The Berry-Esséen bound of sample quantiles for NA sequence

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 $R^B$,

$$Cov(f(X_k,k\in A),g(X_k,k\in B))\le 0.$$

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

$${\xi }_p=inf\{x:F(x)\ge p\}$$

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

$$F_n^{-1}(p)=inf\{x:F_n(x)\ge p\}$$

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

$$(X_1,X_{1+k})\stackrel{d}{=}(X_i,X_{i+k}),i\ge 1,k\ge 1.$$

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 $F^{″}$. Suppose that there exists an ${\epsilon }_0>0$ such that for $x\in [{\xi }_p-{\epsilon }_0,{\xi }_p+{\epsilon }_0]$,

$$\sum _{j=2}^{\mathrm{\infty }}j|Cov[I(X_1\le x),I(X_j\le x)]|<\mathrm{\infty }$$

(1.1)

and

$$Var[I(X_1\le {\xi }_p)]+2\sum _{j=2}^{\mathrm{\infty }}Cov[I(X_1\le {\xi }_p),I(X_j\le {\xi }_p)]={\sigma }^2({\xi }_p)>0.$$

(1.2)

Then

$$\underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(\frac{n^{1/2}({\xi }_{p,n}-{\xi }_p)}{\sigma ({\xi }_p)/f({\xi }_p)}\le t)-\mathrm{\Phi }(t)|=O\left(n^{-1/9}\right),n\to \mathrm{\infty }.$$

(1.3)

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

$$v_p=inf\{x:F(x)\ge p\},$$

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 $F^{″}$. Let $n_0$ be some positive integer. Suppose that there exists an ${\epsilon }_0>0$ such that for $x\in [{\xi }_p-{\epsilon }_0,{\xi }_p+{\epsilon }_0]$

$$|Cov[I(X_1\le x),I(X_j\le x)]|\le C{j}^{-5/2},j\ge n_0$$

(2.1)

and condition (1.2) holds. Then

$$\underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(\frac{n^{1/2}({\xi }_{p,n}-{\xi }_p)}{\sigma ({\xi }_p)/f({\xi }_p)}\le t)-\mathrm{\Phi }(t)|=O(n^{-1/6}logn),n\to \mathrm{\infty }.$$

(2.2)

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 $\mathrm{\forall }\epsilon >0$,

$$P\left(\right|\sum _{i=1}^n{X}_i|>\epsilon )\le 2exp\{-\frac{{\epsilon }^2}{2(2{\mathrm{\Delta }}_n+b\epsilon )}\}.$$

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

$$\sum _{j=b_n}^{\mathrm{\infty }}|Cov(X_1,X_j)|=O\left(b_n^{-\beta }\right)$$

(3.1)

for all $0<b_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{n}^{-1}Var\left(\sum _{i=1}^n{X}_i\right)={\sigma }_1^2>0.$$

(3.2)

Then

$$\underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{Var({\sum }_{i=1}^n{X}_i)}}\le t)-\mathrm{\Phi }(t)|=O(n^{-1/6}logn),n\to \mathrm{\infty }.$$

(3.3)

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 $2k_n+1$ subsets with large block of size $\mu ={\mu }_n$ and small block of size $\nu ={\nu }_n$. Let

$${\mu }_n=\lfloor n^{2/3}\rfloor ,{\nu }_n=\lfloor n^{1/3}\rfloor ,k=k_n=\lfloor \frac{n}{{\mu }_n+{\nu }_n}\rfloor =\lfloor n^{1/3}\rfloor$$

and $Z_{n,i}=X_i/\sqrt{Var({\sum }_{i=1}^n{X}_i)}$. Define ${\eta }_j$, ${\xi }_j$, ${\zeta }_k$ as follows:

$$\begin{array}{c}{\eta }_j=\sum _{i=j(\mu +\nu )+1}^{j(\mu +\nu )+\mu }{Z}_{n,i},0\le j\le k-1,{\xi }_j=\sum _{i=j(\mu +\nu )+\mu +1}^{(j+1)(\mu +\nu )}{Z}_{n,i},0\le j\le k-1,\hfill \\ {\zeta }_k=\sum _{i=k(\mu +\nu )+1}^n{Z}_{n,i}.\hfill \end{array}$$

Denote

$$S_n:=\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{Var({\sum }_{i=1}^n{X}_i)}}=\sum _{j=0}^{k-1}{\eta }_j+\sum _{j=0}^{k-1}{\xi }_j+{\zeta }_k:=S_n^{\prime }+S_n^{\prime \prime }+S_n^{\prime \prime \prime }.$$

