Báo cáo hóa học: " Berry-Esséen bound of sample quantiles for negatively associated sequence" ppt

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
}

n≥1
is a sequence of random variables defined on a fixed probability
space
(
, F , P
)
with a common marginal distribution function F(x )=P(X
1
≤ x). F is a
distribution function (continuous from the right, as usual). For 0 <p < 1, the pth quan-
tile of F is defined as
ξ
p
=inf{x : F(x) ≥ p
}
and is alternately denoted by F
-1
(p). The function F
-1
(t), 0 <t < 1, is called the inverse
function of F. It is easy to check that ξ
p
possesses the following properties:
(i) F(ξ
p
-) ≤ p ≤ F(ξ
p
);
(ii) if ξ
p

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

i=1
I(X
i
≤ x
)
, x Î ℝ,whereI(A)
denotes the indicator function of a set A and ℝ is the real line. For 0 <p < 1, we define
F
−1
n
(p)=inf{x : F
n
(x) ≥ p
}
as the pth quantile of sample.
Recall that a finite family {X
1
, , X
n
} is said to be negatively associated (NA) if for any
disjoint subsets A, B ⊂ {1, 2, , n}, and any real coordinatewise nondecreasing functions
f on R
A
, g on R
B
,
Cov
(
f
(

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
/>© 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,
provided the original work is prope rly cited.

an elegant representation for the sample quantiles in terms of empirical distribution
function based on independent and identically distributed (i.i.d.) random variables. Sen
[2], Babu and Singh [3] and Yoshihara [4] gave the Bahadur representation for the
sample quantiles under j-mixing sequence and a-mixing sequence, respective ly. Sun
[5] established the Bahadur representation for the sample quantiles under a-mixing
sequence with polynomially decaying rate. Ling [6] investigated the Bahadur repre sen-
tation for the sample quantile s under NA sequence. Li et al. [7] investigated the Baha-
dur representation of the sample quantile based on negatively orthant-dependent
(NOD) sequence, which is weaker than NA sequence. Xing and Yang [8] also studied
the Bahadur representation for the sample quantiles under NA sequence. Wang et al.
[9] revised the results of Sun [5] and got a better bound. For more details about Baha-
dur representation, one can refer to Serfling [10].
For a fixed p Î (0, 1), let ξ
p
= F
-1
(p),
ξ
p
,n
= F
−1
n
(p
)
and F(t) be the distribution func-
tion of a standard normal variable. In [[10], p. 81], the Berry-Esséen bound of the sam-
ple quantiles for i.i.d. random variables is given as follows:
Theorem A Let 0<p <1and {X
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


− (t)





= O( n
−1/2
), n →∞
.
In this paper, we investigate the Berry-Esséen bound of the sample quantiles for NA
random variables under some weak conditio ns. The rate of normal approximation is
shown as O(n
-1/9
).
Berry-Esséen theorem, which is known as the rate of convergence in the central limit
theorem, can be found in many monographs such as Shiryaev [11], Petrov [12]. For the
case of i.i.d. random variables, the optimal rate is
O
(
n
−
1
2
)
, and for the case of martin-
gale, the rate is
O
(
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
)

d
=
(
X
i
, X
i+k
)
, i ≥ 1, k ≥ 1
.
Inspired by Serfling [10], Cai and Rou ssas [14,15], Yang [16], Liang and Uña-Álvarez
[19], Lahiri and Sun [20], etc., we obtain Theorem 1.1 in Section 1. Two preliminary
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 2 of 14
lemmas are given in Section 2, and the proof of Theorem 1.1 is given in Section 3.
Next, we give the main result as follows:
Theorem 1.1 Let 0<p <1and {X
n
}
n≥1
be a second-order stationary N A sequence
with common marginal distribution function F and EX
n
=0for n =1,2, Assume
that in a neighborhood of ξ
p
, F possesses a po sitive continuous density f and a bounded
second derivative F″. If there exists an ε
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

