Stochastic comparisons of order statistics in the scale model

Independent random variables Xλ1,…,XλnXλ1,…,Xλn are said to belong to the scale family of distributions if Xλi∼F(λix)Xλi∼F(λix), 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 Xλ1,…,XλnXλ1,…,Xλn is Schur concave (convex) with respect to the vector λλ, if x2r′(x)x2r′(x)(x2r˜′(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 λλ 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 Xλi'sXλi's is stochastically increasing in the vector λλ with respect to majorization thus generalizing the main results in Sun and Zhang (2005) and Khaledi and Kochar (2006).