getdocb6ce. 205KB Jun 04 2011 12:04:53 AM

El e c t ro n ic

Jo ur n

a l o

f P

r

o b_{a b i l i t y}

Vol. 12 (2007), Paper no. 52, pages 1402–1417. Journal URL

http://www.math.washington.edu/~ejpecp/

Edgeworth expansions for a sample sum from a finite

set of independent random variables

∗

Zhishui Hu

Department of Statistics and Finance,

University of Science and Technology of China, Hefei, 230026, China E-mail: huzs@ustc.edu.cn

John Robinson

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia,

E-mail: johnr@maths.usyd.edu.au Qiying Wang

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia,

E-mail: qiying@maths.usyd.edu.au URL: http://www.maths.usyd.edu.au/u/qiying

Abstract

Let{X₁,· · · , XN} be a set of N independent random variables, and let Sn be a sum ofn random variables chosen without replacement from the set{X1,· · ·, XN} with equal prob-abilities. In this paper we give a one-term Edgeworth expansion of the remainder term for the normal approximation ofSn under mild conditions.

Key words: Edgeworth expansion, finite population, sampling without replacement. AMS 2000 Subject Classification: Primary 60F05, 60F15; Secondary: 62E20. Submitted to EJP on October 16, 2006, final version accepted September 24, 2007.

1

Introduction and main results

Let _{X1,· · · , XN} be a set of independent random variables, µk = EXk, 1 ≤ k ≤ N. Let

R= (R1,· · · , RN) be a random vector independent ofX1,· · ·, XN, such thatP(R=r) = 1/N!

for any permutationr = (r1,· · · , rN) of the numbers 1,· · · , N, and put Sn =Pnj=1XRj,1≤

n _≤ N, that is for a sum of n random variables chosen without replacement from the set

{X1,· · · , XN} with equal probabilities.

In the situation that Xk = µk,1 ≤ k ≤ N, are (nonrandom) real numbers, the sample sum

Sn has been studied by a number of authors. The asymptotic normality was established by

Erd¨os and R´enyi (1959) under quite general conditions. The rate in the Erd¨os and R´enyi central limit theorem was studied by Bikelis (1969) and later H¨oglund (1978). An Edgeworth expansion was obtained by Robinson (1978), Bickel and van Zwet (1978), Schneller (1989), Babu and Bai (1996) and later Bloznelis (2000a, b). Extensions toU-statistics and, more generally, symmetric statistics can be found in Nandi and Sen (1963), Zhao and Chen (1987, 1990), Kokic and Weber (1990), Bloznelis and G¨otze (2000, 2001) and Bloznelis (2003).

In contrast to rich investigations for the case Xk = µk,1 ≤ k ≤ N, are (nonrandom) real

numbers, there are only a few results concerned with the asymptotics of generalSn discussed in

this paper. von Bahr (1972) showed that the distribution ofSn/√V arSn may be approximated

by a normal distribution under certain mild conditions. The rate of the normal approximation has currently been established by Zhao, Wu and Wang (2004), in which the paper improved essentially earlier work by von Bahr (1972). Along the lines of Zhao, Wu and Wang (2004), this paper discusses Edgeworth expansions for the distribution ofSn/√V arSn. Throughout the

paper, let

γ12=

1 N

N

X

k=1

EXkEXk2, αj =

1 N

N

X

k=1

(EXk)j, βj =

1 N

N

X

k=1

E(X_kj),

forj= 1,2,3,4, and

p=n/N, q= 1₋p, b= 1₋pα2.

Theorem 1. Suppose thatα1 = 0 and β2 = 1. Then, for all 1≤n < N,

sup

x

¯ ¯ ¯P(Sn/

√

nb_≤x)₋Gn(x)

¯ ¯ ¯

≤ C³∆1n+ (nq)−1

´

+ 3√nq log (nq) exp©

−nq δN

ª

, (1)

where C is an absolute constant,

Gn(x) = Φ(x)−

β3−3pγ12+ 2p2α3

6√nb3/2 Φ′′′(x),

withΦ(x) being a standard normal distribution,

∆1n= (nb)−1α4+

(nb2)−1 N

N

X

k=1

and

δN = 1− sup

δ0b/(9L0)≤|t|≤16 √

nb

¯ ¯ ¯

1 N

N

X

k=1

EeitXk ¯ ¯ ¯,

where _L0 = _N1 P

kE|Xk|3 and δ0 is so small that 192δ02+ 24δ0 ≤1−cos(1/16).