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

$$\begin{array}{rcl}\underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(S_n\le t)-\mathrm{\Phi }(t)|& =& \underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(S_n^{\prime }+S_n^{\prime \prime }+S_n^{\prime \prime \prime }\le t)-\mathrm{\Phi }(t)|\\ \le & \underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(S_n^{\prime }\le t)-\mathrm{\Phi }(t)|+\frac{2{\epsilon }_n}{\sqrt{2\pi }}\\ +P\left(\right|S_n^{\prime \prime }|>{\epsilon }_n)+P\left(\right|S_n^{\prime \prime \prime }|>{\epsilon }_n),\end{array}$$

(3.4)

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

$$\begin{array}{rcl}P\left(\right|S_n^{\prime \prime }|>{\epsilon }_n)& \le & 2exp\{-\frac{M^2{n}^{-1/3}\cdot {log}^{2}n}{2(2C_1{n}^{-1/3}+C_{2}M{n}^{-1/6}{n}^{-1/6}\cdot logn)}\}\\ =& 2exp\{-\frac{M^2\cdot {log}^{2}n}{2(2C_1+C_{2}Mlogn)}\}\\ \le & C_3{n}^{-1}.\end{array}$$

(3.5)

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

$$\begin{array}{rcl}P\left(\right|S_n^{\prime \prime \prime }|>{\epsilon }_n)& \le & 2exp\{-\frac{M^2{n}^{-1/3}\cdot {log}^{2}n}{2(2C_4{n}^{-1/3}+C_{5}M{n}^{-1/2}{n}^{-1/6}\cdot logn)}\}\\ \le & 2exp\{-\frac{M^2{log}^{2}n}{2(2C_4+C_{5}M)}\}\\ \le & C_6{n}^{-1}.\end{array}$$

(3.6)

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

$$\begin{array}{rcl}|\varphi (t)-\psi (t)|& =& |Eexp\left(it\sum _{j=0}^{k-1}{\eta }_j\right)-\prod _{j=0}^{k-1}Eexp(it{\eta }_j)|\\ \le & 4t^2\sum _{0\le i<j\le k-1}\sum _{l_1=1}^{{\mu }_n}\sum _{l_2=1}^{{\mu }_n}|Cov(Z_{n,{\lambda }_i+l_1},Z_{n,{\lambda }_j+l_2})|\\ \le & \frac{C_1{t}^2}{n}\underset{j-i\ge {\nu }_n}{\sum _{1\le i<j\le n}}|Cov(X_i,X_j)|\\ \le & C_2{t}^2\sum _{j\ge {\nu }_n}|Cov(X_1,X_j)|\\ \le & C_3{t}^2{\nu }_n^{-\beta }\le C_4{t}^2{n}^{-\beta /3}.\end{array}$$

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

$$D_{1n}={\int }_{-T}^T\left|\frac{\varphi (t)-\psi (t)}{t}\right|dt\le C{n}^{-\beta /3}\cdot T^2=O\left(n^{-1/6}\right).$$

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

$$\underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(S_n^{\prime }\le t)-\mathrm{\Phi }(t)|=O\left(n^{-1/6}\right).$$

(3.7)

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

$$|Cov(X_1,X_j)|\le C{j}^{-5/2},j\ge n_0$$

(3.8)

and

$$Var(X_1)+2\sum _{j=2}^{\mathrm{\infty }}Cov(X_1,X_j)={\sigma }_0^2>0.$$

Then

$$\underset{-\mathrm{\infty }<t<\mathrm{\infty }}{sup}|P(\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{n}{\sigma }_0}\le t)-\mathrm{\Phi }(t)|=O(n^{-1/6}logn).$$

(3.9)

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

$$\sum _{j=b_n}^{\mathrm{\infty }}|Cov(X_1,X_j)|\le C\sum _{j=b_n}^{\mathrm{\infty }}{j}^{-5/2}=O\left(b_n^{-3/2}\right),$$

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

$$\begin{array}{rcl}\underset{|t|\le L_n}{sup}|G_n(t)-\mathrm{\Phi }(t)|& \le & \underset{|t|\le L_n}{sup}|P[\frac{{\sum }_{i=1}^n{Z}_i}{\sqrt{n}\sigma (n,t)}<-c_{nt}]-\mathrm{\Phi }(-c_{nt})|+\underset{|t|\le L_n}{sup}|\mathrm{\Phi }(t)-\mathrm{\Phi }(c_{nt})|\\ \le & C_1({\sigma }^2({\xi }_p))n^{-\frac{1}{6}}logn+\underset{|t|\le L_n}{sup}|\mathrm{\Phi }(t)-\mathrm{\Phi }(c_{nt})|,\end{array}$$

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]. □