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# The Berry-Esséen bound of sample quantiles for NA sequence

*Journal of Inequalities and Applications*
**volume 2014**, Article number: 79 (2014)

## Abstract

By using the exponential inequality, we investigate the Berry-Esséen bound of sample quantiles for negatively associated (NA) random variables and obtain the rate O({n}^{-1/6}logn). Our result extends the corresponding one obtaining O({n}^{-1/9}).

**MSC:**62F12, 62E20, 60F05.

## 1 Introduction

First, we will recall the definition of negatively associated (NA) random variables.

**Definition 1.1** A finite family \{{X}_{1},\dots ,{X}_{n}\} is said to be negatively associated (NA) if for any disjoint subsets A,B\subset \{1,2,\dots ,n\}, and any real coordinatewise nondecreasing functions *f* on {R}^{A}, *g* on {R}^{B},

A sequence of random variables {\{{X}_{n}\}}_{n\ge 1} is said to be negatively associated (NA) if for every n\ge 2, {X}_{1},{X}_{2},\dots ,{X}_{n} are NA.

The concept of an NA sequence was introduced by Joag-Dev and Proschan [1]. There are many good results of NA random variables. For example, Matula [2] obtained the three series theorem, Su *et al.* [3] gave the moment inequality, Shao [4] investigated the maximal inequality, Yuan *et al.* [5] studied the central limit theorem, Yang [6] and Sung [7] investigated the exponential inequality, *etc.*

In this article, we investigate the Berry-Esséen bound of sample quantiles for NA random variables and obtain the rate O({n}^{-1/6}logn). Our result extends the corresponding one of Yang *et al.* [8] obtaining O({n}^{-1/9}). Let us give some details of the *p* th quantile.

Let {\{{X}_{n}\}}_{n\ge 1} be a sequence of random variables defined on a fixed probability space (\mathrm{\Omega},\mathcal{F},P) with a common marginal distribution function F(x)=P({X}_{1}\le x), where *F* is a distribution function (continuous from the right, as usual). For 0<p<1, the *p* th quantile of *F* is defined as

and is alternately denoted by {F}^{-1}(p). The function {F}^{-1}(t), 0<t<1, is called the inverse function of *F*. With a sample {X}_{1},{X}_{2},\dots ,{X}_{n}, n\ge 1, let {F}_{n} represent the empirical distribution function based on {X}_{1},{X}_{2},\dots ,{X}_{n}, which is defined as {F}_{n}(x)=\frac{1}{n}{\sum}_{i=1}^{n}I({X}_{i}\le x), x\in \mathbb{R}, where I(A) denotes the indicator function of a set *A* and ℝ is the real line. Let 0<p<1, we define

as the *p* th quantile of sample.

Throughout the paper, C,{C}_{1},{C}_{2},\dots ,d denote some positive constants not depending on *n*, which may be different in various places. \lfloor x\rfloor denotes the largest integer not exceeding *x* and second-order stationary means that

For 0<p<1, denote {\xi}_{p}={F}^{-1}(p), {\xi}_{p,n}={F}_{n}^{-1}(p) and \mathrm{\Phi}(t) is the distribution function of a standard normal variable. Yang *et al.* [[8], Theorem 1.1] presented the Berry-Esséen bound of sample quantiles for an NA sequence as follows.

**Theorem 1.1** *Let* 0<p<1 *and* {\{{X}_{n}\}}_{n\ge 1} *be a second*-*order stationary NA sequence with common marginal distribution function* *F* *and* E{X}_{n}=0 *for* n=1,2,\dots . *Assume that in a neighborhood of* {\xi}_{p}, *F* *possesses a positive continuous density* *f* *and a bounded second derivative* {F}^{\u2033}. *Suppose that there exists an* {\epsilon}_{0}>0 *such that for* x\in [{\xi}_{p}-{\epsilon}_{0},{\xi}_{p}+{\epsilon}_{0}],

*and*

*Then*

For the work on Berry-Esséen bounds of sample quantiles, one can refer to Reiss [9] or Chapter 2 of Serfling [10]. Cai and Roussas [11] studied the smooth estimate of quantiles under an association sample, Rio [12] obtained the Berry-Esséen bounds of sample quantiles under a *φ*-mixing sequence, Lahiri and Sun [13] and Yang *et al.* [14] investigated the Berry-Esséen bounds of sample quantiles under an *α*-mixing sequence, *etc.* For more work on Berry-Esséen bounds, we can refer to Chapter 3 of Hall and Heyde [15], Chapter 5 of Petrov [16], Gao *et al.* [17], Chapter 5 of Härdle *et al.* [18], and to the references therein too.