Property (1) improves essentially a result of Mirakhmedov (1983). The related result in Theorem 1 of Mirakhmedov (1983) depends on max1≤k≤NEX_k4. Note that it is frequently the case that

N−1PN

k=1EXk4 is bounded, but max1≤k≤NEXk4 tends to ∞. Also note that Corollary 1 of

Mirakhmedov (1983) requires limt→∞|EeitXk| ≤ ǫ <1. This condition is quite restrictive since

it takes away the most interesting case that theXk are all degenerate.

WhenXk=µk,1≤k≤N, are (nonrandom) real numbers, it is readily seen thatα2 = 1,b=q,

α3 =β3=γ12= _N1 PN_k=1µ3_k,

∆1N + (nq)−1 ≤3 (nq)−1

1 N

N

X

k=1

µ4_k.

In this case, the property (1) reduces to

sup

x |P(Sn/

√_nq

≤x)₋G1n(x)|

≤C(nq)−1 1 N

N

X

k=1

µ4_k+ 3√nq log (nq) exp©

−nq δNª,

whereG1n= Φ(x)+₆p√−_nqq _N1 PNk=1µ3kΦ′′′(x), which gives one of main results in Bloznelis (2000a,

b).

We next give a result complementary to Theorem 1. The result is better than Theorem 1 under certain conditions such as some of theXk’s are non-degenerate random variables and q is close

to 0.

Theorem 2. Suppose thatα1 = 0 and β2 = 1. Then, for all 1≤n≤N,

sup

x

¯ ¯ ¯P(Sn/

√

nb_≤x)₋Gn(x)

¯ ¯

¯ ≤ C∆2n+ 3 √

nlognexp©

−n δ1Nª, (2)

whereCis an absolute constant,Gn(x),L0 andδ0 are defined as in Theorem 1,∆2n= (nb2)−1β4 and

δ1N = 1− sup

δ0b/(9L0)≤|t|≤16 √

nb

1 N

N

X

k=1 ¯ ¯EeitXk

¯ ¯.

In the next section, we prove the main results. Throughout the paper we shall useC, C1, C2, ...

to denote absolute constants whose value may differ at each occurrence. Also,I(A) denotes the indicator function of a set A, ♯(A) denotes the number of elements in the set A, P

k denotes

PN

k=1, and Q

k denotes

QN

k=1. The symboliwill be used exclusively for √

2

Proofs of Theorems

Letµk=EXk and Ψ(t) =Eexp{itSn/√nb}. Recallα1 = 0 andβ2= 1. As in (4) of Zhao, Wu

and Wang (2004),

Ψ(t) = [Bn(p)]−1

Z

|ψ|≤π√nq

Y

k

Eρk(ψ, t)dψ, (3)

whereBn(p) =√2πnqGn(p), Gn(p) =

√

2πC_NnpnqN−n, X_k∗=Xk−pµk and

ρk(ψ, t) =qexp

½

−√ipψ_nq −ipµ√ kt

nb

¾

+pexp

½

iqψ

√_nq +itX_√ k∗ nb

¾

.

The main idea of the proofs is outlined as follows. We first provide the expansions and the basic properties for Q

kEρk(ψ, t) in Lemmas 1–4. In Lemma 5, the idea in von Bahr (1972) is

extended to give an expansion of Ψ(t) for the case n/N _≥1/2. The proofs of Theorems 1 and 2 are finally completed by virtue of the classical Esseen’s smoothing lemma.

