A study on the fractional bilinear transformation and the theory of new distributions

Abstract

This work, in the area of Probability and Mathematical Statistics, has its nucleus based on the Theory of New Distributions, its properties and applications. A sequence of facts is established, ranging from a brief introductory summary, dealing with the need for new distributions, to the proposition of a class of transformations, among which, the well-known Marshal-Olkin, whose expression can be derived. This class, then, is applied according to the aforementioned transformation, to known distributions such as, for example, Exponential,

Weibull, among others. Some properties are studied following the ideas behind a odd log- logistic geometric family, as well as a geometric emphasis associated with the classification of

the risk function on the distributions under analysis and making references to regions where their curves - of the risk functions -, are immersed, according to a criterion developed by Qian, (QIAN, 2012). Before, however, the actual applications, some mathematical properties related to moment calculations are presented making reference to canonical methods, as well as methods under development, using non-canonical techniques and the use of the special Spence functions, (SPENCE, 1809) when solving a particular case, while integrating a function for getting expected value. The applications, an essential part of the work, are interdisciplinary in nature, moving between epidemiological data from the current global sanitary crisis, due to COVID-19, passing through physical systems that demand statistical treatment as, for example, the problem of turbulence, as well as the astrophysics problem concerning sunspots. Times of transitions from hydrodynamic regimes to turbulence are analyzed. These studies play an important role in theoretical science and applications ranging from the construction of airplanes and ships, to biological processes involving the dynamics of blood in the heart.