On the symmetric and skew-symmetric K-distributions

TL;DR

This paper introduces a new four-parameter distribution family called symmetric K-distribution (SKD), which generalizes the K-distribution and includes skewness, with applications in machine learning and Bayesian analysis.

Contribution

It proposes a novel four-parameter distribution family derived from mixture models, extending the K-distribution to include symmetric and skewed forms with detailed properties.

Findings

The SKD family includes the K-distribution as a special case.

Derived distributions of product and ratio of SKD variables.

Both SKD and skew-SKD effectively model complex dynamics in applications.

Abstract

We propose a family of four-parameter distributions that contain the K-distribution as special case. The family is derived as a mixture distribution that uses the three-parameter reflected Gamma distribution as parental and the two-parameter Gamma distribution as prior. Properties of the proposed family are investigated as well; these include probability density function, cumulative distribution function, moments, and cumulants. The family is termed symmetric K-distribution (SKD) based on its resemblance to the K-distribution as well as its symmetric nature. The standard form of the SKD, which often proves to be an adequate model, is also discussed. Moreover, an order statistics analysis is provided as well as the distributions of the product and ratio of two independent and identical SKD random variables are derived. Finally, a generalisation of the proposed family, which enables non-zero skewness values, is investigated, while both the SKD and the skew-SKD are proven capable of describing the complex dynamics of machine learning, Bayesian analysis and other fields through simplified expressions with high accuracy.