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 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, 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 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 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 = 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83 Page 10 of 14 [...]... convergence for negatively associated sequences Sci China A 40, 172–182 (1997) doi:10.1007/BF02874436 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... estimate of quantiles under association Stat Probab Lett 36, 275–287 (1997) doi:10.1016/S0167-7152(97)00074-6 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 Yang, SC: Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples Stat... 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... (2010SQRL016ZD) and Youth Science Research Fund of Anhui University (2009QN011A) Author details 1 School of Mathematical Science, Anhui University Hefei 230039, PR China 2Department 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 (n1/9... 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/83... 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 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 Liang, HY, De Uña-Álvarez, J: A Berry-Esséen type bound in kernel density estimation for strong... 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... density estimation for strong mixing censored samples J Multivar Anal 100, 1219–1231 (2009) doi:10.1016/j.jmva.2008.11.001 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 Chaubey, YP, Dewan, I, Li, J: Smooth estimation of survival and density functions for a stationary associated process using Poisson weights... 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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. 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. 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