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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 . Our result extends the corresponding one obtaining .
MSC:62F12, 62E20, 60F05.
1 Introduction
First, we will recall the definition of negatively associated (NA) random variables.
Definition 1.1 A finite family is said to be negatively associated (NA) if for any disjoint subsets , and any real coordinatewise nondecreasing functions f on , g on ,
A sequence of random variables is said to be negatively associated (NA) if for every , 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 . Our result extends the corresponding one of Yang et al. [8] obtaining . Let us give some details of the p th quantile.
Let be a sequence of random variables defined on a fixed probability space with a common marginal distribution function , where F is a distribution function (continuous from the right, as usual). For , the p th quantile of F is defined as
and is alternately denoted by . The function , , is called the inverse function of F. With a sample , , let represent the empirical distribution function based on , which is defined as , , where denotes the indicator function of a set A and ℝ is the real line. Let , we define
as the p th quantile of sample.
Throughout the paper, denote some positive constants not depending on n, which may be different in various places. denotes the largest integer not exceeding x and second-order stationary means that
For , denote , and 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 and be a second-order stationary NA sequence with common marginal distribution function F and for . Assume that in a neighborhood of , F possesses a positive continuous density f and a bounded second derivative . Suppose that there exists an such that for ,
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 be the market value of an asset over n periods of a time unit, and let be the log-returns. Suppose is a strictly stationary dependent process with marginal distribution function F. Given a positive value p close to zero, the level VaR is
which specifies the smallest amount of loss such that the probability of the loss in market value being large than 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 and be a second-order stationary NA sequence with common marginal distribution function F. Assume that in a neighborhood of , F possesses a positive continuous density f and a bounded second derivative . Let be some positive integer. Suppose that there exists an such that for
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 in (2.2) is better than 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 in (3.9) of Yang et al. [8], defined by , 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 be a NA sequence with , , a.s. . Denote . Then for ,
Lemma 3.2 Let be a stationary NA sequence with and , . Assume that there exists a such that
for all as 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 into subsets with large block of size and small block of size . Let
and . Define , , as follows:
Denote
By Lemma A.3 in Yang et al. [8] with we have
where M is a positive constant.
Combining the definition of NA with the definition of , , we can easily prove that is NA. Together the condition (3.2) with (2.8) of Yang et al. [8], it has . On the other hand, it can be seen that , . 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 . Since , 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 , , we obtain
On the other hand, we take , , in (2.23) of Yang et al. [8] and have .
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 be a stationary NA sequence with and , . Assume that there exists an such that
and
Then
Proof By the condition (3.8), it is checked that
providing as . 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 is a positive constant depending only on . 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