On the symmetric and skew-symmetric K-distributions

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 threeparameter 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.