Journal
FUZZY SETS AND SYSTEMS
Volume 428, Issue -, Pages 58-79Publisher
ELSEVIER
DOI: 10.1016/j.fss.2020.12.022
Keywords
Copula; Transformation; TP2 property; Dependence; Tail dependence coefficients; Concordance order; Geometric convexity; Symmetry
Funding
- Research Council of Lithuania [S-MIP-20-16]
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This paper provides a comprehensive characterization of functions f: [0, 1] -> R+ such that C-f satisfies specific properties, including two special cases and general conditions. Various examples are given to illustrate the obtained results, and properties of such copulas are described in detail.
This paper deals with the problem of characterizing all functions f : [0, 1] -> R+ such that C-f (x, y) = xyf ((1 - x)(1 - y)), x, y is an element of [0, 1] is a bivariate copula. We provide a complete characterization for the two cases: (i) when C-f is, in addition, totally positive of order 2 (TP2) and (ii) when f is twice continuously differentiable. In general, the function f need only be twice differentiable Lebesgue almost everywhere as shown by investigating necessary conditions for C-f to be a copula. The paper also contains numerous examples illustrating obtained results and connections to known facts from the literature. Moreover, several properties of such copulas are described. (C) 2020 Elsevier B.V. All rights reserved.
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