Sklar's theorem in an imprecise setting

Abstract

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.

Defects and transformations of quasi-copulas

“…Imprecise copulas were studied in [30] and [26] (see also [25,36]) in order to construct two-dimensional probability boxes (briefly p-boxes), which are represented by ordered pairs of comparable distribution functions, from two given one-dimensional p-boxes. …”

“…Evidently, both A = i∈I A i and B = i∈I B i are quasi-copulas. Note first that for each i ∈ I and for each rectangle [ The problem whether all imprecise copulas can be obtained in this way (already posed in [26,30]) is still open, namely, whether for each each imprecise copula (A, B) there is a family (C i ) i∈I of copulas such that A = C and B = C.…”

“…This allows us to introduce an equivalence relation on the set of quasi-copulas by grouping quasi-copulas converging to the same copula into an equivalence class. An interesting application of our approach to the so-called imprecise copulas [25,26,30,36] will also be given.…”

Sklar's theorem for minitive belief functions

“…In [13,Prop. 4] it is proven that given two univariate p-boxes (F 1 , F 1 ), (F 2 , F 2 ) and a set of copulas C the couple (F , F ) defined by x 2 ) : x 1 , x 2 ∈ R} and E = D ∪ D c the above bivariate p-box induces a coherent lower probability on E by…”

Copula-based fuzzy clustering of spatial time series

“…Copulas are probability distribution functions with uniform marginals, which can be also seen as aggregation functions with special properties (see Durante & Sempi, 2016;Grabisch et al, 2009). They have been extensively used for modelling uncertainty of different types, from probabilistic methods (see Joe, 2015;Nelsen, 2006) to imprecise probabilities and decision theory (see Yager, 2013;Klement et al, 2014;Montes et al, 2015). Nowadays, copula-based models are also frequently used in many problems from spatial statistics; (see, e.g., Bárdossy & Li, 2008;Durante & Salvadori, 2010;Kazianka & Pilz, 2010;Guthke & Bárdossy, 2017).…”