A family of stochastic pre-orders based on the usual stochastic order

Abstract A new family of stochastic pre-orders is introduced. We focus on the cases in which two variables are not comparable in the usual stochastic order and study how close they are to be comparable in such order. In particular, we require that the difference between the cumulative distribution functions of the random variables does not change sign up to a certain quantile. Then, the higher the order of the quantile, the closer is the relation to the standard usual stochastic order. An analogous definition can be given in terms of upper quantiles as well, and the one associated to the quantile of higher order has to be preferred. We study some properties of the proposed pre-orders and some connections with other stochastic orders. Then, these relations are applied in the context of reliability theory based on residual and past lifetimes. Finally, they are employed in the study of lifetimes of coherent systems under different assumptions on the components’ lifetimes.