An Edgeworth expansion for finite- population U ... - Project Euclid

Bernoulli 6(4), 2000, 729±760

An Edgeworth expansion for ®nitepopulation U-statistics È TZ E 2 M I N D AU G A S B L O Z N E L I S 1 and FR I ED RI C H G O 1

Department of Mathematics, Vilnius University, Naugarduko 24, Vilnius 2006, Lithuania. E-mail: [email protected] 2 FakultaÈ t fuÈr Mathematik, UniversitaÈt Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. E-mail: [email protected] Suppose that U is a U -statistic of degree 2 based on N random observations drawn without replacement from a ®nite population. For the distribution of a standardized version of U we construct an Edgeworth expansion with remainder O(N ÿ1 ) provided that the linear part of the statistic satis®es a CrameÂr type condition. Keywords: central limit theorem; Edgeworth expansion; ®nite population; U -statistic; sampling without replacement

1. Introduction and results Let A ˆ fa1 , . . . , a n g denote a population of size n and let H : A 3 A ! R denote symmetric function of its two arguments. By X 1 , . . . , X N , N < n, we denote random variables with values in A such that X ˆ fX 1 , . . . , X N g represents a random sample from A of size N drawn without replacement, i.e. PfX ˆ Bg ˆ ( nN )ÿ1 for any subset B  A of size N . We shall investigate the second-order asymptotics of the distribution of the statistic X H (X i , X j ): Uˆ 1 N ÿ1 äÿ3=2 g ‡ äÿ3=2 N ÿ1 maxjG9(x)j: x

3

6

By Chebyshev's inequality and the inequality EjË m j  m Ej g2 (X 1 , X 2 )j3, PfjË m j > N ÿ1 äÿ3=2 g < ä9=2 N 3 EjË m j3  äÿ3=2 ã3 N ÿ3=2 ln6 N : Finally, using the bound jG9(x)j  â4 =q ‡ ã2 we obtain Ä  Ä1 ‡ N ÿ1 äÿ3=2 (â4 =q ‡ ã2 ‡ ã3 ):

(3:5)

Therefore, in order to prove the theorem it suf®ces to bound Ä1 . Let k be an integer approximately equal to (N ‡ m)=2. Put I 0 ˆ fm ‡ 1, . . . , N g,  J 0 ˆ Ù n nI 0 , J 1 ˆ J 0 [ fm ‡ 1, . . . , kg and J 2 ˆ J 0 [ fk ‡ 1, . . . , N g. Given A,  de®ne (random) subpopulations A i ˆ fA k , k 2 J i g, i ˆ 0, 1, 2, and let A i be random variables uniformly distributed in A i , i ˆ 0, 1, 2, indpendent of í. Write v1 (a) ˆ

N X

g 2 (a, A j ),

jˆ k‡1

H ˆ N ä=(32q ÿ1 N (È1 ‡ È2 ) ‡ 1),

v2 (a) ˆ

k X

g 2 (a, A j ),

(3:6)

jˆ m‡1

È i ˆ E jv i (Ai )j,

i ˆ 1, 2:

Notice that È1 is a function of the random variables A k‡1 , . . . , A N , and that È2 is a function of A m‡1 , . . . , A k . Step 2. Split the sample as follows. Put X j ˆ A j , for m , j < N. The rest of the sample, X 1 , . . . , X m , is obtained by simple random sampling without replacement from the (random) subpopulation A0 . An application of Prawitz's (1972) smoothing lemma conditionally, given X m‡1 , . . . , X m , or equivalently, given A m‡1 , . . . , A N , gives

An Edgeworth expansion for ®nite-population U-statistics F1 (x‡)


1 ÿ EVP 2

… R

efÿxtg

1 H

efÿxtg

1 H

…

R

737

  t K f 1 (t) dt, H   ÿt K f 1 (t) dt, H

(3:7) (3:8)

where 2K(s) ˆ K 1 (s) ‡ iK 2 (s)=(ðs); see, for example, Bentkus et al. (1997). Here K 1 (s) ˆ Ifjsj < 1g(1 ÿ jsj)

and

K 2 (s) ˆ Ifjsj < 1g(1 ÿ jsj)ðs cot(ðs):

Combining (3.7) and the inversion formula, G(x) ˆ

1 i ‡ lim VP 2 2ð M!1

… j tj< M

^ efÿtxgG(t)

dt , t

(3:9)

we obtain (see, for example, Bentkus et al. 1997) F1 (x‡) ÿ G(x) < EI 1 ‡ EI 2 ‡ EI 3 ,   … 1 ÿ1 t f 1 (t) dt, efÿxtgK 1 I1 ˆ H 2 H R   … i t dt ^ VP efÿxtgK 2 ( f 1 (t) ÿ G(t)) , I2 ˆ 2ð H t R     … i t ~ dt , VP efÿxtg K 2 ÿ 1 G(t) I3 ˆ 2ð H t R