In the proofs of Lemmas 1–4, we assume that ∆1n<1/16 and nq >256, where ∆1n is defined

as in Theorem 1. Throughout this section, we also define,

h(ψ, t) = Y

k

Eρk(ψ, t) and g(ψ, t) =

³

1 +i

3_{f(ψ, t)}

6√N

´

e−(ψ2+t2)/2,

wheref(ψ, t) =A0ψ3+ 3A1ψt2+A2t3, with

A0=

q₋p

√_pq, A1=

(1₋2pα2)√pq

pb , A2 =

β3−3pγ12+ 2p2α3

p1/2_b3/2 . (4)

Lemma 1. For_|ψ_{| ≤}(nq)1/4_{/4 and |t| ≤∆}−1/4

1n /4, we have

|h(ψ, t)₋g(ψ, t)_{| ≤} C[∆1n+ (nq)−1](s4+s8) exp{−s2/3}, (5)

where s2=ψ2+t2.

Proof. Define a sequence of independent random vectors (Uk, Vk), 1≤k≤N, by the conditional

distribution given X_k∗ as follows:

P(Uk=−p/√pq, Vk=−pµk/

p

pb_|X_k∗) =q, P(Uk=q/√pq, Vk=Xk∗/

p

pb_|X_k∗) =p.

Let Wk = ψUk +tVk. As in Lemma 1 of Zhao, Wu and Wang (2004), tedious but simple

calculations show that

EWk = 0,

X

k

EW_k2 =N(ψ2+t2), X

k

EW_k3 =N f(ψ, t),

X

k

EW_k4 _≤ 8X

k

Furthermore, if we letB_n2 =P

Now, by recalling thatWk are independent r.v.s and noting that

h(ψ, t) =EeiPkWk/

√

N _=Eei sP

kWk/Bn_,

it follows from (7)-(8) and the classical result (see, for example, Theorem 8.6 in Bhattacharya and Ranga Rao (1976)) that, for_|ψ_{| ≤}(nq)1/4/4 and _|t_{| ≤}∆−_1n1/4/4,

This proves (5) and hence completes the proof of Lemma 1. ✷

where_|θ1| ≤1 andN−1(ψ2+t2EV_k2)≤1/4.This, together with Taylor’s expansion ofeix, yields

Now (10) follows from (11), (13)-(16) and

X

It follows from (18) and Lemma 1 that for _|ψ_{| ≤}(nq)1/4/4 and _|t_{| ≤}1/4,

|h(ψ, t)₋e−(ψ2+t2)/2_{| ≤}C(∆1n+ (nq)−1)1/2e−ψ

2

/4_{. (20)}

Therefore, by noting

d g(ψ, t)

dt =−tg(ψ, t)− i 6√N

d f(ψ, t)

dt e

−(ψ2

+t2

)/2_,

simple calculations show that

¯ ¯ ¯

d h(ψ, t)

dt −

d g(ψ, t) dt

¯ ¯

¯ ≤ Λ(ψ, t) +t|h(ψ, t)−g(ψ, t)|

+ 1

6√N

¯ ¯ ¯ ¯

d f(ψ, t) dt

¯ ¯ ¯ ¯|

h(ψ, t)₋e−(ψ2+t2)/2_|

≤ C(1 +ψ12)¡

(nq)−1+ ∆1n¢e−ψ

2

/4_,

where we have used (5), (10), (19) and (20). The proof of Lemma 2 is now complete. _✷

Lemma 3. Assume that_|ψ_{| ≤}(nq)1/4_{/4. Then,}

|h(ψ, t)_{| ≤} C∆1ne−(ψ

2

+t2

)/4_{, (21)}

for ∆−_1n1/4/4_{≤ |}t_{| ≤}(∆∗)−1/16, where

∆∗ = N−1X

k

|µk|3/

√

nb+N−1X

k

E_|Xk−pµk|3/(√nb3/2).

Assume that(nq)1/4/4_{≤ |}ψ_{| ≤}π√nq. Then,

|h(ψ, t)_{| ≤} C(nq)−4, (22)

for all _|t_{| ≤} δ0(∆∗)−1, where δ0 is so small that 192δ20+ 24δ0 ≤ 1−cos(1/16). If in addition |t_{| ≤}1/4, then we also have

¯ ¯ ¯

d h(ψ, t) dt

¯ ¯

¯ ≤ C(nq)

−4_{. (23)}

Proof. The proof of this lemma follows directly from an application of Lemmas 1-3 in Zhao, Wu and Wang(2004). The choice of δ0 can be found in the proof of Lemma 2 in Zhao, Wu and