∞
j
=2

Cov[I(X
1
≤ ξ
p
), I(X
j
≤ ξ
p
)] := σ
2
(ξ
p
) > 0
,
(1:2)
then
sup
−∞<t<∞





P

n
1/2
(ξ
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
|Cov(X
1

, X
j+1
)|
1/
3
and supposed that μ(1) <
∞. In addition, they supposed that μ(n)=O(n
-a
)forsomea >0orδ(1) < ∞,where
δ(i)=

∞
j
=i
μ(j)
, then obtained the Berry-Esséen bounds for smooth estimator of a dis-
tribution funct ion. Under the assumptions

∞
j
=n+1
{Cov(X
1
, X
j
)}
1/3
= O(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
j
≤ x)] in the neighborhood of ξ

p
.
2 Preliminaries
Lemma 2.1 Let {X
n
}
n≥1
be a stationary NA sequence with EX
n
=0,|X
n
| ≤ d<∞ for n =
1, 2, . . There exists some b ≥ 1 such that

∞
j
=b
n
|Cov(X
1
, X
j
)| = O(b
−
β
n
)
for all 0<b
n
®

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

⎝

n
i=1
X
i

Var(

n
i=1
X
i
)
≤ t
⎞
⎟
⎠
− (t)







= O(n
−1/9
), 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

=[n
1/3

],
(2:2)
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 3 of 14
and
Z
n,i
= X
i
/

Var(

n
i=1
X
i
)
. Let h
j
, ξ
j
, ζ
j
be defined as follows:
η
j
:=
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
:=
n


i=k
(
μ+ν
)
+1
Z
n,i
.
(2:5)
Write
S
n
:=

n
i=1
X
i

Var(

n
i=1
X
i
)
=

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)|
=sup
−∞<t<∞
|P(S


n
+ S

n
+ S

n
≤ t) − (t)|≤ sup
−∞<t<∞
|P(S

n
≤ t) − (t)
|
+
2n
−
1
9
√
2π
+ P(|S

n
| > n
−
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
9
)
and

P( |S

n
| > n
−
1
9
)
in (2.7). By the conditions |X
i
| ≤ d and
lim inf
n
→∞
n
−1
Var(

n
i=1
X
i
)=σ
2
0
>
0
, 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

n
)

2
≤ C
1

k−1
j=0
Eξ
2
j
≤ C
2
k
n
ν
n
n
≤ C
3
n
μ
n
+ ν
n
ν
n
n
≤ C
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
≤
C
6
n


n
i=k(μ+ν)+1
EX
2
i
≤
C
7
n
(n −k
n
(μ
n
+ ν
n
)) ≤ C
8
μ
n
+ ν
n
n
= O( n
−1/3
)
(2:9)
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

≤ n
2

9
· E(S

n
)
2
= O( n
−1/9
)
,
(2:10)
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 4 of 14
P

|S

n
| > n
−
1
9

≤ n
2
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
j
=0
Var(η

j
), 
n
:=

0≤i<
j
≤k−1
Cov(η
i
, η
j
)
.
Here, we first estimate the growth rate
|
s
2
n
− 1
|
. Since
ES
2
n
=
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
+ S

n
)]

,
by (2.8) and (2.9), it has
|
E(S

n
)
2
− 1| = |E(S

n
+ S

n
)
2
− 2E[S
n
(S

n
+ S

n
)]|
≤ E(S

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
)
2
]

1/2
+2[E(S
2
n
)]
1/2
[E(S

n
)
2
]
1/
2
= O
(
n
−1/3
)
+ O
(
n
−1/6
)
= O
(
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<
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
)
,
but since i ≠ j,|l
i
- l
j
+ 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
j
)
|
≤ C

2

k≥ν
n
|Cov(X
1
, X
k
)| = O(n
−β/3
)=O(n
−1/3
)
(2:14)
following from (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
|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
−1/3
)=O(n
−1/6
)

