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.