Berry-Esséen bound of sample quantiles for negatively associated sequence

Yang Wenzhi, Hu Shuhe, Wang Xuejun, Zhang, Zhang Qinchi

Description

In this paper, we investigate the Berry-Esséen bound of the sample quantiles for the negatively associated random variables under some weak conditions. The rate of normal approximation is shown as O ( n -1/9 ). 2010 Mathematics Subject Classification : 62F12; 62E20; 60F05. In this paper, we investigate the Berry-Esséen bound of the sample quantiles for the negatively associated random variables under some weak conditions. The rate of normal approximation is shown as O ( n -1/9 ). 2010 Mathematics Subject Classification : 62F12; 62E20; 60F05.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 19
• Langue: English

Extrait

Yanget al.Journal of Inequalities and Applications2011,2011:83 http://www.journalofinequalitiesandapplications.com/content/2011/1/83

R E S E A R C HOpen Access BerryEsséen bound of sample quantiles for negatively associated sequence 1 1*1 2 Wenzhi Yang , Shuhe Hu, Xuejun Wangand Qinchi Zhang

* Correspondence: hushuhe@263. net 1 School of Mathematical Science, Anhui University Hefei 230039, PR China Full list of author information is available at the end of the article

Abstract In this paper, we investigate the BerryEsséen bound of the sample quantiles for the negatively associated random variables under some weak conditions. The rate of 1/9 normal approximation is shown asO(n). 2010 Mathematics Subject Classification: 62F12; 62E20; 60F05. Keywords:BerryEss?é?en bound, sample quantile, negatively associated

1 Introduction Assume that {X n } n≥1 is a sequence of random variables defined on a fixed probability space , F ,P with a common marginal distribution functionF(x) =P(X 1 ≤x).Fis a distribution function (continuous from the right, as usual). For 0 <p< 1, thepth quan tile ofFis defined as ξ= inf{x:F(x)≥p 1 1 and is alternately denoted byF(p). The functionF(t), 0 <t< 1, is called the inverse function ofF. It is easy to check thatξ p possesses the following properties:

(i)F(ξ p )≤p≤F(ξ p ); (ii) ifξ p is the unique solutionxofF(x)≤p≤F(x), then for anyε>0, F(ξ−ε)<p<F(ξ+ε) For a sampleX 1 ,X 2 , ...,X n ,n≥1, letF n represent the empirical distribution function  1 n I(X≤x,xÎℝ, whereI(A) based onX 1 ,X 2 ,...,X n , which is defined asF n (x) = i=1 i denotes the indicator function of a setAandℝis the real line. For 0 <p< 1, we define −1 F(p) = inf{x:F n (x)≥p as thepth quantile of sample. Recall that a finite family {X 1 ,...,X n } is said to be negatively associated (NA) if for any disjoint subsetsA,B⊂{1, 2,...,n}, and any real coordinatewise nondecreasing functions A B fonR ,gonR, Covf X k ,k∈A,g X k ,k∈B≤0 A sequence of random variables {X i } i≥1 is said to be NA if for everyn≥2,X 1 ,X 2 ,..., X n are NA. From 1960s, many authors have obtained the asymptotic results for the sample quan tiles, including the wellknown Bahadur representation. Bahadur [1] firstly introduced © 2011 Yang et al; licensee Springer. 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.