On Distributions Generated by Transformation of Scale

This paper concerns the surprisingly wide class of continuous distributions that have densities of the form 2g(t(x)) where g is the density of a symmetric distribution and t is a suitable transformation of scale function. Note the simplicity of the normalising constant and its lack of dependence on the transformation function. It turns out that t should be the inverse of a first iterated symmetric distribution function (or of certain extensions thereof). Transformation of scale distributions have a close link with densities of the form 2π(x)g(x) where π is a skewing function, through a certain transformation of random variables. A particular case of this construction is the Cauchy-Schlömilch transformation recently introduced into statistics by R. Baker. Transformation of scale distributions have a number of attractive tractabilities: modality properties, explicit density-based asymmetry functions and a beautiful Khintchine-type theorem being chief amongst them. Other theoretical properties and one aspect of likelihood inference are also explored.