Stochastic ordering of minima and maxima from heterogeneous bivariate Birnbaum-Saunders random vectors

In this paper, we discuss stochastic comparisons of minima and maxima arising from heterogeneous bivariate Birnbaum-Saunders (BS) random vectors with respect to the usual stochastic order based on vector majorization of parameters. Suppose the bivariate random vectors (X-1, X-2) and (X-1*, X-2*) follow BVBS(alpha(1), beta(1), alpha(2), beta(2),rho) and BVBS(alpha(1)*, beta(1)*, alpha(2)*, beta(2)*,rho) distributions, respectively. Suppose 0 < nu <= 2. We then prove that when alpha(1) = alpha(2) = alpha(1)* = alpha(2)*, (beta(1)*(-1/nu),beta(2)*(-1/nu)) implies X-2: 2* >= st X-1: 2* >= st X-1: 2. These results are subsequently generalized to a wider range of scale parameters. Next, we prove that when beta(1) = beta(2) = beta(1)* = beta(2)*, (1/alpha(1), 1/alpha(2)) >= m (1/alpha(1)*, 1/alpha(2)*) implies X-2: 2 >= st X-2: 2* and X-1: 2 * >= st X-1: 2. Analogous results are then deduced for bivariate log BS distributions as well.