The Berry-Esséen bound of sample quantiles for NA sequence
© Liu et al.; licensee Springer. 2014
Received: 28 July 2013
Accepted: 5 February 2014
Published: 17 February 2014
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.
First, we will recall the definition of negatively associated (NA) random variables.
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 . There are many good results of NA random variables. For example, Matula  obtained the three series theorem, Su et al.  gave the moment inequality, Shao  investigated the maximal inequality, Yuan et al.  studied the central limit theorem, Yang  and Sung  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.  obtaining . Let us give some details of the p th quantile.
as the p th quantile of sample.
For , denote , and is the distribution function of a standard normal variable. Yang et al. [, Theorem 1.1] presented the Berry-Esséen bound of sample quantiles for an NA sequence as follows.
For the work on Berry-Esséen bounds of sample quantiles, one can refer to Reiss  or Chapter 2 of Serfling . Cai and Roussas  studied the smooth estimate of quantiles under an association sample, Rio  obtained the Berry-Esséen bounds of sample quantiles under a φ-mixing sequence, Lahiri and Sun  and Yang et al.  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 , Chapter 5 of Petrov , Gao et al. , Chapter 5 of Härdle et al. , and to the references therein too.
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  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
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. . 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. , was used the Lemma 2.2 of Yang et al. , which requires the condition of mean zero, but in (3.9) of Yang et al. , defined by , satisfies the condition of mean zero. Thus, the mean zero condition of Theorem 1.1 of Yang et al.  is not needed. It coincides with the independent case, which does not need the mean zero condition. For the details, one can see Serfling [, Theorem C, p.81] or Theorem A of Yang et al. .
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 [, Lemma 3.5]
where M is a positive constant.
On the other hand, we take , , in (2.23) of Yang et al.  and have .
Finally, by (3.4)-(3.7), (3.3) holds. □
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. , we have (3.9) finally. □
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. . □
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.
- Joag-Dev K, Proschan F: Negative association of random variables with applications. Ann. Stat. 1983,11(1):286–295. 10.1214/aos/1176346079MathSciNetView ArticleGoogle Scholar
- Matula P: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 1992,15(3):209–213. 10.1016/0167-7152(92)90191-7MathSciNetView ArticleGoogle Scholar
- Su C, Zhao LC, Wang YB: Moment inequalities and weak convergence for negatively associated sequences. Sci. China Ser. A 1997,40(2):172–182. 10.1007/BF02874436MathSciNetView ArticleGoogle Scholar
- Shao QM: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 2000,13(2):343–356. 10.1023/A:1007849609234View ArticleGoogle Scholar
- Yuan M, Su C, Hu TZ: A central limit theorem for random fields of negatively associated processes. J. Theor. Probab. 2003,16(2):309–323. 10.1023/A:1023538824937MathSciNetView ArticleGoogle Scholar
- Yang SC: Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples. Stat. Probab. Lett. 2003,62(2):101–110. 10.1016/S0167-7152(02)00427-3View ArticleGoogle Scholar
- Sung SH: On the exponential inequalities for negatively dependent random variables. J. Math. Anal. Appl. 2011,381(2):538–545. 10.1016/j.jmaa.2011.02.058MathSciNetView ArticleGoogle Scholar
- Yang WZ, Hu SH, Wang XJ, Zhang QC: Berry-Esséen bound of sample quantiles for negatively associated sequence. J. Inequal. Appl. 2011., 2011: Article ID 83Google Scholar
- Reiss RD: On the accuracy of the normal approximation for quantiles. Ann. Probab. 1974,2(4):741–744. 10.1214/aop/1176996617MathSciNetView ArticleGoogle Scholar
- Serfling RJ: Approximation Theorems of Mathematical Statistics. Wiley, New York; 1980.View ArticleGoogle Scholar
- Cai ZW, Roussas GG: Smooth estimate of quantiles under association. Stat. Probab. Lett. 1997,36(3):275–287. 10.1016/S0167-7152(97)00074-6MathSciNetView ArticleGoogle Scholar
- Rio E: Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. Probab. Theory Relat. Fields 1996,104(2):255–282. 10.1007/BF01247840MathSciNetView ArticleGoogle Scholar
- Lahiri SN, Sun S: A Berry-Esseen theorem for samples quantiles under weak dependence. Ann. Appl. Probab. 2009,19(1):108–126. 10.1214/08-AAP533MathSciNetView ArticleGoogle Scholar
- Yang WZ, Hu SH, Wang XJ, Ling NX: The Berry-Esséen type bound of sample quantiles for strong mixing sequence. J. Stat. Plan. Inference 2012,142(3):660–672. 10.1016/j.jspi.2011.09.004MathSciNetView ArticleGoogle Scholar
- Hall P, Heyde CC: Martingale Limit Theory and Its Application. Academic Press, New York; 1980.Google Scholar
- Petrov VV: Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford University Press, New York; 1995.Google Scholar
- Gao JT, Hong SY, Liang H: Berry-Esséen bounds of error variance estimation in partly linear models. Chin. Ann. Math., Ser. B 1996,17(4):477–490.MathSciNetGoogle Scholar
- Härdle W, Liang H, Gao J Springer Series in Economics and Statistics. In Partially Linear Models. Physica-Verlag, New York; 2000.View ArticleGoogle Scholar
- Chen SX, Tang CY: Nonparametric inference of value-at-risk for dependent financial returns. J. Financ. Econom. 2005,3(2):227–255. 10.1093/jjfinec/nbi012View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.