.
(2:15)
For j = 0, 1, , k -1,let
η

j
be the independent random variables and
|s
2
n
− 1| = O(n
−1/6
)+O(n
−1/3
)=O(n
−1/6
)
.
have the same distribution as h
j
, j = 0, 1, ,
k - 1. Define
H
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/s
n
) −(t)| := D
1
+ D
2
+ D
3
.
(2:16)
Let j(t) and ψ(t) be the characteristic functions of
S

n
and H
n
, respectively. By Esséen
inequality [[12], Theorem 5.3], for any T>0,
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 5 of 14
D
1
≤

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
n
) and similar to the proof of Lemma 3.4 of Yang [16], we have
that
|ϕ(t) − ψ(t)| =







E exp
⎛
⎝
it
k−1

j=0
η
j
⎞
⎠
−
k−1

j=0
E exp(itη
j
)






≤ 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
)
|
≤
C
1

t
2
n

1≤i<j≤n
j−i≥ν
n
|Cov(X
i
, X
j
)|
≤ C
2
t
2

j
≥ν
n
|Cov(X
1
, X
j
)|≤C
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
|Cov(X
1
, X
j
)| = O(b

−β
n
)
. Set T = n
(3b - 1)/18
for b ≥ 1, we have by (2.18) that
D
1n
=

T
−
T
|
ϕ(t) − ψ(t)
t
|dt ≤ Cn
−β/3
· T
2
= O(n
−1/9
)
.
(2:19)
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
.
(2:20)
By (2.3) and Lemma A.1,


k−1
j=0
E|η
j
|
3
=

k−1
j=0
E





j(μ+ν)+μ
i=j(μ+ν)+1
Z
n,i




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

k−1
j=0


j(μ+ν)+μ

i=j(μ+ν)+1
E|X
i
|
3
+(

j(μ+ν)+μ
i=j(μ+ν)+1
E|X
i
|
2
)
3/2

≤
C
3
n
3/2

k−1
j
=0
(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
/>Page 6 of 14
since s
n
® 1asn ® ∞ by (2.15). It follows from (2.22) that
sup
−∞<t<∞
|P(H
n

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

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





u + t
s
n

− 

t
s
n





≤ 2sup
−∞<t<∞




P

H
n
s
n

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

which implies that
D
2n
= T sup
−∞<t<∞

|u|≤C/T
|P(H
n
≤ u + t) −P(H
n
≤ t)|d
u
≤
C
1
n
1/6
+
C
2
T
= O( n
−1/6
)+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
=sup
−∞<t<∞
|(t/s
n
) −(t)|
≤ (2π e)

−1/2
(s
n
− 1)I(s
n
≥ 1) + (2π e)
−1/2
(s
−1
n
− 1)I(0 < s
n
< 1
)
≤ (2π e)
−1/2
max(|s
n
− 1|, |s
n
− 1|/s
n
)
≤ C
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

H
n
s
n

≤
t
s
n

− 

t
s
n





= O(n
−1/6
)
.
(2:25)
Therefore, it follows from (2.16), (2.17), (2.19), (2.23), (2.24) and (2.25) that
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
1
, X
j
)| <
∞
. If

V
ar(X
1
)+2

∞
j
=2
Cov(X
1
, X
j
)=σ
2
1
>
0
, then
sup
−∞<t<∞




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
X
i
)
,
σ

2
(n, σ
2
1
)=nσ
2
1
and g(k) = Cov (X
i+k
, X
i
) for k =0,1,
2, For the second-order stationarity process {X
n
}
n≥ 1
with common marginal
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 7 of 14
distribution function, it can be found by the condition

∞
j
=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
γ (j)




=





2n

n−1
j=1
j
n
γ (j) − 2n

∞
j=n
γ (j)




≤ 2

∞
j=1
j|γ (j)| +2n

∞
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

− (t)




