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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
Corresponding

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 

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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.CrossRefGoogle Scholar
4.de Haan, L. & Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
15.Resnick, S. & Stǎricǎ, C. (1997). Smoothing the Hill estimator. Advances in Applied Probability 29: 271293.CrossRefGoogle Scholar
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