A full scale Sklar's theorem in the imprecise setting

TL;DR

This paper extends Sklar's theorem to the imprecise probability setting, introducing new techniques with quasi-distributions, restricted p-boxes, and imprecise copulas, showing that any bivariate distribution can be represented via copulas.

Contribution

It provides a comprehensive extension of Sklar's theorem in the imprecise setting, incorporating quasi-distributions and restricted p-boxes, and demonstrates the universality of copulas for bivariate distributions.

Findings

Extended Sklar's theorem to imprecise probabilities

Developed new techniques for restricted p-boxes

Showed all bivariate distributions can be represented by copulas

Abstract

In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48--66. The main tools we develop in order to do so are: (1) a theory on quasi-distributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) $p$-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) $p$-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladi\v{c} and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) $p$-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual $\sigma$-additive probability space or on an additive space can be obtained as a copula of its margins meaning that its possible extraordinariness depends solely on its margins. This might indicate that copulas are a stronger probability concept than probability itself.