Ordering Results Between Two Finite α-Mixture Models with Components Following Modified Proportional Hazard Rate Model

Abstract

Finite α-mixture models are widely used in reliability engineering and survival analysis to model heterogeneous populations arising from multiple latent failure mechanisms. Comparing the reliability characteristics of such models through stochastic orderings is important for system evaluation, maintenance planning, and reliability-based decision making. The modified proportional hazard and reversed hazard rate models generalize the proportional hazard rate and reversed hazard rate models, respectively, and have attracted considerable attention in reliability theory. However, stochastic comparisons of finite α-mixture models with MPHR- and MPRHR-distributed components have not been investigated in the existing literature. In this work, we consider two finite α-mixture models with modified proportional hazard (reversed hazard) rate components. These mixture models are compared stochastically in the sense of the usual stochastic, hazard rate and reversed hazard rate orders. Sufficient conditions are obtained using vector majorization and chain majorization orders. Examples are provided to illustrate the established results. It is also shown that some of the results based on hazard rate and reversed hazard rate orders can not be extended to the likelihood ratio, relative hazard rate and relative reversed hazard rate orders. The flexibility of the proposed models is validated through aircraft windshield failure-time and active repair time data sets, wherein they successfully identify the underlying heterogeneity structure.