Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-29T05:40:24.746Z Has data issue: false hasContentIssue false

THE SECOND-ORDER REGULAR VARIATION OF ORDER STATISTICS

Published online by Cambridge University Press:  13 December 2013

Qing Liu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China. E-mails: qliu8310@mail.ustc.edu.cn; tmao@ustc.edu.cn; thu@ustc.edu.cn
Tiantian Mao
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China. E-mails: qliu8310@mail.ustc.edu.cn; tmao@ustc.edu.cn; thu@ustc.edu.cn
Taizhong Hu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China. E-mails: qliu8310@mail.ustc.edu.cn; tmao@ustc.edu.cn; thu@ustc.edu.cn

Abstract

Let X1, …, Xn be non-negative, independent and identically distributed random variables with a common distribution function F, and denote by X1:n ≤ ··· ≤ Xn:n the corresponding order statistics. In this paper, we investigate the second-order regular variation (2RV) of the tail probabilities of Xk:n and Xj:n − Xi:n under the assumption that $\bar {F}$ is of the 2RV, where 1 ≤ k ≤ n and 1 ≤ i < j ≤ n.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Balakrishnan, N. & Rao, C.R. (1998). Handbook of statistics 16—order statistics: theory and methods. New York: Elsevier.Google Scholar
2.Balakrishnan, N. & Rao, C.R. (1998). Handbook of statistics 17—order statistics: applications. New York: Elsevier.Google Scholar
3.Bingham, N.H., Goldie, C.M., & Teugels, J.L. (1987). Regular variation. Cambridge: Cambridge University Press.Google Scholar
4.de Haan, L. & Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer.Google Scholar
5.de Haan, L. & Resnick, S. (1996). Second-order regular variation and rates of convergence in extreme-value theory. Annals of Probability 24: 97124.Google Scholar
6.Degen, M., Lambrigger, D.D., & Segers, J. (2010). Risk concentration and diversification: second-order properties. Insurance: Mathematics and Economics, 46: 541546.Google Scholar
7.Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events for finance and insurance. Berlin: Springer-Verlag.Google Scholar
8.Geluk, J.L., de Haan, L., Resnick, S., & Stǎricǎ, C. (1997). Second-order regular variation, convolution and the central limit theorem. Sochastic Processes and Their Applications 69: 139159.Google Scholar
9.Hua, L. (2012). Multivariate extremal dependence and risk measures. PhD thesis, University of British Columbia, Vancouver.Google Scholar
10.Hua, L. & Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks. Insurance: Mathematics and Economics 49: 537546.Google Scholar
11.Jessen, A.H. & Mikosch, T. (2006). Regularly varying functions. Publications de L'Institut Mathématique 80: 171192.CrossRefGoogle Scholar
12.Liu, Q., Mao, T., & Hu, T. (2013). Closure properties of the second-order regular variation under convolutions. submitted.Google Scholar
13.Mao, T. & Hu, T. (2012). Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks. Insurance: Mathematics and Economics 51: 333343.Google Scholar
14.Mao, T. & Hu, T. (2012). Second-order properties of risk concentrations without the condition of asymptotic smoothness. Extremes 16: 383405.Google Scholar
15.Resnick, S. & Stǎricǎ, C. (1997). Smoothing the Hill estimator. Advances in Applied Probability 29: 271293.CrossRefGoogle Scholar