Statistical Methods & Applications

, Volume 20, Issue 4, pp 463–486 | Cite as

Modeling heavy-tailed, skewed and peaked uncertainty phenomena with bounded support

  • C. B. García
  • J. García Pérez
  • J. R. van DorpEmail author


A prevalence of heavy-tailed, peaked and skewed uncertainty phenomena have been cited in literature dealing with economic, physics, and engineering data. This fact has invigorated the search for continuous distributions of this nature. In this paper we shall generalize the two-sided framework presented in Kotz and van Dorp (Beyond beta: other continuous families of distributions with bounded support and applications. World Scientific Press, Singapore, 2004) for the construction of families of distributions with bounded support via a mixture technique utilizing two generating densities instead of one. The family of Elevated Two-Sided Power (ETSP) distributions is studied as an instance of this generalized framework. Through a moment ratio diagram comparison, we demonstrate that the ETSP family allows for a remarkable flexibility when modeling heavy-tailed and peaked, but skewed, uncertainty phenomena. We shall demonstrate its applicability via an illustrative example utilizing 2008 US income data.


Uncertainty modeling Applied probability Income distribution Lorentz curve 


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  1. Aban IB, Meerschaert MM, Panorska AK (2006) Parameter estimation for the truncated Pareto distribution. J Am Stat Assoc 101(473): 270–277MathSciNetzbMATHCrossRefGoogle Scholar
  2. Adler R, Feldman R, Taqqu M (1998) A practical guide to heavy tails. Birkhäuser, BostonzbMATHGoogle Scholar
  3. Arnold BC (1983) Pareto distributions. International Cooperative Publishing House, FairlandzbMATHGoogle Scholar
  4. Barkai E, Metzler R, Klafter J (2000) From continuous time random walks to the fractional Fokker-Planck equation. Phys Rev E61: 132–138MathSciNetGoogle Scholar
  5. Barsky R, Bound J, Kerwin KC, Lupton JP (2002) Accounting for the black-white wealth gap: a nonparametric approach. J Am Stat Assoc 97(459): 663–673zbMATHCrossRefGoogle Scholar
  6. Clementi F, Gallegatib M (2005) Power law tails in the Italian personal income distribution. Phys A Stat Mech Appl 350(2–4): 427–438CrossRefGoogle Scholar
  7. Coelho R, Richmond P, Barrya J, Hutzlera S (2008) Double power laws in income and wealth distributions. Phys A Stat Mech Appl 387(15): 3847–3851CrossRefGoogle Scholar
  8. Douglas EM, Barros AP (2003) Probable maximum precipitation estimation using multifractals: application in the Eastern United States. J Hydrometeorol 4(6): 1012–1024CrossRefGoogle Scholar
  9. Elderton WP, Johnson NL (1969) Systems of frequency curves. Cambridge University Press, LondonzbMATHCrossRefGoogle Scholar
  10. Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, BerlinzbMATHGoogle Scholar
  11. Fernandez C, Steel MFJ (1998) On Bayesian modeling of fat tails and skewness. J Am Stat Assoc 93: 359–371MathSciNetzbMATHCrossRefGoogle Scholar
  12. Gomez HW, Torres FJ, Bolfarine H (2007) Large-sample inference for the epsilon-skew-distribution. Commun Stat Theory Methods 36: 73–81MathSciNetzbMATHCrossRefGoogle Scholar
  13. Hahn ED (2008) Mixture densities for project management activity times: a robust approach to PERT. Eur J Oper Res 188: 450–459zbMATHCrossRefGoogle Scholar
  14. Herrerías-Velasco JM, Herrerías-Pleguezuelo R, van Dorp JR (2008) The generalized two-sided power distribution. J Appl Stat 36(5): 573–587CrossRefGoogle Scholar
  15. Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. Wiley, HobokenzbMATHCrossRefGoogle Scholar
  16. Kotz S, Johnson NL (1985) Moment ratio diagrams, In: Encyclopedia of statistical sciences, vol 5. Wiley, New YorkGoogle Scholar
  17. Kotz S, Kozubowski TJ, Podgórski K (2001) The Laplace distribution and generalizations. Birkhäuser, BostonzbMATHCrossRefGoogle Scholar