Wang(2004). We omit the details. ✷

Lemma 4. Assume that n/N _≥1/2and∆2n≤(nq)−1/25, where∆2n is defined as in Theorem 2. Then, for_|t_{| ≤ 15}1 √nb3/2/_L0,

|h(ψ, t)_{| ≤}exp_{−Ct2_}, (24)

where _L0 = _N1 P

Proof. We only need to note that the condition ∆2n≤(nq)−1/25 implies that

5qα2≤5qβ41/2 ≤b=

1 N

X

k

Var(Xk) +qα2,

that is, _N1 P

kVar(Xk)≥(4/5)b. Then (24) is obtained by repeating the proof of Lemma 4 in

Zhao, Wu and Wang(2004). ✷

Lemma 5. Assume that n/N _≥ 1/2 and ∆2n ≤ 1. Then, for |u| ≤

1 16min

©

(n/β4)1/4,1₈b√n/L0ª,

¯ ¯

¯Eexp{iuSn/ √

n_{} −}exp_{−bu2/2_}

Ã

1 +i

3_u3_b3/2

6√N A2

! ¯ ¯ ¯

≤Cn−1β4(u2+u4+u6b) exp{−0.3b u2}, (25)

where _L0 = _N1 P

kE|Xk|3 is defined as in Theorem 1.

Proof. Write, for 1_≤k_≤N,

fk(u) =Eexp{iuXk/√n}, bk(u) = exp{u2/(2n)}fk(u)−1

and Bj = ((−1)j+1/j)PNk=1b

j

k(u) for 1≤j ≤n. As in von Bahr (1972), we have

exp_{u2/2_}Eexp_{iuSn/√n}=

X

ij≥0,1≤j≤n

n

Y

j=1

(pjBj)ij

ij!

C_N,n,Pn

j=1jij, (26)

where

CN,n,r =

( _¡

N−r n−r

¢. ³

pr¡_N

n

¢´

, r_≤n;

0, r > n.

In view of (28) of Zhao, Wu and Wang (2004), forr >0, CN,n,r ≤1, and forn≥4 andr≤n,

CN,n,r ≥1−r2/n. (27)

To prove (25) by using (26), we need some preliminary results. Writeβjk =EXkj, j = 2,3,4. Recall thatN−1

P

kβ2k= 1. We have thatβ4≥1 and by Taylor’s

expansion, for_|u_{| ≤ 16}1(n/β4)1/4,

exp_{u2/(2n)_}= 1 + 1 2nu

2₊ 1

8n2u 4₊ θ4

n3u

6_{, where}

|θ_{4| ≤}1/24,

fk(u) = 1 +

iu

√

nµk− u2 2nβ2k−

iu3 6n3/2β3k+

θ5u4

n2 β4k, where |θ5| ≤1/24.

Now, by noting that_|µk| ≤ β₄1_k/4 ≤1 +β4k,|β2k| ≤ β₄1_k/2 ≤1 +β4k and |β3k| ≤ β₄3_k/4 ≤1 +β4k,

we obtain that, for_|u_{| ≤ 16}1(n/β4)1/4,

bk(u) = exp{u2/(2n)}fk(u)−1

= _√iu nµk+

u2

2n(1−β2k) + iu3

where_|R1k(u)| ≤(1 +β4k)u4/n2.Furthermore, by noting that

β₄1/_k4_|u_|/√n_≤(N β4)1/4|u|/√n≤1/8

sincen/N _≥1/2 and _|u_{| ≤ 16}1(n/β4)1/4, we have

b2_k(u) = ₋u

2

nµ

2

k+

iu3

n3/2µk(1−β2k) +R2k(u), (29)

b3_k(u) = ₋iu

3

n3/2µ 3

k+R3k(u), (30)

|bj_k(u)_{| ≤} 3(1/2)j−4(1 +β4k)u4/n2, for j≥4, (31)

where _|R2k(u)| ≤ 3(1 +β4k)u4/n2 and |R3k(u)| ≤ 4(1 +β4k)u4/n2. Recalling that Pkµk =

