Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The construction of exible parametric distributions with prescribed properties is a classical problem in statistical distribution theory. In this talk, we present a recently introduced framework for generating broad families of continuous univariate distributions $[1, 2].$ The main idea is to represent the cumulative distribution function in the form  $$F(x) = g^{-1}(h(x)), \quad \quad  x > 0,$$

where $g$ is a suitably chosen generator and $h$ is a parametric part. Many classical distributions arise as special cases of this representation, while the separation between $g$ and $h$ provides a convenient mechanism for studying distributional properties and systematically constructing new models.

We first describe how qualitative properties such as failure-rate behaviour, unimodality and tail behaviour can be characterized through the structural properties of the generator and the parametric part. We then discuss several mechanisms for generating new families, including combinations of di erent parametric parts, combinations of generators, and transformations of the generator. These constructions allow distributions with prescribed aging and tail properties to be obtained within a common framework [2].

More recent developments extend the methodology to additional aging notions, reversed hazard rates, likelihood-ratio properties and stochastic orders. Preservation results under combinations and transformations are obtained, together with connections to actuarial risk measures and quantile-based reliability concepts [3].

Finally, we illustrate the practical potential of the theoretical framework through an application to social network data. A exible model is constructed for node-degree distributions exhibiting pronounced right-skewness, heavy tails, and non-monotone hazard behaviour, and its empirical performance is demonstrated using the Google+ network dataset [4].

The overall aim is to show how the $D^+_g (h)$ framework provides a uni ed link between distribution construction, structural distribution theory, reliability and aging properties, and data-driven statistical modelling.

References
[1] M. V. Koutras and S. D. Dafnis, A new family of continuous univariate distributions, Methodology and Computing in Applied Probability, 27, Article 5, 2025. doi:10.1007/s11009-024-10131-9.
[2] M. V. Koutras and S. D. Dafnis, Methods for generating new families of continuous univariate distributions, Annals of the Institute of Statistical Mathematics, 2026. doi:10.1007/s10463-025-00971-8.
[3] M. V. Koutras, S. D. Dafnis and L. K. Kanellopoulos, Aging and stochastic orders of a wide family of continuous univariate distributions, Methodology and Computing in Applied Probability, 28, Article 104, 2026. doi:10.1007/s11009-026-10331-5.
[4] S. D. Dafnis, M. V. Koutras and S. K. Trousas, A uni ed framework for generating exible distributions for social network analysis, Submitted for publication, 2026.