Stochastic comparisons of parallel systems when components have proportional hazard rates

Authors:
Subhash Kochar;Maochao Xu
Affiliations:
Department of Mathematics and StatisticsPortland State UniversityPortland, OR 97201 E-mail: kochar@pdx.edu;Department of Mathematics and StatisticsPortland State UniversityPortland, OR 97201 E-mail: kochar@pdx.edu
Venue:
Probability in the Engineering and Informational Sciences
Year:
2007

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Abstract

Let X1, …, Xnbe independent random variables with Xihaving survival function Fλi,i = 1, …, n, and let Y1,…,Yn be a random sample with commonpopulation survival distribution Fλ, where λ=Σi=1nλi/n. LetXn:n andYn:n denote the lifetimes of theparallel systems consisting of these components, respectively. Itis shown that Xn:n is greater thanYn:n in terms of likelihood ratioorder. It is also proved that the sample rangeXn:n X1:nis larger than Yn:nY1:n according to reverse hazard rateordering. These two results strengthen and generalize the resultsin Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11],respectively.