P

k(1−β2k) = 0, it follows from (28)–(31) that, for|u| ≤ ₁₆1(n/β4)1/4,

pB1 =p X

k

bk(u) =−

iu3

6√nβ3+θ6β4u

4_{/n, (32)}

p2B2 = (−p2/2) X

k

b2_k(u) = u

2_p

2 α2+ iu3p

2√nγ12+θ7β4u

4_{/n, (33)}

p3B3 = (p3/3) X

k

b3_k(u) =₋ip

2_u3

3√nα3+θ8β4u

4_{/n, (34)}

|pjBj| ≤

X

k

|bj_k(u)_{| ≤}6β4u4(1/2)j−4/n, for j≥4. (35)

where _|θ6| ≤ 2, |θ7| ≤ 3 and |θ8| ≤ 3. By virtue of (35), it is readily seen that, for |u| ≤ 1

16(n/β4)1/4,

n

X

j=4

|pjBj| ≤12β4u4/n, (36)

Noting that_|α_3|+_|β_3|+_|γ_{12| ≤}3_L0 and recalling that ∆2n = (nb2)−1β4 ≤1, it follows easily

from (32)–(34) and (36) that, for _|u_{| ≤ 16}1 min©

(n/β4)1/4,1₈b√n/L0ª,

n

X

j=1

|pjBj| =

1 2p α2u

2_+θ 9

u3_L0

√

n +θ10 u4β4

n

= 1

2p α2u

2_+θ

11b u2, (37)

where_|θ_{9| ≤}3,_|θ_{10| ≤}20 and_|θ_{11| ≤}0.2. Also, if we letL(u) =P3

j=1pjBj−pα2u2/2, we have

|L(u)_{| ≤}0.2bu2,

¯ ¯ ¯ ¯ ¯

L(u) +i u

3_b3/2

6√N A2

¯ ¯ ¯ ¯ ¯

≤8β4u4/n, (38)

whereA2 is defined as in (4). As in the proof of (6), we may obtain

A2₂ =³N−1X

k

EV_k3´2_≤N−2X

k

EV_k2X

k

This together with (38) yields, for_|u_{| ≤ 16}1 (n/β4)1/4,

L2(u)_≤h|u|

3_b3/2

6√N |A2|+ 8 nβ4u

4i2

≤u6bβ4/n+ 128β42u8/n2≤β4(u4+u6b)/n. (39)

We are now ready to prove (25) by using (26). Rewrite (26) as

exp_{u2/2_}Eexp_{iuSn/√n}=I1+I2+I3, (40)

where

I1 = X

n

Y

j=1

(pjBj)ij

ij!

C_N,n,Pn

j=1jij,

I2 =

X

ij≥0,1≤j≤3 3 Y

j=1

(pjBj)ij

ij!

³

C_N,n,P3

j=1jij −1

´

,

I3 =

X

ij≥0,1≤j≤3 3 Y

j=1

(pjBj)ij

ij!

,

where the summation in the expression ofI1 is over all ij ≥0, j= 1,2,3 andij >0 for at least

onej= 4,_{· · ·}, n. As in Mirakhmedov (1983), it follows from (36)-(37) that

|I_{1| ≤} expn

3 X

j=1 |pjBj|

o³

expn

n

X

j=4 |pjBj|

o −1´

≤

n

X

j=4

|pjBj|exp

nXn

j=1 |pjBj|

o

≤ Cn−1β4u4exp{pα2u2/2 + 0.2bu2}.

As forI2, it follows easily from (27) and (37) that

|I_{2| ≤} Cn−1 X

ij≥0,1≤j≤3 3 Y

j=1 |pj_B

j|ij

ij!

³X3

j=1

jij

´2

≤ Cn−1 X

ij≥0,1≤j≤3 3 Y

j=1

i2_j|pjBj|ij

ij!

≤ Cn−1expn

3 X

j=1 |pjBj|

o_X3

j=1 ¡

|pjBj|+|pjBj|2¢

We next estimateI3. Recalling thatb= 1−pα2 and noting thatI3 =e

Combining (40) and all above facts forI1-I3, we obtain

¯

which implies (25). The proof of Lemma 5 is now completed. ✷

After these preliminaries, we are now ready to prove the theorems.

