  In the literature of stochastic orders, one rarely finds results that can be
considered as criteria for the non-comparability of random variables. In this
paper, we provide results that enable researchers to use simple tools to
conclude that two random variables are not comparable with respect to the
convex transform order. The criteria are applied to prove the non-comparability
of parallel systems with components that are either exponential, Weibull or
Gamma distributed, providing a negative answer for a conjecture about
comparability with respect to the convex transform order in a much broader
scope than its initial statement.
