Stochastic Comparisons of Parallel Systems When Components Have Proportional Hazard Rates

Let X 1 , … , X n be independent random variables with X i having survival function F λ i , i = 1, … , n , and let Y 1 , … , Y n be a random sample with common population survival distribution F , where c = ∑ i =1 n λ i / n . Let X n : n and Y n : n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that X n : n is greater than Y n : n in terms of likelihood ratio order. It is also proved that the sample range X n : n − X 1: n is larger than Y n : n − Y 1: n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo 6 and Kochar and Rojo 11, respectively.