Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum–Saunders components

In this paper, we discuss stochastic comparisons of lifetimes of parallel and series systems with independent heterogeneous Birnbaum–Saunders components with respect to the usual stochastic order based on vector majorization of parameters. Specifically, let X1,…,XnX1,…,Xn be independent random variables with Xi∼BS(αi,βi),i=1,…,nXi∼BS(αi,βi),i=1,…,n, and X1∗,…,Xn∗ be another set of independent random variables with Xi∗∼BS(αi∗,βi∗),i=1,…,n. Then, we first show that when α1=⋯=αn=α1∗=⋯=αn∗, (β1,…,βn)⪰m(β1∗,…,βn∗) implies Xn:n≥stXn:n∗ and (1β1,…,1βn)⪰m(1β1∗,…,1βn∗) implies X1:n∗≥stX1:n. We subsequently generalize these results to a wider range of the scale parameters. Next, we show that when β1=⋯=βn=β1∗=⋯=βn∗, (1α1,…,1αn)⪰m(1α1∗,…,1αn∗) implies Xn:n≥stXn:n∗ and X1:n∗≥stX1:n. Finally, we establish similar results for the log Birnbaum–Saunders distribution.