RESEARC H Open Access
Berry-Esséen bound of sample quantiles for
negatively associated sequence
Wenzhi Yang
1
, Shuhe Hu
1*
, Xuejun Wang
1
and Qinchi Zhang
2
* Correspondence: hushuhe@263.
net
1
School of Mathematical Science,
Anhui University Hefei 230039, PR
China
Full list of author information is
available at the end of the article
Abstract
In this paper, we investigate the Berry-Esséen bound of the sample quantiles for the
negatively associated random variables under some weak conditions. The rate of
normal approximation is shown as O(n
-1/9
).
2010 Mathematics Subject Classification: 62F12; 62E20; 60F05.
Keywords: Berry-Ess?é?en bound, sample quantile, negatively associated
1 Introduction
Assume that {X
n
}

is the unique solution x of F (x-) ≤ p ≤ F(x), then for any ε >0,
F( ξ
p
− ε) < p < F(ξ
p
+ ε)
.
For a sample X
1
, X
2
, , X
n
, n ≥ 1, let F
n
represent the empirical distribution function
based on X
1
, X
2
, , X
n
,whichisdefinedas
F
n
(x)=
1
n

n

X
k
, k ∈ A
)
, g
(
X
k
, k ∈ B
))
≤ 0
.
A sequence of random variables {X
i
}
i≥1
is said to be NA if for every n ≥ 2, X
1
, X
2
, ,
X
n
are NA.
From 1960s, many authors have obtained the asymptotic results for the sample quan-
tiles, including the well-known Bah adur representation. Bahadur [1] firstly intro duced
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
http://www.journalofinequalitiesandapplications.com/content/2011/1/83
© 2011 Yang et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License (http://creativecomm ons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in a ny medium,

n
}
n≥1
be a sequence of i.i.d. random variables. Sup-
pose that in a neighborhood of ξ
p
, F possesses a positive continuous density f and a
bounded second derivative F″. Then
sup
−∞<t<∞





P

n
1/2
(ξ
p,n
− ξ
p
)
[p(1 −p)]
1/2
/f (ξ
p
)
≤ t

n
−
1
4
log n
)
[[13], Chapter 3]. For other papers about Berry-Esséen
bound, for example, under the association sample, Cai and Roussas [14,15] studied the
Berry-Esséen bounds for the smooth estimator of quantiles and the smooth estimator
of a distribution function, respectively; Yang [16] obtained the Berry-Esséen bound of
the regression weighted estimator for NA sequence; Wang and Zhang [17] provided
the Berry-Esséen bound for linear negative quadrant-dependent (LNQD) sequence;
Liang and Baek [18] gave the Berry-Essée n bounds for density estimates under NA
sequence; Liang and Uña-Álvarez [19] studie d the Berry-Es séen bound in kernel den-
sity estimation for a-mixing censored sample; Lahiri and S un [20] obtained the Berry-
Esséen bound of the sample quantiles for a-mixing random variables, etc.
Throughout the paper, C, C
1
, C
2
, C
3
, , d denote some positive cons tants not
depending on n, which may be different in various places. ⌊x⌋ denotes the largest inte-
ger not exceeding x, and the second-order stationarity means that
(
X
1
, X
1+k

0
>0 such that for × Î [ξ
p
- ε
0
, ξ
p
+ ε
0
],

∞
j
=2
j|Cov[I(X
1
≤ x ), I(X
j
≤ x )]| < ∞
,
(1:1)
and
V
ar[I(X
1
≤ ξ
p
)] + 2

∞

p,n
− ξ
p
)
σ (ξ
p
)/f (ξ
p
)
≤ t

− (t)





= O( n
−1/9
), n →∞
.
(1:3)
Remark 1.1 Assumption (1.2) is a general condition, see for example Cai and Roussas
[14]. For the stationary sequences of associated and negatively associated, Cai and
Roussas [15] gave the notation
μ(n)=

