Abstract: 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 goodnessof- fit testing for a general class of non-standard laws. We suggest a novel datadriven 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.

Key words and phrases: Central limit theorem, consistent estimation, goodness-of-fit testing, Laplace transform, stable distribution, Tweedie distribution.