Joint weak hazard rate order under non-symmetric copulas

A weak version of the joint hazard rate order, useful to stochastically compare not independent random variables, has been recently defined and studied in Belzunce et al. (2016). Comparison of hazard rates for dependent random variables, Statistics 50, 630-648. In the present paper, further results on this order are proved and discussed. In particular, some statements dealing with the relationships between the joint weak hazard rate order and other stochastic orders are generalized to the case of non symmetric copulas, and its relations with some multivariate aging notions (studied in Bassan and Spizzichino (2005), Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93, 313–339) are presented. For this purpose, the new notions of Generalized Supermigrative and Generalized Submigrative copulas are defined. Other new results, examples and discussions are provided as well.