Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Dependence Modeling

Ed. by Puccetti, Giovanni

1 Issue per year

Covered by:
SCOPUS
MathSciNet
zbMATH



Emerging Science

Open Access
Online
ISSN
2300-2298
See all formats and pricing
More options …

Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables

K. Müller
  • Corresponding author
  • University of Rostock, Institute of Mathematics, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ W.-D. Richter
  • Corresponding author
  • University of Rostock, Institute of Mathematics, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-02-22 | DOI: https://doi.org/10.1515/demo-2016-0002

Abstract

A measure-of-cone representation of skewed continuous ln,p-symmetric distributions, n ∈ N, p > 0, is proved following the geometric approach known for elliptically contoured distributions. On this basis, distributions of extreme values of n dependent random variables are derived if the latter follow a joint continuous ln,p-symmetric distribution. Light, heavy, and extremely far tails as well as tail indices are discussed, and new parameters of multivariate tail behavior are introduced.

Keywords: measure-of-cone representation; p-generalized Laplace and Gaussian distributions; skewed ln,psymmetric distribution; tail index, light/ heavy center of distribution

References

  • [1] Arellano-Valle, R. B. and del Pino, G. (2004). From symmetric to asymmetric distributions: a unified approach. In Genton, M. G., editor, Skew-elliptical Distributions and Their Applications: a Journey Beyond Normality, pages 113–128. Chapman & Hall/CRC, Boca Raton, FL. Google Scholar

  • [2] Arellano-Valle, R. B. and Genton, M. G. (2007). On the exact distribution of linear combinations of order statistics from dependent random variables. J. Multivariate Anal. 98(10), 1876–1894. CrossrefWeb of ScienceGoogle Scholar

  • [3] Arellano-Valle, R. B. and Genton, M. G. (2008). On the exact distribution of themaximumof absolutely continuous dependent random variables. Stat. Probab. Lett. 78(1), 27–35. CrossrefGoogle Scholar

  • [4] Arellano-Valle, R. B. and Richter, W.-D. (2012). On skewed continuous ln,p-symmetric distributions. Chil. J. Stat. 3(2), 193– 212. Google Scholar

  • [5] Arslan, O. and Genç, A. I. (2003). Robust location and scale estimation based on the univariate generalized t (GT) distribution. Commun. Stat. Theory Methods 32(8), 1505–1525. CrossrefGoogle Scholar

  • [6] Batún-Cutz, J., González-Farías, G., and Richter, W.-D. (2013). Maximum distributions for l2,p-symmetric vectors are skewed l1,p-symmetric distributions. Stat. Probab. Lett. 83(10), 2260–2268. CrossrefWeb of ScienceGoogle Scholar

  • [7] Cambanis, S., Huang, S., and Simons, G. (1981). On the theory of elliptically contoured distributions. J. Multivariate Anal. 11(3), 368–385. CrossrefGoogle Scholar

  • [8] Castillo, E. (2012). Extreme Value Theory in Engineering. Elsevier, London. Google Scholar

  • [9] David, H. A. and Nagaraja, H. N. (2003). Order Statistics. Wiley, New York, 3rd edition. Google Scholar

  • [10] Demarta, S. and McNeil, A. J. (2005). The t copula and related copulas. Int. Stat. Rev. 73(1), 111–129. CrossrefGoogle Scholar

  • [11] Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. Google Scholar

  • [12] Fortin, J.-Y. and Clusel, M. (2015). Applications of extreme value statistics in physics. J. Phys. A Math. Theor. 48(18). Google Scholar

  • [13] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics. Krieger Publishing Co., Inc., Melbourne, FL, 2nd edition. Google Scholar

  • [14] Galambos, J., Lechner, J., and Simiu, E., editors (1994). Extreme Value Theory and Applications. Springer, Berlin. Google Scholar

  • [15] Gumbel, E. J. (1958). Statistics of Extremes. Columbia University Press, New York. Google Scholar

  • [16] Günzel, T., Richter, W.-D., Scheutzow, S., Schicker, K., and Venz, J. (2012). Geometric approach to the skewed normal distribution. J. Stat. Plann. Inference 142(12), 3209–3224. Google Scholar

  • [17] Gupta, S. S. and Pillai, K. S. (1965). On linear functions of ordered correlated normal random variables. Biometrika 52(3/4), 367–379. CrossrefGoogle Scholar

  • [18] Jamalizadeh, A. and Balakrishnan, N. (2010). Distributions of order statistics and linear combinations of order statistics from an elliptical distribution as mixtures of unified skew-elliptical distributions. J. Multivar. Anal. 101(6), 1412–1427. Web of ScienceCrossrefGoogle Scholar

  • [19] Kella,O. (1986). On the distribution of themaximumof bivariate normal random variableswith generalmeans and variances. Commun. Stat. Theory Methods 15(11), 3265–3276. CrossrefGoogle Scholar

  • [20] Leadbetter, M., Lindgren, G., and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York. Google Scholar

  • [21] Loperfido, N. (2002). Statistical implications of selectively reported inferential results. Stat. Probab. Lett. 56(1), 13–22. CrossrefGoogle Scholar

