Directionally convex comparison of correlated first passage times (Q1610842)

The authors study the following shock model: Consider \(m\) components subjected to shocks that occur randomly in time according to some point process. In particular, the point process can be a renewal process. Each shock kills, with some probability, some of the components. Denote by \((T_1,T_2,\ldots,T_m)\) the vector of the components lifetimes. The authors show, under some conditions, that if the interarrival times between shocks decrease in the convex order, then the vector \((T_1,T_2,\ldots,T_m)\) decreases in the directionally convex stochastic order. The authors use their results to obtain some computable distributional bounds for the above vector \((T_1,T_2,\ldots,T_m)\) of first passage times when its distribution cannot be expressed explicitly.