  In this paper, we investigate stochastic comparisons of parallel systems, and
obtain two characterization results in this regard. First, we compare a
parallel system with independent heterogeneous components to a parallel system
with homogeneous components, and establish some certain assumptions under which
the hazard rate and usual stochastic orders between the lifetimes of two
parallel systems are equivalent. Next, we turn our attention to two parallel
systems with their component lifetimes following multiple-outlier model and
prove that under some specified assumptions, the $p$-larger order between the
vectors of scale parameters is equivalent to the hazard rate order as well as
the usual stochastic order between the lifetimes of these systems. The results
established here are applicable to compute an upper bound for the hazard rate
function and a lower bound for the survival function of a parallel systems
consisting of heterogeneous components.
