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Characterization of pre-idempotent Copulas

Author

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  • Chamnan Wongtawan
  • Sumetkijakan Songkiat

    (Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand)

Abstract

Copulas CC for which (CtC)2=CtC{({C}^{t}C)}^{2}={C}^{t}C are called pre-idempotent copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified Sobolev norm, with respect to which the class ℛ{\mathcal{ {\mathcal R} }} of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of ℛ{\mathcal{ {\mathcal R} }}. Identifying copulas with Markov operators on L2{L}^{2}, the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated σ\sigma -algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.

Suggested Citation

  • Chamnan Wongtawan & Sumetkijakan Songkiat, 2023. "Characterization of pre-idempotent Copulas," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-12, January.
    • Handle: RePEc:vrs:demode:v:11:y:2023:i:1:p:12:n:1
    DOI: 10.1515/demo-2023-0106
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    References listed on IDEAS

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    1. Piotr Mikusiński & Michael Taylor, 2010. "Some approximations of n-copulas," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 72(3), pages 385-414, November.
    2. Karl Siburg & Pavel Stoimenov, 2010. "A measure of mutual complete dependence," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 71(2), pages 239-251, March.
    3. Nattakarn Chaidee & Tippawan Santiwipanont & Songkiat Sumetkijakan, 2016. "Patched approximations and their convergence," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(9), pages 2654-2664, May.
    4. Printechapat, Tanes & Sumetkijakan, Songkiat, 2018. "Factorizable non-atomic copulas," Statistics & Probability Letters, Elsevier, vol. 143(C), pages 86-94.
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