Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal...

Area of Science:

• Statistical modeling
• Probability theory
• Applied mathematics

Background:

• Flexible tail behavior in probabilistic models is crucial for engineering and earth science applications.
• Classical distributions like Weibull and lognormal have limitations in modeling extreme events.

Purpose of the Study:

• Introduce nonlinear normalizing transformations based on Kaniadakis' deformed functions.
• Develop new probability distributions (κ-Weibull, κ-lognormal) with enhanced tail behavior.
• Apply these models to real-world data, including precipitation time series and material strength.

Main Methods:

• Utilized Kaniadakis' deformed lognormal and exponential functions for nonlinear transformations.
• Applied the deformed exponential transform to a censored autoregressive model for precipitation data.
• Investigated the connection between the κ-Weibull distribution and weakest-link scaling theory.
• Introduced the κ-lognormal distribution and calculated its generalized mean.

Main Results:

• The deformed exponential transform successfully generates skewed data from normal variates.
• The κ-Weibull distribution is suitable for modeling material strength, linking to weakest-link scaling.
• The κ-lognormal distribution is a viable model for the permeability of random porous media.
• K-deformations effectively modify tails of classical distributions.

Conclusions:

• Kaniadakis' κ-deformations provide a powerful tool for developing flexible probabilistic models.
• These new distributions offer improved analysis of spatiotemporal data with skewed distributions.
• The research opens new avenues for modeling complex phenomena in science and engineering.