Characterization of a subclass of Tweedie distributions by a property of generalized stability

Abstract: We introduce a class of distributions originating from an exponential family and having a property related to the strict stability property. A characteristic function representation for this family is obtained and its properties are investigated. The proposed class relates to stable distributions and includes Inverse Gaussian distribution and Levy distribution as special cases. Due to its origin, the proposed distribution has a sufficient statistic. Besides, it combines stability property at lower scales with an exponential decay of the distribution's tail and has an additional flexibility due to the convenient parametrization. Apart from the basic model, certain generalizations are considered, including the one related to geometric stable distributions.