≤ sup

−∞<t<∞




P


n
i=1
X
i
σ
n
≤
σ (n, σ
2
1
)
σ
n
t

− 

σ (n, σ
2
1
)
σ

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

+ D
2
.
(2:29)
Obviously, if b
n
® ∞ as n ® ∞, then it follows from

∞
j
=2
j|Cov(X
1
, X
j
)| <
∞
that

∞
j=b
n
|Cov(X
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
→
∞
yield that
lim
n
→∞
σ
2

n
/σ
2
(n, σ
2
1
)=
1
.Thus,by
Lemma 2.1,
D
1
= O
(
n
−1/9
).
(2:30)
By (2.28) again and similar to the proof of (2.24), it follows
D
2
≤ C




σ
2
n
σ

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


= O(n
−1
)
.
(2:31)
Finally, by (2.29), (2.30) and (2.31), (2.27) holds true. □
Remark 2.1 UndertheconditionsofLemma2.2,wehave(27).Furthermore,bythe
proof of Lemma 2.2, we can obtain that
sup
−∞<t<∞




P


n
i=1
X
i
√
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
p
)/f (ξ
p
) and
G
n
(t )=P(n

1/2
(ξ
p
,n
− ξ
p
)/A ≤ t)
.
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 8 of 14
Let L
n
= (log n log log n)
1/2
, we have
sup
|t|>L
n
|G
n
(t ) − (t)| =max

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
)+1− G
n
(L
n
)+1− (L
n
)
≤ P(|ξ

p
,n
− ξ
p
|≥AL
n
n
−1/2
)+1− (L
n
).
(3:1)
Since
1 −(x) ≤
(2π)
−1/2
x
e
−x
2
/
2
, 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
p
|≥A(log n log log n)
1/2
n
−1/2
) ≤ P(|ξ
p
,n

− ξ
p
| >ε
n
)
and
P( |ξ
p
,n
− ξ
p
| >ε
n
)=P(ξ
p
,n
>ξ
p
+ ε
n
)+P(ξ
p
,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


n
i=1
I(X
i
>ξ
p
+ ε

n
) > n(1 − p)

= P


n
i=1
(V
i
− EV
i
) > nδ
n1

,
where V
i
= I (X
i
> ξ
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
) ≥ nδ
n2

,
where W
i
= I (X
i

> ξ
p
- ξ
n
)andδ
n2
= p - F(ξ
p
- ε
n
). It is easy to see that {V
i
- EV
i
}
1≤ i≤ n
.and{W
i
- EV
i
}
1≤ i≤ n
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
)
2
≤
n
.ByLemmaA.2,we
have that
P( ξ
p,n

>ξ
p
+ ε
n
) ≤ 2 exp

−
nδ
2
n1
2(2 + δ
n1
)

,
P( ξ
p,n
<ξ
p
− ε
n
) ≤ 2 exp

−
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
, δ
n2
))

.
(3:3)
Since F (x) is continuous at ξ
p

with F’ (ξ
p
)>0,ξ
p
is the unique solution of F (x-) ≤ p
≤ F (x) and F (ξ
p
)=p. By the assumption on f’(x) and Taylor’s expansion,
F( ξ
p
+ ε
n
) −p = F(ξ
p
+ ε
n
) −F(ξ
p
)=f (ξ
p
)ε
n
+ 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
/>Page 9 of 14
Therefore, we can get that for n large enough,
f (ξ
p
)ε
n
2
=
f (ξ
p
)(A −ε
0
)(log n log log n)
1/2
2n
1/2

≤ F(ξ
p
+ ε
n
) −p,
f (ξ
p
)ε
n
2
=
f (ξ
p
)(A −ε
0
)(log n log log n)
1/2
2
n
1/2
≤ p −F(ξ
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
(
2+max
(
δ
n1
, δ
n2
))

