Preservation of certain dependent structures under bivariate homogeneous Poisson shock models

A bivariate Poisson shock model resulting from two devices receiving shocks from two independent sources is shown to preserve certain bivariate dependent structures such as total positivity of order 2 (TP2), stochastic increasing (SI), right tail increasing (RTI) etc. However, when two devices are subjected to the same source of shocks it is observed through a counter example that some of these preservation results do not hold any more. In such cases sufficient conditions are given under which the bivariate random vector denoting the life lengths of two devices is shown to have the above-mentioned bivariate dependent structures.