ] Hy I hxj,...,x j y ] Hy dy E I hx,...,x,x n x ] Hx I hx,...,x,x n y ] Hy I hxj,...,x j x ] Hx I hxj,...,x j y ] Hy dxdy E I hx,...,x,x n x ] Hx I hx,...,x,x n y ] Hy E I hxj,...,x j x ] Hx I hxj,...,x j y ] Hy dxdy I hx,...,x,x n x ] Hx I hx,...,x,x n y ] Hy I hxj,...,x j x ] Hx I hxj,...,x j y ] Hy dxdy Hx Hx / Hy Hy / n / / Hx Hx Hy Hy dxdy n n σ0 n. Here we used the following estiate: I hxj,...,x j x ] Hx n n Hx Hx.

17 Non-unifor Berry Esseen Bounds for Weighted U-Statistics 67 Thus, we obtain and hence L l, 64 σ 0 n.9 L l 44 σ0 n,.0 by.7 and.8. This proves.4 by.5,.6, and.0, as desired. This copletes the proof of Lea.4. Acknowledgeents The research of Q.-M. Shao is partly supported by Hong Kong RGC GRF 6070, 044. References. Borovshikh, Y.V., Weber, N.C.: Asyptotic distributions for a class of generalized L-statistics. Bernoulli 6, Cai, Z.W.: Moderate deviations and large deviations of generalized L-statistics. Chin. Ann. Math., Ser. A, Chen, L.H.Y., Goldstein, L., Shao, Q.M.: Noral Approxiation by Stein s Method. Springer, Heidelberg 0 4. Chen, L.H.Y., Shao, Q.M.: Noral approxiation for nonlinear statistics using a concentration inequality approach. Bernoulli, Helers, R., Janssen, P., Serfling, R.J.: Berry Esseen bound and bootstrap results for generalized L-statistics. Scand. J. Stat. 7, Helers, R., Ruygaart, F.H.: Asyptotic norality of generalized L-statistics with unbounded scores. J. Stat. Plan. Inference 9, Major, P.: Asyptotic distributions for weighted U-statistics. Ann. Probab., O Neil, K.A., Redner, R.A.: Asyptotic distributions of weighted U-statistics of degree two. Ann. Probab., Rifi, M., Utzet, F.: On the asyptotic behavior of weighted U-statistics. J. Theor. Probab., Shapiro, C.P., Hubert, L.: Asyptotic norality of perutation statistics derived fro weighted sus of bivariate functions. Ann. Stat. 7, Serfling, R.: Approxiation Theores of Matheatical Statistics. Wiley, New York 980. Serfling, R.: Generalized L-, M-andR-statistics. Ann. Stat., Serfling, R.: Robust estiation via generalized L-statistics: theory, applications, and perspectives. In: Advances on Methodological and Applied Aspects of Probability and Statistics pp Taylor & Francis, London 998