On Families of Distributions with Shape Parameters

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 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 non‐uniform random variables. I also treat briefly the issues of multi‐variate extension, of distributions on subsets of and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal. © 2014 The Authors. International Statistical Review © 2014 International Statistical Institute