Proof of Theorem 1. Without loss of generality, assume that nq >256 and ∆1n<1/16. Write

T−1_{= ∆}

whereA2 is defined as in Lemma 1. We shall prove,

lemma, simple calculations show that

sup

x

¯ ¯ ¯P(Sn/

√

nb_≤x)₋Gn(x)

¯ ¯ ¯

≤ ³ Z

|t|≤1/4

+

Z

1/4≤|t|≤T1 +

Z

T1≤|t|≤T

|Ψ(t)₋gn(t)|

|t_| dt+Cm T−

1

≤ C(∆1n+ (nq)−1) + 3√nq log(nq) exp©−nq δNª,

whereT1= min{δ0(∆∗)−1, T}, which implies (1) and hence Theorem 1.

We next prove (41)-(43). Throughout the proof, we writes2 _=ψ2_+t2_.

Consider (42) first. Note that gn(t) = √1_2πR_−∞∞ g(ψ, t)dψ.It is readily seen that

|Ψ(t)₋gn(t)| ≤II1+II2+II3+II4, (44)

where

II1 = [Bn(p)]−1

Z

|ψ|≤(nq)1/4_/4|

h(ψ, t)₋g(ψ, t)_|dψ,

II2 = [Bn(p)]−1

Z

(nq)1/4_/4

≤|ψ|≤π√nq|

h(ψ, t)_|dψ,

II3 = [Bn(p)]−1

Z

|ψ|≥(nq)1/4_/4

³

1 +|f(ψ, t)| 6√N

´

e−s2/2dψ,

II4 = ¯ ¯

¯[Bn(p)]

−1

−(2π)−1/2¯¯ ¯

Z ∞

−∞

³

1 +|f(ψ, t)| 6√N

´

e−s2/2dψ.

To estimateIIj, j= 1,2,3,4, we first recall that, by (18), for allψ andt,

|f(ψ, t)_{| ≤} 3s3√N(∆1n+ (nq)−1)1/2 ≤

√

N s3, (45)

and by virtue of Stirling’s formula,

1_≤√2π/Bn(p)≤1 + 1/nq. (46)

In view of (45) and (46), it is readily seen that

II3+II4 ≤C(nq)−1(1 +t6)e−t

2

/3_{. (47)}

By using (22), we have

II2≤C(nq)−3. (48)

As forII1, if|t| ≤min{∆1−n1/4/4, δ(∆∗)−1}, Lemma 1 implies that

II1 ≤C(∆1n+ (nq)−1)(1 +t8)e−t

2

/4_{; (49)}

if ∆−_1n1/4/4_{≤ |}t_{| ≤}δ(∆∗)−1, then it follows from (21) and (45) that

II1 ≤ Z

|ψ|≤(nq)1/4

/4|

h(ψ, t)_|dψ+

Z

|ψ|≤(nq)1/4

/4 ³

1 +|f(ψ, t)| 6√N

´

e−s2/2dψ

≤ C∆1ne−t

2

/4_+C

1(1 +|t|3)e−t

2

/2 _≤C∆ 1ne−t

2

Taking (47)-(50) into (44), we obtain the required (42).

Taking these estimates into (51), we obtain for_|t_{| ≤}1/4,

implies that if δ0(∆∗)−1 ≤ |t| ≤T, thenδ0b/(9L0)≤ |t|/√nb≤16√nb and hence

which yields (43). The proof of Theorem 1 is complete. ✷

Proof of Theorem 2. Without loss of generality, assume ∆2n ≤ 1. We first prove the property

(2) forn/N _≥1/2 and ∆2n≤(nq)−1/25. proof of Theorem 1, it follows from Esseen’s smoothing lemma that

whereGn(p) =√2πC_NnpnqN−n. This, together with the fact thatδ0b/(5L0)≤ |t|/√nb≤16√nb

whenever T₂∗ _{≤ |}t_{| ≤}T∗, implies that

|Ψ(t)_{| ≤} √2π(Gn(p))−1

Y

k

¡

q+p_|EeitXj/√nb_|¢

≤ √2π(Gn(p))−1exp

n

pX

k

¡

|EeitXj/√nb_{| −1}¢o

≤ 3√nexp_{−nδ1N},

where we have used the inequality √π/2 _≤ √nqGn(p) < 1 (see, for instance, Lemma 1 in

H¨oglund(1978)). This proves (57) forq >0. Ifq = 0, thenN =nhence by the independence of Xk,

|Ψ(t)_| = Y

k

|EeitXj/√nb_{| ≤exp}n X

k

¡

|EeitXj/√nb_{| −1}¢o

≤exp_{−nδ1N}.

