A note on metrics induced by copulas

doi:10.1016/j.fss.2011.08.003

Fuzzy Sets and Systems

Volume 191, 16 March 2012, Pages 122-125

https://doi.org/10.1016/j.fss.2011.08.003 Get rights and content

Abstract

In this short note, we provide a continuous triangular norm that is not a copula and that generates a metric. Thus, the 9th open problem collected in Alsina et al. [Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Company, Singapore, 2006] has a negative answer.

Highlights

► The metrics induced by copulas are discussed. ► We provide a continuous t-norm that is not a copula and that generates a metric. ► The 9th open problem in a recent monograph of Alsina et al. is solved.

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The idea of constructing a distance function from fuzzy logic connectives such as t-norms, and its dual t-conorms, was originally introduced by Alsina in [1], wherein it was shown that this distance function turns out to be a metric if the t-norm is a copula. The converse, however, need not be true, i.e., t-norms that are not copulas can give rise to a metric, for instance, continuous non-strict Archimedean t-norms (see [2]). Alsina, in his work [4] dealing with the construction of metrics obtained from quasi-copulas, has also shown that various concepts of dependencies between random variables can be expressed in terms of the proposed metrics.
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Short CommunicationA note on metrics induced by copulas

Abstract

Highlights

Inform. Control

On some metrics induced by copulas

Associative Functions: Triangular Norms and Copulas

Short Communication
A note on metrics induced by copulas