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Generalized Diagonal Band Copulas with Two-Sided Generating Densities

Author

Listed:
  • Samuel Kotz

    (Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052)

  • Johan René van Dorp

    (Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052)

Abstract

Copulas are joint continuous distributions with uniform marginals and have been proposed to capture probabilistic dependence between random variables. Maximum-entropy copulas introduced by Bedford and Meeuwissen (Bedford, T., A. M. H. Meeuwissen. 1997. Minimally informative distributions with given rank correlations for use in uncertainty analysis. J. Statist. Comput. Simulation 57 (1--4) 143--175) provide the option of making minimally informative assumptions given a degree-of-dependence constraint between two random variables. Unfortunately, their distribution functions are not available in a closed form, and their application requires the use of numerical methods. In this paper, we study a subfamily of generalized diagonal band (GDB) copulas, separately introduced by Ferguson (Ferguson, T. F. 1995. A class of symmetric bivariate uniform distributions. Statist. Papers 36 (1) 31--40) and Bojarski (Bojarski, J. 2001. A new class of band copulas---Distributions with uniform marginals. Technical report, Institute of Mathematics, Technical University of Zielona Góra, Zielona Góra, Poland). Similar to Archimedean copulas, GDB copula construction requires a generator function. Bojarski's GDB copula generator functions are symmetric probability density functions. In this paper, symmetric members of a two-sided framework of distributions introduced by van Dorp and Kotz (van Dorp, J. R., S. Kotz. 2003. Generalizations of two-sided power distributions and their convolution. Comm. Statist.: Theory and Methods 32 (9) 1703--1723) shall be considered. This flexible setup allows for derivations of GDB copula properties resulting in novel convenient expressions. A straightforward elicitation procedure for the GDB copula dependence parameter is proposed. Closed-form expressions for specific examples in the subfamily of GDB copulas are presented, which enhance their transparency and facilitate their application. These examples closely approximate the entropy of maximum-entropy copulas. Application of GDB copulas is illustrated via a value-of-information decision analysis example.

Suggested Citation

  • Samuel Kotz & Johan René van Dorp, 2010. "Generalized Diagonal Band Copulas with Two-Sided Generating Densities," Decision Analysis, INFORMS, vol. 7(2), pages 196-214, June.
    • Handle: RePEc:inm:ordeca:v:7:y:2010:i:2:p:196-214
    DOI: 10.1287/deca.1090.0162
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    References listed on IDEAS

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    2. Andy Lewin & Ekundayo Shittu & Thomas Mazzuchi & Rene Dorp, 2021. "The correlation of cost and schedule variance in satellite programs: level of effort versus discrete cost accounts," Environment Systems and Decisions, Springer, vol. 41(2), pages 248-266, June.
    3. L. Robin Keller & Ali Abbas & J. Eric Bickel & Vicki M. Bier & David V. Budescu & John C. Butler & Enrico Diecidue & Robin L. Dillon-Merrill & Raimo P. Hämäläinen & Kenneth C. Lichtendahl & Jason R. W, 2012. "From the Editors ---Brainstorming, Multiplicative Utilities, Partial Information on Probabilities or Outcomes, and Regulatory Focus," Decision Analysis, INFORMS, vol. 9(4), pages 297-302, December.
    4. Tianyang Wang & James S. Dyer, 2012. "A Copulas-Based Approach to Modeling Dependence in Decision Trees," Operations Research, INFORMS, vol. 60(1), pages 225-242, February.
    5. L. Robin Keller & Kelly M. Kophazi, 2012. "From the Editors ---Copulas, Group Preferences, Multilevel Defenders, Sharing Rewards, and Communicating Analytics," Decision Analysis, INFORMS, vol. 9(3), pages 213-218, September.
    6. Donatella Vicari & Johan Ren� van Dorp, 2013. "On a bounded bimodal two-sided distribution fitted to the Old-Faithful geyser data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 40(9), pages 1965-1978, September.
    7. Tim Bedford & Alireza Daneshkhah & Kevin J. Wilson, 2016. "Approximate Uncertainty Modeling in Risk Analysis with Vine Copulas," Risk Analysis, John Wiley & Sons, vol. 36(4), pages 792-815, April.
    8. L. Robin Keller & Kelly M. Kophazi, 2010. "From the Editors..," Decision Analysis, INFORMS, vol. 7(2), pages 151-154, June.

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