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.
KeywordsBerry-Esséen bound sample quantile NA random variables
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.
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