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

Dependence Modeling

Ed. by Puccetti, Giovanni

1 Issue per year

Covered by:

Open Access
See all formats and pricing
More options …

Exact distributions of order statistics of dependent random variables from ln,p-symmetric sample distributions, n ∈ {3,4}

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-2016-0001


Integral representations of the exact distributions of order statistics are derived in a geometric way when three or four random variables depend on each other as the components of continuous ln,psymmetrically distributed random vectors do, n ∈ {3,4}, p > 0. Once the representations are implemented in a computer program, it is easy to change the density generator of the ln,p-symmetric distribution with another one for newly evaluating the distribution of interest. For two groups of stock exchange index residuals, maximum distributions are compared under dependence and independence modeling.

Keywords: density generator; extreme value statistics; geometric measure representation; p-generalized Gaussian and Laplace distributions; financial data analysis


  • [1] 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

  • [2] 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

  • [3] 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

  • [4] Arnold, B. C., Castillo, E., and Sarabia, J. M. (2007). Variations on the classical multivariate normal theme. J. Stat. Plann. Inference 137(11), 3249–3260. Google Scholar

  • [5] 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

  • [6] Box, G. E. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley Publishing Company, Reading, Mass. Google Scholar

  • [7] Clauset, A., Shalizi, C. R., and Newman, M. (2009). Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703. CrossrefWeb of ScienceGoogle Scholar

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

  • [9] Dietrich, T., Kalke, S., and Richter, W.-D. (2013). Stochastic representations and a geometric parametrization of the twodimensional Gaussian law. Chil. J. Stat. 4(2), 27–59. Google Scholar

  • [10] Fang, K.-T., Kotz, S., and Ng, K.-W. (1990). Symmetric Multivariate And Related Distributions. Chapman and Hall, London. Google Scholar

  • [11] Gómez, E., Gómez-Villegas, M. A., and Marín, J. M. (1998). A multivariate generalization of the power exponential family of distributions. Commun. Stat. Theory Methods 27(3), 589–600. CrossrefGoogle Scholar

  • [12] 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

  • [13] Gupta, A. K. and Song, D. (1997). lp-norm spherical distributions. J. Stat. Plann. Inference 60(2), 241–260. CrossrefGoogle Scholar

  • [14] 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

  • [15] Jondeau, E., Poon, S.-H., and Rockinger, M. (2007). FinancialModelingUnder Non-Gaussian Distributions. Springer, London. Google Scholar

  • [16] Kalke, S. and Richter, W.-D. (2013). Simulation of the p-generalized Gaussian distribution. J. Statist. Comput. Simulation 83(4), 639–665. Google Scholar

  • [17] Kalke, S., Richter, W.-D., and Thauer, F. (2013). Linear combinations, products and ratios of simplicial or spherical variates. Comm. Stat. Theory Methods 42(3), 505–527. CrossrefGoogle Scholar

  • [18] 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

  • [19] 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

  • [20] Nadarajah, S. (2003). The Kotz-type distribution with applications. Statistics 37(4), 341–358. CrossrefGoogle Scholar

  • [21] Nadarajah, S. (2005). A generalized normal distribution. J. Appl. Stat. 32(7), 685–694. CrossrefGoogle Scholar

  • [22] Nadarajah, S. (2006). Acknowledgement of priority: The generalized normal distribution. J. Appl. Stat. 33(9), 1031–1032. CrossrefGoogle Scholar

  • [23] Osiewalski, J. and Steel, M. F. (1993). Robust Bayesian inference in lq-spherical models. Biometrika 80(2), 456–460. Google Scholar

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

  • [25] Richter, W.-D. (2008a). On l2,p-circle numbers. Lith. Math. J. 48(2), 228–234. Web of ScienceCrossrefGoogle Scholar

  • [26] Richter, W.-D. (2008b). On the Pi-function for nonconvex l2,p-circle discs. Lith. Math. J. 48(3), 332–338. CrossrefWeb of ScienceGoogle Scholar

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

  • [28] Richter, W.-D. (2014a). Classes of standard Gaussian random variables and their generalizations. In Knif, J. and Pape, B., editors, Contributions to Mathematics, Statistics, Econometrics, and Finance, pages 53–69. University of Vaasa. Google Scholar

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

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

  • [31] 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 diMatematica (S.I.M.), University of Basilicata, Italy. Google Scholar

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

  • [33] Sarabia, J. M. and Gómez-Déniz, E. (2008). Construction of multivariate distributions: a review of some recent results. SORT 32(1), 3–36. Google Scholar

  • [34] Silverman, H. (1994). The value of reproving. PRIMUS 4(2), 151–154. CrossrefGoogle Scholar

  • [35] Subbotin, M. (1923). On the law of frequency of error. Rec. Math. Moscou 31(2), 296–301. Google Scholar

  • [36] Theodossiou, P. (1998). Financial data and the skewed generalized t distribution. Manage. Sci. 44(12), 1650–1661. 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-2016-0001.

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.

Wolf-Dieter Richter
Journal of Statistical Distributions and Applications, 2017, Volume 4, Number 1
Eckhard Liebscher and Wolf-Dieter Richter
Risks, 2016, Volume 4, Number 4, Page 44
Klaus Müller and Wolf-Dieter Richter
Journal of Statistical Computation and Simulation, 2017, Volume 87, Number 5, Page 933

Comments (0)

Please log in or register to comment.
Log in