Stochastic Comparisons for Multivariate Shock Models

Consider two devices subjected to shocks arriving according to two identically defined counting processes. Let N1 and N2 be the random numbers of shocks until failure of the two devices, respectively, and let T1 and T2 be their random lifetimes. Conditions such that stochastic orders between N1 and N2 are preserved by T1 and T2 have been investigated in recent literature. Here we study this problem in a multivariate setting, considering systems of non-independent components, and we extend some known results to multivariate stochastic orders. Two kinds of multi-variate generalizations are considered; the case that each one of the components is subjected to its own fond of shocks and the case that all the components of the same system are subjected to a common font of shocks.