  We study failure rate monotonicity and generalized convex transform
stochastic ordering properties of random variables, with a concern on
applications. We are especially interested in the effect of a tail weight
iteration procedure to define distributions, which is equivalent to the
characterization of moments of the residual lifetime at a given instant. For
the monotonicity properties, we are mainly concerned with hereditary properties
with respect to the iteration procedure providing counter-examples showing
either that the hereditary property does not hold or that inverse implications
are not true. For the stochastic ordering, we introduce a new criterium, based
on the analysis of the sign variation of a suitable function. This criterium is
then applied to prove ageing properties of parallel systems formed with
components that have exponentially distributed lifetimes.
