Probability density functions covering stretched exponential and stretched Cauchy distributions

Abstract A class of probability density functions with three parameters in terms of Mittag-Leffler functions is presented. The proposed distributions include stretched exponential and stretched Cauchy distributions as special cases, and stretched exponential distributions cover Gaussian and Laplace distributions. Heavy-tailed distributions are clarified as a subclass of the proposed distribution. For heavy-tailed distributions, a variance does not exist, nor does even a mean. Instead of the variance and kurtosis defined for stretched exponential distributions, a class of light-tailed distributions, we present two characteristic numbers, fractional order moment and centrality, applicable for heavy-tailed distributions to describe degree of variation of the corresponding random variable. The fractional order moment is defined as an absolute central moment of fractional power while the centrality is prescribed as the product of the full width at the half maximum and the peak value of the probability density function. Numerical simulations show the effectiveness of the proposed perspective and methods.