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This question is about an application of the Sklar's Extension Theorem, whose proof can be found in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look backward and forward." Institute of Mathematical Statistics Lecture Notes-Monograph Series, 28, 1–14. [106, 133].

I will firstly briefly (and, due to space constraints, non exhaustively) summarise the content of the Sklar's Extension Theorem and then I will present my question.


Review of Sklar's Extension Theorem

For simplicity, I will focus on 3-dimensional copulas. Note that the Sklar's Extension Theorem applies to any dimension $N\geq 1$.

Let $\mathcal{I}\equiv \mathcal{I}_1\times \mathcal{I}_2 \times \mathcal{I}_3\subseteq [0,1]^3$, where $\mathcal{I}_n$ is such that $\{0,1\}\in \mathcal{I}_n$ for each $n=1,2,3$.

An $3$-dimensional subcopula is a function $\bar{C}:\mathcal{I}\rightarrow [0,1]$ such that:

  1. $\bar{C}$ is non decreasing.

  2. $\bar{C}(u) = 0$ for any $u \in \mathcal{I}$ that has at least one component equal to 0.

  3. $\bar{C}(u) = u_n$ for any $u \in \mathcal{I}$ that has all components, except the $n$-th, equal to 1.

A $3$-dimensional copula is a $3$-dimensional subcopula for which $\mathcal{I}=[0,1]^3$ .

Sklar's Lemma: Let $\bar{C}$ be a $3$-dimensional subcopula with domain $\mathcal{I}$. Then, there exists a proper $3$-dimensional copula $C$ such that $C(u) = \bar{C}(u)$ for all $u\in \mathcal{I}$.


Question: when extending the copula, can we make sure that the CDF associated with the copula satisfies some desired support restrictions?

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  • $\begingroup$ Could you please explain what you might mean by "support restrictions"? Copulas have nothing to do with the actual support of the multivariate distributions they model: that part is handled by the initial transformation from the distribution to a distribution supported on $[0,1]^n.$ $\endgroup$
    – whuber
    Oct 29 '21 at 22:20
  • $\begingroup$ You might want to include a link to the referenced article (projecteuclid.org/download/pdf_1/euclid.lnms/1215452606), I think it's open access. $\endgroup$ Oct 29 '21 at 22:26

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