Constructing families of symmetric dependence functions

Włodzimierz Wysocki

Kybernetika (2012)

  • Volume: 48, Issue: 5, page 977-987
  • ISSN: 0023-5954

Abstract

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We construct two pairs ( 𝒜 F [ 1 ] , 𝒜 F [ 2 ] ) and ( 𝒜 ψ [ 1 ] , 𝒜 ψ [ 2 ] ) of ordered parametric families of symmetric dependence functions. The families of the first pair are indexed by regular distribution functions F , and those of the second pair by elements ψ of a specific function family ψ . We also show that all solutions of the differential equation d y d u = α ( u ) u ( 1 - u ) y for α in a certain function family α s are symmetric dependence functions.

How to cite

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Wysocki, Włodzimierz. "Constructing families of symmetric dependence functions." Kybernetika 48.5 (2012): 977-987. <http://eudml.org/doc/251414>.

@article{Wysocki2012,
abstract = {We construct two pairs $(\mathcal \{A\}^\{[1]\}_\{F\}, \mathcal \{A\}^\{[2]\}_\{F\})$ and $(\mathcal \{A\}^\{[1]\}_\{\psi \}, \mathcal \{A\}^\{[2]\}_\{\psi \})$ of ordered parametric families of symmetric dependence functions. The families of the first pair are indexed by regular distribution functions $F$, and those of the second pair by elements $\psi $ of a specific function family $\mathbb \{\psi \}$. We also show that all solutions of the differential equation $\frac\{\{\mathrm \{d\}\}y\}\{\{\mathrm \{d\}\}u\}=\frac\{\alpha (u)\}\{u(1-u)\}y$ for $\alpha $ in a certain function family $\{\mathbb \{\alpha \}\}_\{\rm s\}$ are symmetric dependence functions.},
author = {Wysocki, Włodzimierz},
journal = {Kybernetika},
keywords = {archimax copula; copula; dependence function; generator of a dependence function; archimax copula; copula; dependence function; generator of a dependence function},
language = {eng},
number = {5},
pages = {977-987},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Constructing families of symmetric dependence functions},
url = {http://eudml.org/doc/251414},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Wysocki, Włodzimierz
TI - Constructing families of symmetric dependence functions
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 5
SP - 977
EP - 987
AB - We construct two pairs $(\mathcal {A}^{[1]}_{F}, \mathcal {A}^{[2]}_{F})$ and $(\mathcal {A}^{[1]}_{\psi }, \mathcal {A}^{[2]}_{\psi })$ of ordered parametric families of symmetric dependence functions. The families of the first pair are indexed by regular distribution functions $F$, and those of the second pair by elements $\psi $ of a specific function family $\mathbb {\psi }$. We also show that all solutions of the differential equation $\frac{{\mathrm {d}}y}{{\mathrm {d}}u}=\frac{\alpha (u)}{u(1-u)}y$ for $\alpha $ in a certain function family ${\mathbb {\alpha }}_{\rm s}$ are symmetric dependence functions.
LA - eng
KW - archimax copula; copula; dependence function; generator of a dependence function; archimax copula; copula; dependence function; generator of a dependence function
UR - http://eudml.org/doc/251414
ER -

References

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  5. Hürlimann, W., Properties and measures of dependence for the archimax copula., Adv. Appl. Statist. 5 (2005), 125-143. MR2204874
  6. Hutchinson, T. P., Lai, C. D., Continuous Bivariate Distributions. Emphasising Applications., Rumsby Sci. Publ., Adelaide 1990. Zbl1170.62330MR1070715
  7. Joe, H., Multivariate Models and Dependence Concepts., Chapman and Hall, London 1997. Zbl0990.62517MR1462613
  8. Nelsen, R. B., An Introduction to Copulas., Springer, New York 1999. Zbl1152.62030MR1653203
  9. Pickands, J., Multivariate extreme value distributions., Bull. Int. Statist. Inst. 49 (1981), 859-879. Zbl0518.62045MR0820979
  10. Wysocki, W., When a copula is archimax., Statist. Probab. Lett. (2012), to appear. 

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