On hazard rate ordering of dependent variables

Shanthikumar and Yao (1991) introduced some new stochastic order relations to compare the components of a bivariate random vector (X1, X2). As they point out in their paper, even if according to their hazard rate (or likelihood ratio) ordering, the marginal distributions may not be ordered accordingly. We introduce some new concepts where the marginal distributions preserve the corresponding stochastic orders. Also a relation between the bivariate scale model and the introduced bivariate hazard rate ordering is established.