= O(n

−1
)
.
(3:4)
Next, we define
σ
2
(n, t)=Var(Z
1
)+2

∞
j
=2
Cov(Z
1
, Z
j
)
,
where Z
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
), I(X
j
≤ ξ
p
)]
,
we will estimate the convergence rate of |s
2

(n , t)-s
2
(ξ
p
)|. By the condition (1.1),
we can see that s
2
(ξ
p
)<∞. Since that F possesses a positi ve continuous density f and
a bounded second derivative F’,for|t| ≤ L
n
=(logn log log n)
1/2
, we will obtain by
Taylor’s expansion that
|Var(Z
1
) −Var[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
)]|
≤ f (ξ
p
) ·|t|An
−1/2
+ o(|t|An
−1/2
)
+|F(ξ
p

)+F(ξ
p
+ tAn
−1/2
)|·[f (ξ
p
) ·|t|An
−1/2
+ o(|t|An
−1/2
)
]
= O
((
log n log log n
)
1/2
n
−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
≤ ξ
p
+ tAn
−1/2
) − I(X
j
≤ ξ
p
)|

=[F(ξ
p
+ tAn
−1/2
) − F(ξ
p
)]I(t ≥ 0) + [F(ξ
p
) − F(ξ
p
+ tAn
−1/2
)]I(t < 0)
= O
((
log n log log n
)
1/2
n
−1/2
)
,
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
1
≤ ξ
p
)I(X
j
≤ ξ
p
)]|
+|E[I(X

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

j
≤ ξ
p
)]|
+|E[I(X
1
≤ ξ
p
+ tAn
−1/2
)]E[I(X
j
≤ ξ
p
+ tAn
−1/2
)]
−E[I(X
1
≤ ξ
p
+ 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
)
1/2
n
−1/2
)
.
(3:6)
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 10 of 14
Consequently, by the conditions (1.1) and (3.5), (3.6), for |t| ≤ L

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

≤ C
1
(log n log log n)
1/2
n
−1/2
+ C
2
n
1/5
(log n log log n)
1/2
n
−1/2
+ o(n
−1/5
)
= o
(
n
−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
p
+ tAn
−1/2
)

= P

n
1/2
(p −F ( ξ

p
+ tAn
−1/2
))
σ (n, t)
≤

n
i=1
Z
i
√
nσ (n, t)

.
Thus,
G
n
(t )=P


n
i=1
Z
i
√
nσ
(
n, t
)

≥−c
nt

=1−P


n
i=1
Z
i
√
nσ
(
n, t
)
< −c
nt

,
where
c
nt
=
n
1/2
(F ( ξ
p
+ tAn
−1/2
) −p)

σ
(
n, t
)
.
It is easy to check that
(t) − G
n
(t )=(t) − 1+P


n
i=1
Z
i
√
nσ (n, t)
< −c
nt

= 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
)+(t) − (c
nt
)
.
(3:8)
By (3.7), it has that
lim
σ
2
(n,t)

σ
2
(ξ
p
)
→
1
as n ® ∞, which implies that 0 <s
2
(n, t) for |t| ≤
L
n
and n large enough. Obviously, {Z
i
} is a second-order stationary NA sequence.
Thus, for a fixed t,|t| ≤ L
n
, by the Lemma 2.2, (2.32) in Remark 2.1 and (3.7), it has
for n large enough that




P


n
i=1
Z
i

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





≤ C(σ
2
(n, t))n
−1/9
= C( σ
2
(ξ
p
)+o(n
−1/5
))n
−1/
9
≤ C
1
n
−1/9
,
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>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
nt

− (−c
nt
)




≤ C
1

n
−1/9
.
By (3.8) and the inequality above, we can get that for n large enough,
sup
|t|≤L
n
|G
n
(t ) − (t)|
≤ sup
|t|≤L
n




P


n
i=1
Z
i
√
nσ (n, t)
< −c
nt

− (−c

nt
)




