A new family of tempered distributions

Abstract

Tempered distributions have received considerable attention, both from a theoretical point of view and in several important application fields. The most popular choice is perhaps the Tweedie model, which is obtained by tempering the Positive Stable distribution. Through tempering, we suggest a very flexible four-parameter family of distributions that generalizes the Tweedie model and that could be applied to data sets of non-negative observations with complex (and difficult to accommodate) features. We derive the main theoretical properties of our proposal, through which we show its wide application potential. We also embed our proposal within the theory of Lévy processes, thus providing a strengthened probabilistic motivation for its introduction. Furthermore, we derive a series expansion for the probability density function which allows us to develop algorithms for fitting the distribution to data. We finally provide applications to challenging real-world examples taken from international trade.