Threshold copulas and positive dependence

Starting with a notion of positive dependence P and with the family of the lower threshold copulas C-t associated with a bivariate distribution having copula C, we define different notions of positive dependence for C, reflecting the dependence properties of the copulas C-t for some t. Then, we analyze some structural aspects of lower threshold copulas and of the given definitions. Furthermore we consider several specific cases arising from relevant special choices of P (e.g., PQD, LTD, TP2 and PLR). Our analysis, in particular, allows us to present a number of relevant examples and counter-examples, which can be useful in the study of the tail dependence for a bivariate distribution. (C) 2008 Elsevier B.V. All rights reserved.