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

TL;DR

This paper introduces a new class of distributions derived from exponential families that exhibit a property related to stability, encompassing well-known distributions like Inverse Gaussian and Levy, with applications in modeling and statistical analysis.

Contribution

The paper characterizes a subclass of Tweedie distributions with a generalized stability property, providing a characteristic function, properties, and extensions including geometric stable distributions.

Findings

Includes Inverse Gaussian and Levy distributions as special cases

Features a tail decay combining stability and exponential properties

Provides a parametrization with a sufficient statistic

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.