An investigation is presented of how a comprehensive choice of five most important measures of concordance (namely Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta, and their weaker counterpart Spearman's footrule) relate to non-exchangeability, i.e., asymmetry on copulas. Besides these results, the method proposed also seems to be new and may serve as a raw model for exploration of the relationship between a specific property of a copula and some of its measures of dependence structure, or perhaps the relationship between various measures of dependence structure themselves.
The relation between non-exchangeability and radial asymmetry of copulas
University of Ljubljana, School of Economics and Business（卢布尔雅那大学经济商学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学物理学院）
An investigation is presented of how a comprehensive choice of four most important measures of concordance (namely Spearman's rho, Kendall's tau, Spearman's footrule, and Gini's gamma) relate to the fifth one, i.e., the Blomqvist's beta. In order to work out these results we present a novel method of estimating the values of the four measures of concordance on a family of copulas with fixed value of beta. These results are primarily aimed at the community of practitioners trying to find the right copula to be employed on their data. However, the proposed method as such may be of independent in…
In the present paper we propose and study estimators for a wide class of bivariate measures of concordance for copulas. These measures of concordance are generated by a copula and generalize Spearman's rho and Gini's gamma. In the case of Spearman's rho and Gini's gamma the estimators turn out to be the usual sample versions of these measures of concordance.
In the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions $d\ge$ 3, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions and we provide a necessary condition for a copula to be minimal: The sufficient conditions are related to the extreme negative dependence concept of d-countermonotonicity and the necessary condition is related to the collection of all copu…
我们给出了在任意维数下，当变元被置换时，copula与自身之间的最大距离，推广了Nelsen（2007），Klement and Mesiar（2006）关于二维的结果。此外，我们建立了一个子集的$[0,1]^d$，在其中可能达到这个界限。对于这个子集中的每一个点，我们提出了一个copula和一个置换，在这个点的距离是最大的。在这个过程中，我们看到这个子集取决于维度是偶数还是奇数。
Methods are developed for checking and completing systems of bivariate and multivariate Kendall's tau concordance measures in applications where only partial information about dependencies between variables is available. The concept of a concordance signature of a multivariate continuous distribution is introduced; this is the vector of concordance probabilities for margins of all orders. It is shown that every attainable concordance signature is equal to the concordance signature of a unique mixture of the extremal copulas, that is the copulas with extremal correlation matrices consisting exc…
When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930's, copulas have been studied since 1950's, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshall's copulas, have been studied since 1960's. However, the point of non-exchange…
Working with shuffles we establish a close link between Kendall's tau, the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well-known that Spearman's rho of a bivariate copula A is a rescaled version of the volume of the area under the graph of A, in this contribution we show that the other famous concordance measure, Kendall's tau, allows for a simple geometric interpretation as well - it is inextricably linked to the surface area of A.
Quantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpre…
The exact region determined by Spearman's footrule, Gini's gamma and Kendall's tau
University of Ljubljana, School of Economics and Business（卢布尔雅那大学经济商学院）；Institute of Mathematics, Physics and Mechanics（数学物理力学研究所）；University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学物理学院）；University of Zagreb, Faculty of Science, Department of Mathematics（萨格勒布大学理学院数学系）；University of Ljubljana, Faculty of Civil and Geodetic Engineering（卢布尔雅那大学土木与大地工程学院）
