Stochastic Orders in Reliability

Abstract

Stochastic orders enable global comparison of two distributions in terms of their characteristics. Specifically, for a given characteristic A, stochastic order says that the distribution of X has lesser (greater) A than the distribution of Y. For example, one may use hazard rate or mean residual life for such a comparison. In this chapter, we discuss various stochastic orders useful in reliability modelling and analysis.The stochastic order treated here are the usual stochastic order, hazard rate order, mean residual life order, harmonic mean residual life order, renewal and harmonic renewal mean residual life orders, variance residual life order, percentile residual life order, reversed hazard rate order, mean inactivity time order, variance inactivity time order, the total time on test transform order, the convex transform (IHR) order, star (IHRA) order, DMRL order, superadditive (NBU) order, NBUE order, NBUHR and NBUHRA orders and MTTF order. The interpretation of ageing concepts, preservation properties with reference to convolution, mixing and coherent structures are also discussed in relation to each of these orders. Implications among the different orders are also presented. Examples of the stochastic orders and counter examples where certain implications do not hold are also provided. Some special models used in reliability like proportional hazard and reverse hazard models, mean residual life models and weighted distributions have been discussed in earlier chapters. Some applications of these stochastic models are reviewed as well.