This implies that (57) still holds forq= 0. We have now completed the proof of (57) and hence (2) forn/N _≥1/2 and ∆2n≤(nq)−1/25.

Note thatβ4 ≥1, ∆1n ≤17 ∆2n and b≥q≥1/2 if n/N ≤1/2. We have that ∆1n+ (nq)−1≤

42 ∆2n, whenever n/N ≤ 1/2 or ∆2n > (nq)−1/25. Based on this fact, by using a similar

argument to that above and that in the proof of Theorem 1, we may obtain (2) forn/N _≤1/2 or ∆2n>(nq)−1/25, as well. The details are omitted. The proof of Theorem 2 is now complete. ✷

References

[1] Babu, G. J. and Bai, Z. D. (1996). Mixtures of global and local Edgeworth expansions and their applications. J. Multivariate. Anal.59 282-307. MR1423736

[2] von Bahr, B. (1972). On sampling from a finite set of independent random variables. Z. Wahrsch. Verw. Geb. 24279–286. MR0331471

[3] Bhattacharya, R. N. and Ranga Rao, R. (1976). Normal approximation and asymptotic expansions. Wiley, New York. MR0436272

[4] Bickel, P. J. and von Zwet, W. R. (1978). Asymptotic expansions for the power of distribution-free tests in the two-sample problem. Ann. Statist.6 937-1004. MR0499567

[5] Bikelis, A. (1969). On the estimation of the remainder term in the central limit theo-rem for samples from finite populations. Studia Sci. Math. Hungar. 4 345-354 in Russian. MR0254902

[6] Bloznelis, M. (2000a). One and two-term Edgeworth expansion for finite population sample mean. Exact results, I. Lith. Math. J. 40(3)213-227. MR1803645

[8] Bloznelis, M. (2003). Edgeqorth expansions for studentized versions of symmetric finite population statistics. Lith. Math. J.43(3)221-240. MR2019541

[9] Bloznelis, M. and G¨otze, F. (2000). An Edgeworth expansion for finite population U -statistics.Bernoulli6 729-760. MR1777694

[10] Bloznelis, M. and G¨otze, F. (2001). Orthogonal decomposition of finite population statistic and its applications to distributional asymptotics. Ann. Statist.29899-917. MR1865345

[11] Erd¨os, P. and Renyi, A. (1959). On the central limit theorem for samples from a finite population. Fubl. Math. Inst. Hungarian Acad. Sci.4 49-61. MR0107294

[12] H¨oglund, T.(1978). Sampling from a finite population. A remainder term estimate.Scand. J. Statistic.5 69-71. MR0471130

[13] Kokic, P. N. and Weber, N. C. (1990). An Edgeworth expansion for U-statistics based on samples from finite populations. Ann. Probab. 18390-404. MR1043954

[14] Mirakjmedov, S. A. (1983). An asymptotic expansion for a sample sum from a finite sample.

Theory Probab. Appl. 28(3)492-502.

[15] Nandi, H. K. and Sen, P. K. (1963). On the properties of U-statistics when the observa-tions are not independent II: unbiased estimation of the parameters of a finite population.

Calcutta Statist. Asso. Bull 12993-1026. MR0161418

[16] Robinson, J. (1978). An asymptotic expansion for samples from a finite population. Ann. Statist.6 1004-1011. MR0499568

[17] Schneller, W. (1989). Edgeworth expansions for linear rank statistics. Ann. Statist. 17 1103–1123. MR1015140

[18] Zhao, L.C. and Chen, X. R. (1987). Berry-Esseen bounds for finite populationU-statistics.

Sci. Sinica. Ser. A30 113-127. MR0892467

[19] Zhao, L.C. and Chen, X. R. (1990). Normal approximation for finite populationU-statistics.

Acta Math. Appl. Sinica6 263-272. MR1078067