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 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 non-uniform random variables. I also treat briefly the issues of multi-variate extension, of distributions on subsets of inline image and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal