Dependence properties of bivariate copula families

Motivated by recently investigated results on dependence measures and robust risk models, this paper provides an overview of dependence properties of many well known bivariate copula families, where the focus is on the Schur order for conditional distributions, which has the fundamental property that minimal elements characterize independence and maximal elements characterize perfect directed dependence. We give conditions on copulas that imply the Schur ordering of the associated conditional distribution functions. For extreme value copulas, we prove the equivalence of the lower orthant order, the Schur order for conditional distributions, and the pointwise order of the associated Pickands dependence functions. Further, we provide several tables and figures that list and illustrate various positive dependence and monotonicity properties of copula families, in particular from classes of Archimedean, extreme value, and elliptical copulas. Finally, for Chatterjee's rank correlation, which is consistent with respect to the Schur order for conditional distributions, we give some new closed-form formulas in terms of the parameter of the underlying copula family

Dependence properties of bivariate copula families

Motivated by recently investigated results on dependence measures and robust risk models, this article provides an overview of dependence properties of many well known bivariate copula families, where the focus is on the Schur order for conditional distributions, which has the fundamental property that minimal elements characterize independence and maximal elements characterize perfect directed dependence. We give conditions on copulas that imply the Schur ordering of the associated conditional distribution functions. For extreme-value copulas, we prove the equivalence of the lower orthant order, the Schur order for conditional distributions, and the pointwise order of the associated Pickands dependence functions. Furthermore, we provide several tables and figures that list and illustrate various positive dependence and monotonicity properties of copula families, in particular, from classes of Archimedean, extreme-value, and elliptical copulas. Finally, for Chatterjee’s rank correlation, which is consistent with the Schur order for conditional distributions, we give some new closed-form formulas in terms of the parameter of the underlying copula family

Passive Films On Stainless-Steels in Aqueous-Media

The purpose of this paper is to provide a syntheses of experimental results regarding films formed on the surface of stainless steels. Such syntheses are attempted for the environments most studied. In each case the overview is presented with reference to the most important papers. Conflicting data are also presented and discussed. Based on the results of the prior studies, a four region model is proposed to describe the surface passive film and its breakdown