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THE SECOND-ORDER REGULAR VARIATION OF ORDER STATISTICS

  • Qing Liu (a1), Tiantian Mao (a1) and Taizhong Hu (a1)
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.

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1.Balakrishnan, N. & Rao, C.R. (1998). Handbook of statistics 16—order statistics: theory and methods. New York: Elsevier.
2.Balakrishnan, N. & Rao, C.R. (1998). Handbook of statistics 17—order statistics: applications. New York: Elsevier.
3.Bingham, N.H., Goldie, C.M., & Teugels, J.L. (1987). Regular variation. Cambridge: Cambridge University Press.
4.de Haan, L. & Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer.
5.de Haan, L. & Resnick, S. (1996). Second-order regular variation and rates of convergence in extreme-value theory. Annals of Probability 24: 97124.
6.Degen, M., Lambrigger, D.D., & Segers, J. (2010). Risk concentration and diversification: second-order properties. Insurance: Mathematics and Economics, 46: 541546.
7.Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events for finance and insurance. Berlin: Springer-Verlag.
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.
9.Hua, L. (2012). Multivariate extremal dependence and risk measures. PhD thesis, University of British Columbia, Vancouver.
10.Hua, L. & Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks. Insurance: Mathematics and Economics 49: 537546.
11.Jessen, A.H. & Mikosch, T. (2006). Regularly varying functions. Publications de L'Institut Mathématique 80: 171192.
12.Liu, Q., Mao, T., & Hu, T. (2013). Closure properties of the second-order regular variation under convolutions. submitted.
13.Mao, T. & Hu, T. (2012). Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks. Insurance: Mathematics and Economics 51: 333343.
14.Mao, T. & Hu, T. (2012). Second-order properties of risk concentrations without the condition of asymptotic smoothness. Extremes 16: 383405.
15.Resnick, S. & Stǎricǎ, C. (1997). Smoothing the Hill estimator. Advances in Applied Probability 29: 271293.
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Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
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