∞
j
=n

−(r−1)
)
for
some r>1or

∞
n
=1
n
7
Cov(X
1
, X
n
) <
∞
, Chaubey et al. [21] studied the smooth esti-
mation of survival and densit y functions for a stationary-associat ed process using Pois-
sonweights.Inthispaper,forx Î [ξ
p
- ε
0
, ξ
p
+ ε
0
], the assumption (1.1) has some
restriction on the covariances of Cov[I(X
1
≤ x), I(X

n
®
∞ as n ® ∞. If
lim inf
n
→∞
n
−1
Var(

n
i=1
X
i
)=σ
2
0
> 0
,
then
sup
−∞<t<∞







P

), n →∞
.
(2:1)
Proof We employ Bernstei n’s big-block and small-block procedure. Partition the set
{1, 2, , n}into2k
n
+ 1 subsets with large blocks of size μ = μ
n
and small block of size
υ = υ
n
. Define
μ
n
=[n
2/3
], ν
n
=[n
1/3
], k = k
n
:=

n
μ
n
+ ν
n


:=
j(μ+ν)+μ

i=j
(
μ+ν
)
+1
Z
n,i
,0≤ j ≤ k −1
,
(2:3)
ξ
j
:=
(j+1)(μ+ν)

i=j
(
μ+ν
)
+μ+1
Z
n,i
,0≤ j ≤ k − 1
,
(2:4)
ζ
k


k−1
j=0
η
j
+

k−1
j=0
ξ
j
+ ζ
k
:= S

n
+ S

n
+ S

n
.
(2:6)
By Lemma A.3, we can see that
sup
−∞<t<∞
|P(S
n
≤ t) − (t)|

1
9
)+P(|S

n
| > n
−
1
9
).
(2:7)
Firstly , we estimate
E(S

n
)
2
and
E(S

n
)
2
, which will be used to estimate
P( |S

n
| > n
−
1

, it is easy to see that
|
Z
n,i
|≤
C
1
√
n
.AndE(ξ
j
)
2
≤ Cυ
n
/
n follows from EZ
n,i
= 0 and Lemma A.1. Combining the definition of NA with the
definition of ξ
j
, j = 0, 1, , k - 1, we can easily prove that {ξ
0
, ξ
1
, , ξ
k-1
} is NA. There-
fore, it follows from (2.2), (2.4), (2.6) and Lemma A.1 that
E(S

4
ν
n
μ
n
= O(n
−1/3
)
.
(2:8)
On the other hand, we can get that
E(S

n
)
2
≤
C
5
n
E


n
i=k(μ+ν)+1
X
i

2
≤

from (2.5),
lim inf
n
→∞
n
−1
Var(

n
i=1
X
i
)=σ
2
0
>
0
,|X
i
| ≤ d and Lemma A.1. Consequently,
by Markov’s inequality, (2.8) and (2.9),
P

|S

n
| > n
−
1
9

· E(S

n
)
2
= O( n
−1/9
)
.
(2:11)
In the following, we will estimate
sup
−
∞
<
t
<
∞
|P(S

n
≤ t) − (t)
|
. Define
s
2
n
:=

k

1
and
E(S

n
)
2
= E[S
n
− (S

n
+ S

n
)]
2
=1+E(S

n
+ S

n
)
2
− 2E[S
n
(S

n


n
)
2
+ E(S

n
)
2
+2[E(S

n
)
2
]
1/2
[E(S

n
)
2
]
1/2
+2[E(S
2
n
)]
1/2
[E(S


n
−1/6
)
.
(2:12)
Notice that
s
2
n
= E( S

n
)
2
− 2
n
.
(2:13)
With l
j
= j(μ
n
+ υ
n
),
2
n
=2

0≤i<

+ l
1
- l
2
| ≥ υ
n
, it has that
|
2
n
|≤2

1≤i<j≤n
j−i≥ν
n
|Cov(Z
n,i
, Z
n,j
)|≤
C
1
n

1≤i<j≤n
j−i≥ν
n
|Cov(X
i
, X

i
)=σ
2
0
>
0
and

∞
j
=b
n
|Cov(X
1
, X
j
)| = O(b
−
β
n
)
, b ≥ 1. So, by (2.12),
(2.13) and (2.14), we can get that
|
s
2
n
− 1| = O(n
−1/6
)+O(n

k
−1
j
=0
η

j
. It can be found that
sup
−∞<t<∞
|P(S

n
≤ t) −(t)|
≤ sup
−∞<t<∞
|P(S

n
≤ t) − P(H
n
≤ t)| +sup
−∞<t<∞
|P(H
n
≤ t) − (t/s
n
)|
+sup
−

T
−T
|
φ(t) − ψ(t)
t
|dt + T sup
−∞<t<∞

|u|≤
C
T
|P(H
n
≤ u + t) −P(H
n
≤ t)|d
u
:= D
1
n
+ D
2
n
.
(2:17)
With l
j
= j(μ
n
+ υ