(3:10)

where VP means also that one should take lim M!1 if necessary. Combining (3.8) and (3.9), we obtain a bound for G(x) ÿ F1 (ÿx) similar to (3.10). We shall bound F1 (x‡) ÿ G(x) only. To this end, we prove that 1=2

jEI 1 j ‡ jE(I 2 ‡ I 3 )j  N ÿ1 (â4 =q ‡ äÿ1 (äÿ1 ‡ q ÿ1 ) ‡ äÿ2 q ÿ2 (ã2 ‡ ã2 ) ‡ ã4 ):

(3:11)

The analogous bound for G(x) ÿ F1 (xÿ) can be derived in the same way. Using these bounds, (3.5) and the inequality ä > r (see Lemma 2.1), we obtain the estimate of the theorem. in the remaining part of the prooof we verify (3.11). Step 3: Estimate for jEI 1 j. We shall replace the random bound H in the integral I 1 by a non-random one and K 1 (t= H) by 1. We have jEI 1 j < jEI 4 j ‡ EI 5 , where   … t ÿ1 f 1 (t) dt, efÿtxgK 1 Z ˆ ft 2 R : jtj < H 1 g, I4 ˆ H H Z   … … t j f 1 (t)j dt < H ÿ1 K1 j f 1 (t)j dt: I 5 ˆ H ÿ1 H H1 2,

where Z 1 . . . , Z j are independent and centred ransom variables. We apply this inequality to  the sum æ m (A k ) ± cf. (4.15) ± conditionally given A, !3=2 n n X X 3 3 2 j g2 (A k , A l )j ‡ pq g2 (A k , A l ) : Ef m‡1,:::, ng jæ m (A k )j  pq lˆ m‡1

lˆ m‡1

Finally, using HoÈlder's inequality, we bound the second sum above by !3=2 n n X X 2 g 2 (a, A l ) < (n ÿ m)1=2 j g2 (a, A l )j3 , a 2 A, lˆ m‡1

lˆ m‡1

thus arriving at (5.4).

h

Lemma 5.2. For each 0 , d , ð and x, y 2 R, and â(x) de®ned in Section 3.1, we have   2ð 4 ‡ 1 y2 , where v[d] ( y) ˆ 1 ‡ pq jâ(x ‡ y)j2 < u[d] (x)v[d] ( y), d È(d) and where the function u[d] id de®ned in (4.8). Proof. In the case where jxj > ð ‡ d, we have u[d] (x) ˆ 1 and the desired inequality follows from the simple bound jâ(x ‡ y)j < 1. In the case where jxj , ð ‡ d, we apply the mean value theorem to obtain jcos(x ‡ y) ÿ cos(x)j < jE sin(x ‡ è1 y) yj < (jxj ‡ j yj)j yj < cx 2 ‡ (cÿ1 ‡ 1) y 2 :

(5:6)

In the last step we applied the inequality jxyj < cx 2 ‡ cÿ1 y 2, with c . 0. Combining (5.6) and the indentity jâ(x ‡ y)j2 ˆ 1 ÿ 2 pq(1 ÿ cos(x ‡ y)), we obtain

756

M. Bloznelis and F. GoÈtze jâ(x ‡ y)j2 < 1 ÿ 2 pq(1 ÿ cos(x) ÿ cx 2 ÿ (cÿ1 ‡ 1) y 2 ):

Now invoking (5.15), we obtain jâ(x ‡ y)j2 < w1 ‡ w2 ,

w1 ˆ 1 ÿ pq(È(d) ÿ 2c)x 2 ,

w2 ˆ (cÿ1 ‡ 1)2 pqy 2 :

But 1 ÿ pqx 2 È(d) > d=ð, for jxj < ð ‡ d. Hence, w1 . d=ð and therefore w1 ‡ w2 < h w1 (1 ‡ ðd ÿ1 w2 ). Choosing c ˆ È(d)=4 completes the proof of the lemma. Lemma 5.3. Assume that â2 ˆ 1. For every s, t 2 R and 0 , d , ð, we have     t2 b1 b21 2 2 ‡ s (1 ÿ 2c[d] ), ; 2 : c[d] ˆ max E Z (A1 )I [d] (A1 ) > N d d Here Z(a) ˆ tg 1 (a) ‡ s and I [d] (a) ˆ If H 1 j g1 (a)j , dg, for a 2 A. Remark. A similar inequality was used by HoÈglund (1978), where the constatnt (corresponding to c[d] ) was not speci®ed. For our purposes the dependence of c d on the parameters b1 and d is important and thus we include the proof. Proof. Denote K [d] ˆ fk : I [d] (a k ) ˆ 0g. Clearly, for r . 0, X X r r ÿr 1< j g1 (a k ) H 1 =dj r < nâ r âÿ jK [d] j ˆ 3 b1 d : k2K [d]

(5:7)

k2K [d]

