Algebraic properties of copulas defined from matrices

Cécile Amblard 1 Stephane Girard 2, * Ludovic Menneteau 3
* Corresponding author
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : We propose a new family of copulas, defined by: $$ S_{\phi}(u,v)= \;^t\phi(u) A\phi(v),\;\; (u,v)\in[0,1]^2, $$ where $\phi$ is a function from $[0,1]$ to ${\mathbb R}^p$ and $A$ is a $p\times p$ matrix. Let us remark that if $p=2$ and $A$ is a diagonal matrix, then $S_\phi$ reduces to the family proposed in~\cite{Amblard05}. As a consequence, $S_\phi$ can be seen as an extension of this former family to arbitrary matrices. First, we shall give sufficient conditions on $A$ and $\phi$ to obtain copulas. Then, we shall establish the dependence and symmetry properties of this family of copulas. Finally, we shall study the stability properties of $S_\phi$ with respect to the operator $*$ (presented for instance in~\cite{Nelsen99}, p. 194) as well as other algebraic properties.
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Cécile Amblard, Stephane Girard, Ludovic Menneteau. Algebraic properties of copulas defined from matrices. Workshop on Copulae in Mathematical and Quantitative Finance, Jul 2012, Cracovie, Poland. ⟨hal-00990969⟩

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