Analytical mechanisms of shape formation of a generalized family of distributions

Abstract

This paper introduces a generalized family of distributions designed to model complex empirical data, addressing the limitations of traditional mixture distributions such as an increased number of parameters and difficulties in analytical investigation. The authors analyze the analytical conditions for the existence of the probability density function (PDF) and inflection points of the cumulative distribution function (CDF) for this generalized family. This analysis helps in understanding the mechanisms shaping the distribution's form and its application limitations based on the chosen base distribution and parameter values. The study delves into the specific case where the base function is the CDF of the exponential distribution, demonstrating that the generalized distribution can have no more than one inflection point. In contrast, for normal and Weibull base distributions, more than one inflection point can exist when 0<ω≤1, indicating a more complex shape. The paper notes that when ω>1, the generalized CDF exhibits good flexibility while maintaining a sigmoid shape. The inclusion of a scaling factor K further enhances its applicability for approximating experimental dependencies with varying saturation levels. Computational experiments suggest that these models offer higher accuracy in approximating experimental data with asymmetric dynamics compared to classical logistic models, especially when the constraint ω>1 is applied.