≤ 4t
2

0≤i<j≤k−1
μ
n

l
1
=1
μ
n

l
2
=1
|Cov(Z
n,λ
i
+l
1
, Z
n,λ
j
+l
2

3
t
2
n
−β/3
(2:18)
by (2.2) and the conditions of stationary,
lim inf
n
→∞
n
−
1
Var(

n
i=1
X
i
)=σ
2
0
>
0
and

∞
j
=b
n

It follows from the Berry-Esséen inequality [[12], Theorem 5.7], that
sup
−∞<t<∞
|P(H
n
/s
n
≤ t) − (t)|≤
C
s
3
n

k−1
j=0
E|η

j
|
3
=
C
s
3
n

k−1
j=0
E|η
j

3
≤
C
1
n
3/2

k−1
j=0
E





j(μ+ν)+μ
i=j(μ+ν)+1
X
i




3
≤
C
2
n
3/2


(μ + μ
3/2
) ≤
C
4
kμ
3/2
n
3/2
= O(n
−1/6
).
(2:21)
Combining (2.20) with (2.21), we obtain that
sup
−∞<t<∞
|P(
H
n
s
n
≤ t) − (t)| = O(n
−1/6
)
,
(2:22)
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
http://www.journalofinequalitiesandapplications.com/content/2011/1/83
Page 6 of 14
since s

n





+sup
−∞<t<∞




P

H
n
s
n
≤
t
s
n

− (
t
s
n
)



P

H
n
s
n
≤ t

− (t)| +sup
−∞<t<∞






u + t
s
n

− 

t
s
n

|
= O( n
−1/6
)+O

)+O(n
−1/9
)=O(n
−1/9
),
(2:23)
where T = n
(3b - 1)/18
. It is known that [[12], Lemma 5.2],
sup
−∞<x<∞
|(px) −(x)|≤
(p −1)I(p ≥ 1)
(
2πe
)
1/2
+
(p
−
1
− 1)I(0 < p < 1)
(
2πe
)
1/2
.
Thus, by (2.15),
D
3

1
max(|s
n
− 1|, |s
n
− 1|/s
n
) ·(s
n
+ 1) (note that s
n
→ 1)
≤ C
2
|s
2
n
− 1| = O(n
−1/6
),
(2:24)
and by (2.22),
D
2
=sup
−∞<t<∞





<∞
|P(S

n
≤ t) − (t)| = O(n
−1/9
)+O(n
−1/6
)=O(n
−1/9
)
.
(2:26)
Finally, by (2.7), (2.10), (2.11) and (2.26), (2.1) holds true. □
Lemma 2.2 Let {X
n
}
n≥1
be a second-order stationary NA sequence with common mar-
ginal distribution function and EX
n
=0,|X
n
| ≤ d< ∞, n = 1,2, We give an assumption
such that

∞
j
=2
j|Cov(X

P


n
i=1
X
i
√
nσ
1
≤ t

− (t)




= O(n
−1/9
), n →∞
.
(2:27)
Proof Define
σ
2
n
=Var(

n
i

=1
j|γ (j)| <
∞
that
|σ
2
n
− σ
2
(n, σ
2
1
)| =




nγ (0) + 2n

n−1
j=1

1 −
j
n

γ (j) − nγ (0) − 2n

∞
j=1


∞
j=n
|γ (j)|
≤ 4

∞
j
=1
j|γ (j)| = O(1).
(2:28)
On the other hand,
sup
−∞<t<∞




P


n
i=1
X
i
σ (n, σ
2
1
)
≤ t

− 

σ (n, σ
2
1
)
σ
n
t





+sup
−∞<t<∞






σ (n, σ
2
1
)
σ
n
t


1
, X
j
)|≤
1
b
n

∞
j=b
n
j|Cov(X
1
, X
j
)| = o(b
−1
n
)
.
(2.28) and the fact
σ
2
(n, σ
2
1
)=nσ
2
1
→




σ
2
n
σ
2
(n, σ
2
1
)
− 1




=
C
σ
2
(n, σ
2
1
)


σ
2
n

nσ
1
≤ t

− (t)




