Stochastic comparisons of order statistics in the scale model

Independent random variables X λ 1 , … , X λ n are said to belong to the scale family of distributions if X λ i ∼ F ( λ i x ) , for i=1,…,n, where F is an absolutely continuous distribution function with hazard rate r and reverse hazard rate r ˜ . We show that the hazard rate (reverse hazard rate) of a series (parallel) system consisting of components with lifetimes X λ 1 , … , X λ n is Schur concave (convex) with respect to the vector λ , if x 2 r ′ ( x ) ( x 2 r ˜ ′ ( x ) ) is decreasing (increasing). We also show that if xr(x) is increasing in x, then the survival function of the parallel system is increasing in the vector λ with respect to p-larger order, an order weaker than majorization. We prove that all these new results hold for the scaled generalized gamma family as well as the power-generalized Weibull family of distributions. We also show that in the case of generalized gamma and power generalized Weibull distribution, under some conditions on the shape parameters, the vector of order statistics corresponding to X λ i ʹs is stochastically increasing in the vector λ with respect to majorization thus generalizing the main results in Sun and Zhang (2005) and Khaledi and Kochar (2006).