On likelihood ratio ordering of parallel systems with exponential components

Abstract

Let T(λ1,...,λ n ) be the lifetime of a parallel system consisting of exponential components with hazard rates λ1,...,λ n , respectively. For systems with only two components, Dykstra et. al. (1997) showed that if (λ1, λ2) majorizes (γ1, γ2), then, T(λ1, λ2) is larger than T(γ1, γ2) in likelihood ratio order. In this paper, we extend this theorem to general parallel systems. We introduce a new partial order, the so-called d-larger order, and show that if (λ1,...,λ n ) is d-larger than (γ1,...,γ n ), then T(λ1,...,λ n ) is larger than T(γ1,...,γ n ) in likelihood ratio order.