Probability in the Engineering & Informational Sciences
Random variables -- Mathematical models, Stochastic analysis, Mathematical statistics
Let Χ1, … Χn be independent random variables with Χᵢ having survival function Fλᵢ, i=1, … , n, and let Y₁, … , Yn be a random sample with common population survival distribution Fλ, where λ = Σᵢ=₁nλᵢl n. Let Χn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Χn:n - Χ₁:n is larger than Yn n:n - Y₁:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo  and Kochar and Rojo , respectively.
Kochar, S., & Xu, M. (2007). Stochastic comparisons of parallel systems when component have proportional hazard rates. Probability In The Engineering & Informational Sciences [serial online]. October 2007;21(4):597-609.