≤ C( σ
2
1
)n
−1/9
, n →∞
,
(2:32)
where
C(σ
2
1
)
is a positive constant depending only on
σ
2
1
.
3 Proof of the main result
Proof of Theorem 1.1 The proof is inspired by the proofs of Theorem A and Theorem
C of Serfling [[10], pp. 77-84]. Denote A = s (ξ

sup
t<−L
n
|G
n
(t ) − (t)|,sup
t>L
n
|G
n
(t ) − (t)|

≤ max{G
n
(−L
n
)+(−L
n
), 1 −G
n
(L
n
)+1− (L
n
)
}
≤ G
n
(−L
n

, x > 0 it follows
1 −(L
n
) ≤
(2π)
−1/2
L
n
e
−log n log log n/2
= O(n
−1
)
.
(3:2)
Let ε
n
=(A - ε
0
) (log n log log n)
1/2
n
-1/2
, where 0 <ε
0
<A. Seeing that
P( |ξ
p
,n
− ξ

,n
<ξ
p
− ε
n
)
,
by Lemma A.4 (iii), we obtain
P( ξ
p,n
>ξ
p
+ ε
n
)=P(p > F
n
(ξ
p
+ ε
n
)) = P(1 −F
n
(ξ
p
+ ε
n
) > 1 − p
)
= P


+ ξ
n
) and δ
n1
= F(ξ
p
+ ε
n
)-p. Likewise,
P( ξ
p,n
<ξ
p
− ε
n
) ≤ P(p ≤ F
n
(ξ
p
− ε
n
)) = P


n
i=1
(W
i
− EW
i

are still NA sequences. Obviously, |V
i
- EV
i
| ≤ 1,

n
i
=1
E(V
i
− EV
i
)
2
≤
n
,|W
i
- EW
i
| ≤ 1,

n
i
=1
E(W
i
− EW
i


−
nδ
2
n2
2
(
2+δ
n2
)

.
Consequently,
P( |ξ
p,n
− ξ
p
| >ε
n
) ≤ 4 exp

−
n[min(δ
n1
, δ
n2
)]
2
2(2+max(δ
n1

+ o(ε
n
),
p −F( ξ
p
− ε
n
)=F(ξ
p
) −F(ξ
p
− ε
n
)=f (ξ
p
)ε
n
+ o(ε
n
)
.
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
http://www.journalofinequalitiesandapplications.com/content/2011/1/83
Page 9 of 14
Therefore, we can get that for n large enough,
f (ξ
p
)ε
n
2

p
− ε
n
)
.
Note that max(δ
n1
, δ
n2
) ® 0. as n ® ∞. So with (3), for n large enough,
P( |ξ
p,n
− ξ
p
| >ε
n
) ≤ 4 exp

−
f
2
(ξ
p
)(A − ε
0
)
2
log n log log n
8
(

i
= I [X
i
≤ ξ
p
+ tAn
-1/2
]-EI [X
i
≤ ξ
p
+ tAn
-1/2
]. Seeing that
σ
2
(ξ
p
)=Var[I(X
1
≤ ξ
p
)] + 2

∞
j
=2
Cov[I(X
1
≤ ξ

p
)]|
= |Var[I(X
1
≤ ξ
p
+ tAn
−1/2
)] − Var[I(X
1
≤ ξ
p
)]|
= |F(ξ
p
+ tAn
−1/2
) −F(ξ
p
)+[F
2
(ξ
p
) −F
2
(ξ
p
+ tAn
−1/2
)]|

(3:5)
Similarly, for j ≥ 2 and |t| ≤ L
n
,
|E[I(X
1
≤ ξ
p
+ tAn
−1/2
)I(X
j
≤ ξ
p
+ tAn
−1/2
)] − E[I(X
1
≤ ξ
p
+ tAn
−1/2
)I(X
j
≤ ξ
p
)]
|
≤ E|I(X
j

,
Therefore, by a similar argument, for j ≥ 2 and |t| ≤ L
n
,
|Cov(Z
1
, Z
j
) − Cov[I(X
1
≤ ξ
p
), I(X
j
≤ ξ
p
)]|
≤|E[I(X
1
≤ ξ
p
+ tAn
−1/2
)I(X
j
≤ ξ
p
+ tAn
−1/2
)] − E[I(X

