Sklar's Theorem in an Imprecise Setting

TL;DR

This paper extends Sklar's theorem to imprecise probabilistic models using p-boxes and sets of copulas, linking it to stochastic ordering with uncertainty.

Contribution

It introduces a generalized version of Sklar's theorem for imprecise marginals and copulas, advancing the theoretical framework for uncertain dependence modeling.

Findings

Extended Sklar's theorem to p-boxes and sets of copulas

Connected imprecise copula models with stochastic ordering

Provided theoretical foundations for uncertain dependence analysis

Abstract

Sklar's theorem is an important tool that connects bidimensional distribution functions with their marginals by means of a copula. When there is imprecision about the marginals, we can model the available information by means of p-boxes, that are pairs of ordered distribution functions. Similarly, we can consider a set of copulas instead of a single one. We study the extension of Sklar's theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.