  18. Kotz S, van Dorp JR (2004) Beyond beta: other continuous families of distributions with bounded support and applications. World Scientific Press, SingaporezbMATHCrossRefGoogle Scholar
  19. Kotz S, van Dorp JR (2005) A link between two-sided power and asymmetric laplace distributions: with applications to mean and variance approximations. Stat Probabil Lett 71: 382–394MathSciNetCrossRefGoogle Scholar
  20. Lévy P (1925) Calcul des probabilitiés, 2nd part, Chap. VI. Gauthier-Villars, ParisGoogle Scholar
  21. Levy H, Duchin R (2004) Asset return distribution and the investment horizon, explaining contradictions. J Portfolio Manage 30(3): 47–62CrossRefGoogle Scholar
  22. Lu SL, Molz FJ (2001) How well are hydraulic conductivity variations approximated by additive stable processes?. Adv Environ Res 5(1): 39–45CrossRefGoogle Scholar
  23. McFall Lamm Jr R (2003) Asymmetric returns and optimal hedge fund portfolios. J Alternat Invest, pp 9–21Google Scholar
  24. McCulloch J (1996) Financial applications of stable distributions. In: Madfala G, Rao CR (eds) Statistical methods in finance. Elsevier, Amsterdam, pp 393–425Google Scholar
  25. Miyazima S, Yamamoto K (2006) Power-law behaviors in high income distribution. Pract Fruits Econophys 5: 344–348CrossRefGoogle Scholar
  26. Nagahara Y (1999) The PDF and CDF of Pearson Type IV distributions and the ML estimation of the parameters. Stat Probabil Lett 43: 251–264MathSciNetzbMATHCrossRefGoogle Scholar
  27. Pareto V (1964) Cours d’Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino. Librairie Droz, Geneva 299–345Google Scholar
  28. Press WH, Flannery BP, Teukolsky SA, Vettering WT (1989) Numerical recipes in pascal. Cambridge University Press, CambridgezbMATHGoogle Scholar
  29. Resnick S (1997) Heavy tail modeling and teletraffic data. Ann Stat 25: 1805–1869MathSciNetzbMATHCrossRefGoogle Scholar
  30. Samorodnitsky G, Taqqu M (2004) Stable non-Gaussian random processes. Chapman & Hall, New YorkGoogle Scholar
  31. Sarabia JM (2008) Parametric Lorenz curves: models and applications, Modeling Income Distributions and Lorenz Curves. Economic studies in inequality. In: Chotikapanich D (eds) Social exclusion and well-being vol 4. Springer, Berlin, pp 167–190Google Scholar
  32. Singh A, van Dorp JR, Mazzuchi TA (2007) A novel assymetric distribution with power tails. Commun Stat Theory Methods 36(2): 235–252MathSciNetzbMATHCrossRefGoogle Scholar
  33. Solomon S, Levy M (2000) Market ecology, Pareto wealth distribution and leptokurtic returns in microscopic simulation of the LLS stock market model. In: Proceedings of complex behavior in economics, Aix en Provence (Marseille), France, May 4–6Google Scholar
  34. Stuart A, Ord JK (1994) Kendall’s advanced theory of statistics. Wiley, New YorkGoogle Scholar
  35. Van Dorp JR, Kotz S (2002) The standard two sided power distribution and its properties: with applications in financial engineering. Am Stat 56(2): 90–99MathSciNetzbMATHCrossRefGoogle Scholar
  36. Van Dorp JR, Kotz S (2003) Generalizations of two-sided power distributions and their convolution. Commun Stat Theory Methods 32(9): 1703–1723MathSciNetzbMATHCrossRefGoogle Scholar
  37. Zabell SL (2008) On Student’s 1908 Article “The Probable Error of a Mean”. J Am Stat Assoc 103(481): 1–7MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • C. B. García
    • 1
  • J. García Pérez
    • 2
  • J. R. van Dorp
    • 3
    Email author
  1. 1.Departamento de Métodos Cuantitativos para la Economía y la Empresa, Facultad de Ciencias Económicas y EmpresarialesUniversity of GranadaGranadaSpain
  2. 2.Departamento de Economía AplicadaUniversity of AlmeríaAlmeríaSpain
  3. 3.Department of Engineering Management and Systems EngineeringThe George Washington UniversityWashingtonUSA

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