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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


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


  • [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.

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© 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

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