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  • Author: Wolfgang Trutschnig x
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Abstract

We solve a recent open problem about a new transformation mapping the set of copulas into itself. The obtained mapping is characterized in algebraic terms and some limit results are proved.

Abstract

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.

Abstract

We derive a new (lower) inequality between Kendall’s τ and Spearman’s ρ for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.

Abstract

Motivated by the nice characterization of copulas A for which d(A, At) is maximal as established independently by Nelsen [11] and Klement & Mesiar [7], we study maximum asymmetry with respect to the conditioning-based metric D1 going back to Trutschnig [12]. Despite the fact that D1(A, At) is generally not straightforward to calculate, it is possible to provide both, a characterization and a handy representation of all copulas A maximizing D1(A, At). This representation is then used to prove the existence of copulas with full support maximizing D1(A, At). A comparison of D1- and d-asymmetry including some surprising examples rounds off the paper.

Abstract

The problem of quantifying the overlap of Hutchinsonian niches has received much attention lately, in particular in quantitative ecology, from where it also originates. However, the niche concept has the potential to also be useful in many other application areas, as for example in economics. We are presenting a fully nonparametric, robust solution to this problem, along with exact shortcut formulas based on rank-statistics, and with a rather intuitive probabilistic interpretation. Furthermore, by deriving the asymptotic sampling distribution of the estimators, we are proposing the first asymptotically valid inference method, providing confidence intervals for the niche overlap. The theoretical considerations are supplemented by simulation studies and a real data example.