Moreover, value-at-risk (VaR) is a popular measure of the market risk associated with an asset or a portfolio of assets. It has been chosen by the Basel Committee on Banking Supervision as a benchmark risk measure and has been used by financial institutions for asset management and minimization of risk. Let {\{{X}_{t}\}}_{t=1}^{n} be the market value of an asset over *n* periods of a time unit, and let {Y}_{t}=log({X}_{t}/{X}_{t-1}) be the log-returns. Suppose {\{{Y}_{t}\}}_{t=1}^{n} is a strictly stationary dependent process with marginal distribution function *F*. Given a positive value *p* close to zero, the 1-p level VaR is

which specifies the smallest amount of loss such that the probability of the loss in market value being large than {v}_{p} is less than *p*. So, the study of VaR is a specific application of the *p* th quantile. For more details, one can refer to Chen and Tang [19] and the references therein.

In this paper, by the exponential inequality and properties of NA random variables, we go on studying the Berry-Esséen bound of sample quantiles for an NA sequence and get a better rate of normal approximation. For the details, see Theorem 2.1 in Section 2. Some preliminaries and the proof of Theorem 2.1 are presented in Section 3.

## 2 Main result

**Theorem 2.1** *Let* 0<p<1 *and* {\{{X}_{n}\}}_{n\ge 1} *be a second*-*order stationary NA sequence with common marginal distribution function* *F*. *Assume that in a neighborhood of* {\xi}_{p}, *F* *possesses a positive continuous density* *f* *and a bounded second derivative* {F}^{\u2033}. *Let* {n}_{0} *be some positive integer*. *Suppose that there exists an* {\epsilon}_{0}>0 *such that for* x\in [{\xi}_{p}-{\epsilon}_{0},{\xi}_{p}+{\epsilon}_{0}]

*and condition* (1.2) *holds*. *Then*

**Remark 2.1** Obviously, the condition (2.1) of Theorem 2.1 is relatively stronger than (1.1) of Theorem 1.1, but the normal approximation rate O({n}^{-1/6}logn) in (2.2) is better than O({n}^{-1/9}) in (1.3). So our result Theorem 2.1 extends Theorem 1.1 of Yang *et al.* [8]. It is pointed out that the condition of mean zero in Theorem 1.1 should be removed. In fact, the process of estimating (3.9) on page 12 of Yang *et al.* [8], was used the Lemma 2.2 of Yang *et al.* [8], which requires the condition of mean zero, but {Z}_{i} in (3.9) of Yang *et al.* [8], defined by {Z}_{i}=I[{X}_{i}\le {\xi}_{p}+tA{n}^{-1/2}]-EI[{X}_{i}\le {\xi}_{p}+tA{n}^{-1/2}], satisfies the condition of mean zero. Thus, the mean zero condition of Theorem 1.1 of Yang *et al.* [8] is not needed. It coincides with the independent case, which does not need the mean zero condition. For the details, one can see Serfling [[10], Theorem C, p.81] or Theorem A of Yang *et al.* [8].

## 3 Some preliminaries and the proof of Theorem 2.1

First, we give some preliminaries, which will be used to prove our Theorem 2.1.

**Lemma 3.1** [[6], Lemma 3.5]

*Let* {\{{X}_{n}\}}_{n\ge 1} *be a NA sequence with* E{X}_{n}=0, |{X}_{n}|\le b, *a*.*s*. n=1,2,\dots . *Denote* {\mathrm{\Delta}}_{n}={\sum}_{i=1}^{n}E{X}_{i}^{2}. *Then for* \mathrm{\forall}\epsilon >0,

**Lemma 3.2** *Let* {\{{X}_{n}\}}_{n\ge 1} *be a stationary NA sequence with* E{X}_{n}=0 *and* |{X}_{n}|\le d<\mathrm{\infty}, n=1,2,\dots . *Assume that there exists a* \beta \ge 3/2 *such that*