  • [22] Majumdar, S. N. and Krapivsky, P. L. (2002). Extreme value statistics and traveling fronts: application to computer science. Phys. Rev. E 65(3), 036127. CrossrefGoogle Scholar

  • [23] McDonald, J. B. and Newey, W. K. (1988). Partially adaptive estimation of regression models via the generalized t distribution. Econometric Theory 4(3), 428–457. CrossrefGoogle Scholar

  • [24] McNeil, A. J., Frey, R., and Embrechts, P. (2005). Quantitative RiskManagement. Concepts, Techniques, and Tools. Princeton University Press, Princeton, NJ. Google Scholar

  • [25] Moszynska, M. and Richter,W.-D. (2012). Reverse triangle inequality. Antinorms and semi-antinorms. Stud. Sci.Math. Hung. 49(1), 120–138. Web of ScienceGoogle Scholar

  • [26] Müller, K. and Richter,W.-D. (2015). Exact extreme value, product, and ratio distributions under non-standard assumptions. AStA Adv. Stat. Anal. 99(1), 1–30. CrossrefGoogle Scholar

  • [27] Müller, K. and Richter, W.-D. (2016). Exact distributions of order statistics of dependent random variables from ln,psymmetric sample distributions, n 2 f3, 4g. Depend. Model. 4, 1-29 . Google Scholar

  • [28] Nagaraja, H. N. (1982). Record values and extreme value distributions. J. Appl. Probab. 19(1), 233–239. CrossrefGoogle Scholar

  • [29] Pfeifer, D. (1989). Einführung In Die Extremwertstatistik. (Introduction to Extreme Value Statistics). B.G. Teubner, Stuttgart. Google Scholar

  • [30] Reiss, R.-D. (1989). Approximate Distributions of Order Statistics. With Applications to Nonparametric Statistics. Springer- Verlag, New York. Google Scholar

  • [31] Reiss, R.-D., Haßmann, S., and Thomas, M. (1994). XTREMES: Extreme value analysis and robustness. In Galambos, J., Lechner, J., and Simiu, E., editors, Extreme value theory and applications., pages 175–187. Springer, New York. Google Scholar

  • [32] Reiss, R.-D. and Thomas, M. (1997). Statistical Analysis of ExtremeValues.With Applications to Insurance, Finance, Hydrology and Other Fields. With CD-ROM. Birkhäuser, Basel. Google Scholar

  • [33] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. Google Scholar

  • [34] Richter, W.-D. (1985). Laplace-Gauß integrals, Gaussian measure asymptotic behaviour and probabilities of moderate deviations. Z. Anal. Anwendungen 4(3), 257–267. Google Scholar

  • [35] Richter, W.-D. (1991). Eine geometrische Methode in der Stochastik. In Rostocker Mathematisches Kolloquium volume 44, pages 63–72. Google Scholar

  • [36] Richter, W.-D. (2007). Generalized spherical and simplicial coordinates. J. Math. Anal. Appl. 336(2), 1187–1202. CrossrefGoogle Scholar

  • [37] Richter, W.-D. (2009). Continuous ln,p-symmetric distributions. Lith. Math. J. 49(1), 93–108. CrossrefWeb of ScienceGoogle Scholar

  • [38] Richter,W.-D. (2013). Geometric and stochastic representations for elliptically contoured distributions. Comm. Stat. Theory Methods 42(4), 579–602. CrossrefGoogle Scholar

  • [39] Richter, W.-D. (2014). Geometric disintegration and star-shaped distributions. J. Stat. Distrib. Appl. 1:20. Google Scholar

  • [40] Richter, W.-D. (2015a). Convex and radially concave contoured distributions. J. Probab. Stat. 2015. Article ID 165468. CrossrefGoogle Scholar

  • [41] Richter, W.-D. (2015b). Norm contoured distributions in R2. In Lecture notes of Seminario Interdisciplinare di Matematica. Vol. XII, volume 12, pages 179–199. Potenza: Seminario Interdisciplinare di Matematica (S.I.M.), University of Basilicata, Italy. Google Scholar

  • [42] Richter, W.-D. and Venz, J. (2014). Geometric representations of multivariate skewed elliptically contoured distributions. Chil. J. Stat. 5(2), 71–90. Google Scholar

  • [43] Shibata, T. (1994). Application of extreme value statistics to corrosion. J. Res. Nati. Inst. Stand. Technol., 99(4), 327–327. CrossrefGoogle Scholar

About the article

Received: 2015-10-12

Accepted: 2016-02-05

Published Online: 2016-02-22


Citation Information: Dependence Modeling, Volume 4, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2016-0002.

Export Citation

© 2016 K. Müller and W.-D. Richter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Wolf-Dieter Richter
Journal of Statistical Distributions and Applications, 2017, Volume 4, Number 1
[2]
Wolf-Dieter Richter and Kay Schicker
Journal of Probability and Statistics, 2017, Volume 2017, Page 1

Comments (0)

Please log in or register to comment.
Log in