  Many flexible families of positive random variables exhibit non-closed forms
of the density and distribution functions and this feature is considered
unappealing for modelling purposes. However, such families are often
characterized by a simple expression of the corresponding Laplace transform.
Relying on the Laplace transform, we propose to carry out parameter estimation
and goodness-of-fit testing for a general class of non-standard laws. We
suggest a novel data-driven inferential technique, providing parameter
estimators and goodness-of-fit tests, whose large-sample properties are
derived. The implementation of the method is specifically considered for the
positive stable and Tweedie distributions. A Monte Carlo study shows good
finite-sample performance of the proposed technique for such laws.