+sup
|t|≤L
n
|(t) − (c
nt
)
|
≤ Cn
−1/9
+sup
|
t
|
≤L
n
|(t) − (c
nt
)|.
(3:9)
On the other hand,
sup
|t|≤L
n
|(t) − (c

nt
)|≤ sup
−∞<t<∞




(t) − 

σ (ξ
p
)
σ (n, t)
t





+sup
|
t
|
≤L
n







σ (ξ
p
)
σ (n, t)
t

− (c
nt
)




:= H
1
+ H
2
.
(3:10)
By (3.7) again and similar to the proof of (2.31), we have
H
1
≤ C




σ
2

(n, t)
σ
2
(ξ
p
)
− 1




= Cσ
−2
(ξ
p
)|σ
2
(n, t) − σ
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

(ξ
p
)tAn
−1/2
+
1
2
F

(ξ

p,t
)(tAn
−1/2
)
2
tAn
−1/2
= t
σ (ξ
p
)
σ
(
n, t
)
+ t
2
A
2
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
of ξ
p
, we get for n large enough that
H
2
=sup
|
t
|
≤L
n







σ (ξ
p
)
σ (n, t)
t

− (c
nt
)




≤ 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
. Therefore, it follows from (3.9),
(3.10), (3.11) and (3.14) that
sup
|
t
|
≤L
n
|G
n
(t ) − (t)| = O(n

−1/9
)
.
(3:15)
Finally, the desired result (1.3) follows from (3.1), (3.2), (3.4) and (3.15) immediately.
□
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 12 of 14
Appendix
Lemma A.1 [[22], Theorem 1] Let {X
n
}
n≥1
be NA random variables, EX
i
=0,E|X
i
|
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
=1
EX
2
i
)
p/2
}
.
Lemma A.2 [[16], Lemma 3.5] Let {X
n
}
n≥1

be a NA sequence with EX
i
=0,|X
i
| ≤ b,
a.s. i = 1, 2, , Denote

n
=

n
i=1
EX
2
i
. Then for ∀ ε >0,
P( |

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
(i) F
-1
(F(x)) ≤ x,-∞ <x < ∞;
(ii) F (F
-1
(t)) ≥ t,0<t <1;
(ii) F (x) ≥ t if and only if × ≥ F
-1
(t).
Acknowledgements
The authors are most grateful to the Editor Charles E. Chidume and an anonymous referee for the careful reading of
the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper.

Supported by the NNSF of China (11171001, 61075009), HSSPF of the Ministry of Education of China (10YJA910005),
Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Talents youth Fund of Anhui Province
Universities (2010SQRL016ZD) and Youth Science Research Fund of Anhui University (2009QN011A).
Author details
1
School of Mathematical Science, Anhui University Hefei 230039, PR China
2
Department of Statistics and Finance
University of Science and Technology of China Hefei 230026, PR China
Authors’ contributions
Under some weak conditions, the Berry-Esséen bound of the sample quantiles for NA sequence is presented as O (n
-
1/9
). All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 4 January 2011 Accepted: 10 October 2011 Published: 10 October 2011
References
1. Bahadur, RR: A note on quantiles in large samples. Ann. Math. Stat. 37, 577–580 (1966). doi:10.1214/aoms/1177699450
2. Sen, PK: On Bahadur representation of sample quantile for sequences of -mixing random variables. J. Multivar. Anal. 2,
77–95 (1972). doi:10.1016/0047-259X(72)90011-5
3. Babu, GJ, Singh, K: On deviations between empirical and quantile processes for mixing random variables. J. Multivar.
Anal. 8, 532–549 (1978). doi:10.1016/0047-259X(78)90031-3
4. Yoshihara, K: The Bahadur representation of sample quantile for sequences of strongly mixing random variables. Stat.
Probab. Lett. 24, 299–304 (1995). doi:10.1016/0167-7152(94)00187-D
5. Sun, SX: The Bahadur representation for sample quantiles under weak dependence. Stat. Probab. Lett. 76, 1238–1244
(2006). doi:10.1016/j.spl.2005.12.021
6. Ling, NX: The Bahadur representation for sample quantiles under negatively associated sequence. Stat. Probab. Lett. 78,
2660–2663 (2008). doi:10.1016/j.spl.2008.03.026
7. Li, XQ, Yang, WZ, Hu, SH, Wang, XJ: The Bahadur representation for sample quantile under NOD sequence. J.