≤|E[I(X
1
≤ ξ
p
+ tAn
−1/2
)I(X
j
≤ ξ
p
+ tAn
−1/2
)]
−E[I(X
1
≤ ξ
p
+ tAn
−1/2
)I(X
j
≤ ξ
p
)]|
+|E[I(X
1
≤ ξ
p
+ tAn
−1/2

+ tAn
−1/2
)]E[I(X
j
≤ ξ
p
)]|
+|E[I(X
1
≤ ξ
p
+ tAn
−1/2
)]E[I(X
j
≤ ξ
p
)] − E[I(X
1
≤ ξ
p
)]E[I(X
j
≤ ξ
p
)]|
= O
((
log n log log n
)

]
j=2
|Cov(Z
1
, Z
j
) −Cov[I(X
1
≤ ξ
p
), I(X
j
≤ ξ
p
)]|
+2

∞
j=[n
1/5
]+1
|Cov(Z
1
, Z
j
)| +2

∞
[j=n
1/5

−1/5
)
.
(3:7)
By Lemma A.4 (iii) again, it has
G
n
(t )=P(ξ
p,n
≤ ξ
p
+ tAn
−1/2
)=P[p ≤ F
n
(ξ
p
+ tAn
−1/2
)
]
= P

np ≤

n
i=1
I(X
i
≤ξ


n
i=1
Z
i
√
nσ
(
n, t
)
≥−c
nt

=1−P


n
i=1
Z
i
√
nσ
(
n, t
)
< −c
nt

,
where

= P


n
i=1
Z
i
√
nσ (n, t)
< −c
nt

− [1 − (t)]
= P


n
i=1
Z
i
√
nσ
(
n, t
)
< −c
nt

− (−c
nt




P


n
i=1
Z
i
√
nσ (n, t)
< −c
nt

− (−c
nt
)




≤ sup
−∞<x<∞




P


−1/9
,
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
http://www.journalofinequalitiesandapplications.com/content/2011/1/83
Page 11 of 14
where C
1
does not depend on t for |t| ≤ L
n
. Therefore, for n large enough, we have
sup
|
t
|
≤L
n




P


n
i=1
Z
i
√
nσ (n, t)
< −c


n
i=1
Z
i
√
nσ (n, t)
< −c
nt

− (−c
nt
)




+sup
|t|≤L
n
|(t) − (c
nt
)
|
≤ Cn
−1/9
+sup
|
t
|

+sup
|
t
|
≤L
n






σ (ξ
p
)
σ (n, t)
t

− (c
nt
)




:= H
1
+ H
2
.

(ξ
p
)| = o(n
−1/5
)
.
(3:11)
By Taylor’s expansion again, we obtain that
c
nt
= t ·
A
σ (n, t)
·
F( ξ
p
+ tAn
−1
/
2
) −F(ξ
p
)
tAn
−1/2
= t ·
A
σ (n, t)
·
F

F

(ξ
p,t
)
2σ
(
n, t
)
n
−1/2
,
(3:12)
where ξ
p,t
lies between ξ
p
and ξ
p
+ Atn
-1/2
. It is known that [[12], Lemma 5.2],
sup
x
|(x + q) − (x)|≤|q|·(2π)
−1/2
, for everyq
.
(3:13)
Therefore, by (3.12), (3.13) and the condition that F’ is bounded in a neighborhood

≤ CL
2
n
n
−1/2
= O(log n ·log log n ·n
−1/2
)
,
(3:14)
since s
2
( ξ
p
)<∞ and
lim
n
→∞
σ
2
(n,t)
σ
2
(ξ
p
)
=
1
for |t| ≤ L
n

p
<
∞, where i = 1, 2, , n and p ≥ 2. Then, there exists some constant c
p
depending only on
p such that
E|

n
i
=1
X
i
|
p
≤ c
p
{

n
i
=1
E|X
i
|
p
+(

n
i

n
i=1
X
i
| >ε) ≤ 2 exp{−
ε
2
2
(
2
n
+ bε
)
}
.
Lemma A.3 [[23], Lemma 2] Let × and Y be random variables, then for any a >0,
sup
t
|P(X + Y ≤ t) −(t)|≤sup
t
|P(X ≤ t) − (t)| +
a
√
2π
+ P(|Y| > a)
.
Lemma A.4 [[10], Lemma 1.1.4] Let F(x) be a right-continuous distribution function.
The inverse function F
-1
(t), 0 <t <1,is nondecreasing and left-continuous, and satisfies

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