Contributions to the class of beta-generated distributions

• Published in: Communications in statistics. Theory and methods 2023-02, Vol.52 (4), p.1101-1117
• Main Authors: Visagie, I. J. H., de Waal, D. J., Makgai, S. L., Bekker, A.
• Format: Article
• Language: English
• Subjects: Baseline distribution; beta distribution; Bivariate analysis; bivariate normal distribution; Composite functions; Distribution functions; generator; independence; Normal distribution; parameter space; Probability distribution functions
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1080/03610926.2021.1923748

The beta generator technique entails constructing a univariate distribution function as a composite function of two distribution functions. The success of this technique in the univariate setting has prompted research into the possibility of generalization to the bivariate case. Such a generalization, using copulas, has already been proposed in the literature. In this paper, we construct bivariate distribution functions by passing a bivariate distribution function as an argument to the univariate beta distribution function. The class of distributions obtained is identical to an existing class of distributions; however, the elementary elements of the two classes differ (i.e., some distributions are simple to construct using one of the techniques considered and difficult to construct using the other). This paper provides a rigorous derivation of the parameter space of the beta-generated distributions, as well as a result relating to the dependence structure of the marginals. Finally, a practical example is included demonstrating the use of a beta-generated distribution in the modeling of observed losses in the energy market.