Necessary and sufficient conditions for stochastic orders between (n − r + 1)-out-of-n systems in proportional hazard (reversed hazard) rates model

Consider two (n − r + 1)-out-of-n systems, one with independent and non-identically distributed components and another with independent and identically distributed components. When the lifetimes of components follow the proportional hazard rates model, we establish a necessary and sufficient condition for the usual stochastic order to hold between the lifetimes of these two systems. For the special case of r = 2, some generalized forms of this …