Furthermore, since E Z 2 (A1 ) ˆ t 2 N ÿ1 ‡ s 2 , we have E Z 2 (A1 )2 I [d] (A1 ) ˆ

t2 ‡ s 2 ÿ W nÿ1 , N

Wˆ

X

Z 2 (a k ):

(5:8)

k2K [d]

The inequality (a ‡ b)2 < 2a2 ‡ 2b2 implies W < 2W 1 ‡ 2W 2 , where X t2 2=3 W2 ˆ t 2 g 21 (a k ) < n2=3 â3 jK [d] j1=3 : W 1 ˆ s 2 jK [d] j, N k2K [d]

In the last step we applied HoÈlder's inequality to obtain 0 12=3 X X g 21 (a k ) < @ j g 31 (a k )jA jK [d] j1=3 : k2K [d]

k2K [d]

Now, (5.7) (with r ˆ 2) implies W 1 < s 2 nc d . Furthermore, (5.7) (with r ˆ 3) implies W 2 < t 2 N ÿ1 nc d . These inequalities, combined with (5.8), complete the proof. h Lemma 5.4. Assume that â2 ˆ 1 and that (3.3) holds. For jtj < H 1 and jsj < ðô, the inequalities (4.9) hold true. Proof. Throughout this proof we use the notation introduced in Section 4. Fix B  Ù m . By Lemma 5.2,

An Edgeworth expansion for ®nite-population U-statistics ZB ˆ

Y

jâ(z k ‡ tæ m (A k ))j < ç1 ç2 ,

k2B

ç21 ˆ

Y

757 Y

ç22 ˆ

u[1] (z k ),

k2 B

v[1] (tæ m (A k )):

k2B

Using the inequality 1 ‡ x < expfxg, we obtain ç22 < expf pqt 2 k B jBj=N g and therefore, (1 ÿ I B )ç2 < g B (t). Finally, combining the inequalities Z B < 1 and Z B < ç1 ç2, we obtain Z B ˆ I B Z B ‡ (1 ÿ I B ) Z B < I B ‡ (1 ÿ I B )ç1 ç2 < I B ‡ Ø B , thus proving the ®rst inequality in (4.9). Here the random variables Ø B ˆ ç1 g B (t), k B and I B are de®ned in (4.7) and the function g B (t) is given by (4.8). Clearly, Lemma 5.2 (with x ˆ z k and y ˆ 0) implies the second inequality of (4.9). To prove the third and last one, observe that by HoÈlder's inequality we have E i1 ,:::,i4 Ø B  (E i1 ,:::,i4 Ø2B )1=2 and thus, it suf®ces to show E i1 ,:::,i4 Ø2B < F 2B ,

for every i1 , . . . , i4 2 Ù n nB:

(5:9)

To prove (5.9) note that the inequalities jtj < H 1 and jsj < ðô imply u[1] (z k ) < w(A k ),

w(A k ) ˆ 1 ÿ

pq È(1)z2k I [1] (A k ), 2

k 2 Ù n,

where we denote I [1] (a) ˆ If H 1 j g 1 (a)j , 1g. Therefore, Ø2B ˆ g 2B (t)ç21 < g 2B (t)ç,

where ç ˆ

Y

w(A k ):

(5:10)

k2 B

Denote D1 ˆ fi1 , . . . , i4 g and D2 ˆ Ù n nD1 . By Theorem 4 of Hoeffding (1963), j Bj

E i1 ,:::,i4 ç < w ,

where w ˆ 1 ÿ

pq È(1)Ã , 2

à ˆ

1 X 2 z I [1] (A k ): jD2 j k2D k

Below we construct the following lower bound for Ã, ! 2 9 s 1 t2 ‡ : Ã > q N 10

(5:12)

Combining (5.11), (5.12) and the inequality 1 ‡ x < expfxg expfÿ0:45 pqÈ(1)(t 2 ‡ s 2 =q)jBjN ÿ1 g. Now (5.9) follows from (5.10). Let us prove (5.12). We have à ˆ

n 1 Ez2 I [1] (A1 ) ÿ M, nÿ4 1 nÿ4

(5:11)

2

Mˆ

X k2 D1

we

z2k I [1] (A k ):

obtain

ç
v2 È(u), 2   ð1=2 n < s N (1 ÿ s) nÿ N (2ðs(1 ÿ s)n)1=2 < 1, N 2

0 < u < ð, with s ˆ

N , n

(5:15) (5:16)

where 1 < N < n.

Acknowledgements The research for this paper was supported by Sonderforschungsbereich 343 in Bielefeld. The authors would like to thank V. Bentkus for discussions and comments.

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Zhao, L.C. and Chen, X.R. (1987) Berry±Esseen bounds for ®nite-population U -statistics. Sci. Sinica Ser. A, 30, 113±127. Zhao, L.C. and Chen, X.R. (1990) Normal approximation for ®nite-population U -statistics. Acta Math. Appl. Sinica, 6, 263±272. Received November 1997 and revised June 1999