Tweedie, Bar-Lev, and Enis class of leptokurtic distributions as a candidate for modeling real data

Modeling real life data is often a demanding task and plethora of distributional models have been proposed in the statistical literature in an attempt to describe different data sets in a better way than those used to describe them. In this article, we establish a broad pool of families of parametric distributions previously used in the literature. This pool, which includes 23 parametric models of distributions, is implemented to test the fit of its models to 17 data sets having different characteristics. In doing so, we will mainly pay attention to a three-parameter model that includes the class of natural exponential families generated by positive stable distributions. Indeed, this is the class we wish to pinpoint in this article and highlight its importance for modeling real data sets. The class is shown to be rather competitive alternative to some well-known parametric models in the pool especially when applied to leptokurtic data sets is available. Appropriate R codes which include all parametric models in the pool are provided in a supplementary file for further applications and implementations for other data sets. Supplemental data for this article is available online at https://doi.org/10.1080/23737484.2021.1880988.