*for all*
0<{b}_{n}\to \mathrm{\infty}
*as*
n\to \mathrm{\infty}
*and*

*Then*

*Proof* By taking the same notation as that in the proof of Lemma 2.1 of Yang *et al.* [8], we partition the set \{1,2,\dots ,n\} into 2{k}_{n}+1 subsets with large block of size \mu ={\mu}_{n} and small block of size \nu ={\nu}_{n}. Let

and {Z}_{n,i}={X}_{i}/\sqrt{Var({\sum}_{i=1}^{n}{X}_{i})}. Define {\eta}_{j}, {\xi}_{j}, {\zeta}_{k} as follows:

Denote

By Lemma A.3 in Yang *et al.* [8] with a=2{\epsilon}_{n}=2M{n}^{-1/6}logn we have

where *M* is a positive constant.

Combining the definition of NA with the definition of {\xi}_{j}, j=0,1,\dots ,k-1, we can easily prove that \{{\xi}_{0},{\xi}_{1},\dots ,{\xi}_{k-1}\} is NA. Together the condition (3.2) with (2.8) of Yang *et al.* [8], it has E{({S}_{n}^{\prime \prime})}^{2}\le {C}_{1}{n}^{-1/3}. On the other hand, it can be seen that |{\xi}_{j}|\le {C}_{2}{n}^{-1/6}, j=0,1,\dots ,k-1. Thus, we take *M* large enough and apply Lemma 3.1, and we obtain for *n* large enough

Meanwhile, by (2.9) of Yang *et al.* [8], it follows E{({S}_{n}^{\prime \prime \prime})}^{2}\le {C}_{4}{n}^{-1/3}. Since |{Z}_{n,i}|\le {C}_{5}{n}^{-1/2}, by Lemma 3.1 again, one has for *n* large enough

Similar to the proof of (2.18) in Yang *et al.* [8], by (3.1), it can be seen that

Combining the above inequality with T={n}^{\frac{2\beta -1}{12}}, \beta \ge 3/2, we obtain

On the other hand, we take T={n}^{\frac{2\beta -1}{12}}, \beta \ge 3/2, in (2.23) of Yang *et al.* [8] and have {D}_{2n}=O({n}^{-1/6}).

Consequently, by the proof of (2.26) of Yang *et al.* [8], it is easy to check that

Finally, by (3.4)-(3.7), (3.3) holds. □

**Lemma 3.3** *Let* {\{{X}_{n}\}}_{n\ge 1} *be a stationary NA sequence with* E{X}_{n}=0 *and* |{X}_{n}|\le d<\mathrm{\infty}, n=1,2,\dots . *Assume that there exists an* {n}_{0} *such that*

*and*

*Then*

*Proof* By the condition (3.8), it is checked that

providing {b}_{n}\to \mathrm{\infty} as n\to \mathrm{\infty}. So by (3.8), the condition (3.1) of Lemma 3.2 holds. Combining Lemma 3.2 with the proof of Lemma 2.2 of Yang *et al.* [8], we have (3.9) finally. □

*Proof of Theorem 2.1* By taking the same notation as that in the proof of Theorem 1.1 in Yang *et al.* [8], one checks the proof of (3.9) in Yang *et al.* [8] and obtains by Lemma 3.3

where {C}_{1}({\sigma}^{2}({\xi}_{p})) is a positive constant depending only on {\sigma}^{2}({\xi}_{p}). Therefore, (2.2) follows by the same steps as those in the proof of Theorem 1.1 of Yang *et al.* [8]. □

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## Acknowledgements

The authors are deeply grateful to the editor and two anonymous referees whose insightful comments and suggestions have contributed substantially to the improvement of this paper. This work was supported by the NNSF of China (11171001, 11201001, 11301004, 11326172), Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03), Talents Youth Fund of Anhui Province Universities (2012SQRL204), Higher Education Talent Revitalization Project of Anhui Province (2013SQRL005ZD) and the Academic and Technology Leaders to Introduction Projects of Anhui University.

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Liu, T., Zhang, Z., Hu, S. *et al.* The Berry-Esséen bound of sample quantiles for NA sequence.
*J Inequal Appl* **2014, **79 (2014). https://doi.org/10.1186/1029-242X-2014-79

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DOI: https://doi.org/10.1186/1029-242X-2014-79

### Keywords

- Berry-Esséen bound
- sample quantile
- NA random variables