On a class of distributions defined by the relationship between their density and distribution functions

Jones, M. C. (2007). On a class of distributions defined by the relationship between their density and distribution functions. Communications in Statistics - Theory and Methods, 36(10) pp. 1835–1843.

DOI: https://doi.org/10.1080/03610920601126464

Abstract

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F-alpha(x)(1 - F(x))(beta), alpha,beta epsilon R, is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and-for the first time-transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom, and, newly, the distribution corresponding to alpha = beta = 5/2. The logistic distribution is central to some of the developments of the article.