Algebraic properties of copulas defined from matrices

Abstract

International audienceWe 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