Nonparametric Stat. 23(1), 59–65 (2011). doi:10.1080/10485252.2010.486033
8. Xing, GD, Yang, SC: A remark on the Bahadur representation of sample quantiles for negatively associated sequences. J.
Korean Stat. Soc. 40(3), 277–280 (2011). doi:10.1016/j.jkss.2010.10.006
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 13 of 14
9. Wang, XJ, Hu, SH, Yang, WZ: The Bahadur representation for sample quantiles under strongly mixing sequence. J. Stat.
Plan. Inf. 141, 655–662 (2010)
10. Serfling, RJ: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)
11. Shiryaev, AN: Probability, 2nd edn. Springer, New York (1989)
12. Petrov, VV: Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford University Press
Inc., New York (1995)
13. Hall, P: Martingale Limit Theory and Its Application. Academic Press, Inc., New York (1980)
14. Cai, ZW, Roussas, GG: Smooth estimate of quantiles under association. Stat. Probab. Lett. 36, 275–287 (1997).
doi:10.1016/S0167-7152(97)00074-6
15. Cai, ZW, Roussas, GG: Berry-Esséen bounds for smooth estimator of a distribution function under association. J.
Nonparametric Stat. 11,79–106 (1999). doi:10.1080/10485259908832776
16. Yang, SC: Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples. Stat.
Probab. Lett. 62, 101–110 (2003)
17. Wang, JF, Zhang, LX: A Berry-Esséen theorem for weakly negatively dependent random variables and its applications.
Acta Math. Hung. 110(4):293–308 (2006). doi:10.1007/s10474-006-0024-x
18. Liang, HY, Baek, J: Berry-Esséen bounds for density estimates under NA assumption. Metrika. 68, 305–322 (2008).
doi:10.1007/s00184-007-0159-y
19. Liang, HY, De Uña-Álvarez, J: A Berry-Esséen type bound in kernel density estimation for strong mixing censored
samples. J. Multivar. Anal. 100, 1219–1231 (2009). doi:10.1016/j.jmva.2008.11.001
20. Lahiri, SN, Sun, S: A Berry-Esséen theorem for samples quantiles under weak dependent. Ann. Appl. Probab. 19(1),
108–126 (2009). doi:10.1214/08-AAP533
21. Chaubey, YP, Dewan, I, Li, J: Smooth estimation of survival and density functions for a stationary associated process
using Poisson weights. Stat. Probab. Lett. 81, 267–276 (2011). doi:10.1016/j.spl.2010.10.010
22. Su, C, Zhao, LC, Wang, YB: Moment inequalities and weak convergence for negatively associated sequences. Sci. China
A. 40, 172–182 (1997). doi:10.1007/BF02874436

23. Chang, MN, Rao, PV: Berry-Esséen bound for the Kaplan-Meier estimator. Commun. Stat. Theory Methods. 18(12),
4647–4664 (1989). doi:10.1080/03610928908830180
doi:10.1186/1029-242X-2011-83
Cite this article as: Yang et al.: Berry-Esséen bound of sample quantiles for negatively associated sequence.
Journal of Inequalities and Applications 2011 2011:83.
Submit your manuscript to a
journal and benefi t from:
7 Convenient online submission
7 Rigorous peer review
7 Immediate publication on acceptance
7 Open access: articles freely available online
7 High visibility within the fi eld
7 Retaining the copyright to your article
Submit your next manuscript at 7 springeropen.com
Yang et al. Journal of Inequalities and Applications 2011, 2011:83
/>Page 14 of 14