On Families of Distributions with Shape Parameters

Abstract

Univariate continuous distributions are one of the fundamental components on which statistical modelling, ancient and modern, frequentist and Bayesian, multi-dimensional and complex, is based. In this article, I review and compare some of the main general techniques for providing families of typically unimodal distributions on R with one or two, or possibly even three, shape parameters, controlling skewness and/or tailweight, in addition to their all-important location and scale parameters. One important and useful family is comprised of the 'skew-symmetric' distributions brought to prominence by Azzalini. As these are covered in considerable detail elsewhere in the literature, I focus more on their complements and competitors. Principal among these are distributions formed by transforming random variables, by what I call 'transformation of scale'-including two-piece distributions-and by probability integral transformation of nonuniform random variables. I also treat briefly the issues of multi-variate extension, of distributions on subsets of R and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal.

A Sample Selection Model with Skew‐normal Distribution

“…We emphasize that the type of tail behaviour of the asymmetric distribution, as characterized by Jones (), will not affect the interpretation of both the inherent skewness and skewness due to selection. Briefly, asymmetric distributions with the same tail behaviour in each direction (e.g.…”

A new method for generating distributions with an application to exponential distribution

“…Recently Aljarrah et al (2014) used quantile functions to generate T-X family of distributions. For more survey about methods to generating distributions see Lee et al (2013) and Jones (2014).…”

Natural (Non‐)Informative Priors for Skew‐symmetric Distributions

“…Given these needs, there exists a plethora of distinct proposals for skew distributions in the literature; for a recent and extensive overview of the state of the art, we refer the reader to the discussion paper of Jones (). A popular class of such distributions are the skew‐symmetric densities of the form with f the symmetric density (to be skewed), G any symmetric, univariate, absolutely continuous cumulative distribution function (cdf) and ω an odd function (Azzalini & Capitanio, ; Wang et al, ).…”

“…Given these needs, there exists a plethora of distinct proposals for skew distributions in the literature; for a recent and extensive overview of the state of the art, we refer the reader to the discussion paper of Jones (2015). A popular class of such distributions are the skew-symmetric densities of the form…”