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, . . ., X_n be independent random variables with X_i∼BS(α_i, β_i), i=1, . . ., n, and X₁^*,. . .,X_n^* be another set of independent random variables with X_i^*∼BS(α_i^*,β,_i^*),i = 1,. . .,n. Then, we first show that when α₁=⋯=α_n=α₁^*=⋯=α_n^*, (β₁,. . .,β_n)≽m(β₁^*,. . .,β_n^*) implies X_n:n≥_stX_n:n^* and (1/β₁,. . .,1/β_n)≽_m(1/β₁^*,. . .,1/β_n^*) implies X_1:n^*≥_stX_1:n. We subsequently generalize these results to a wider range of the scale parameters. Next, we show that when β₁=⋯=β_n=β₁^*=⋯=β_n^*, (1/α₁,. . .,1/α_n)≽_m(1/α₁^*,. . .,1/α_n^*) implies X_n:n≥_stX_n:n^* and X_1:n^*≥_stX_1:n. Finally, we establish similar results for the log Birnbaum-Saunders distribution.