Statistica Sinica 24 (2014), 749-771

GENERATING DISTRIBUTIONS

BY TRANSFORMATION OF SCALE

M. C. Jones

The Open University

Abstract: This paper investigates the surprisingly wide and practicable
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
invertible transformation of scale function which introduces skewness. Note
the simplicity of the normalising constant and its lack of dependence on
the transformation function. It turns out that the key requirement is that
Π = t^{-1} satisfies Π(y) - Π(-y) = y for all y; Π thus belongs to a class of
functions that includes first iterated symmetric distribution functions but is
also much wider than that. Transformation of scale distributions have a link
with ‘skew-g’ densities of the form 2π(x)g(x), where π = Π′ is a skewing
function, by using Π to transform random variables. A particular case of
the general construction is the Cauchy-Schl\"omilch transformation recently
introduced into statistics by Baker (2008); another is the long extant family of
‘two-piece’ distributions. Transformation of scale distributions have a number
of further attractive tractabilities, modality properties, explicit density-based
asymmetry functions, a beautiful Khintchine-type theorem and invariant
entropy being chief amongst them. Inferential questions are considered briefly.

Key words and phrases: Asymmetry function, Cauchy-Schl\"omilch transformation, invariant entropy, iterated distribution function, Khintchine theorem, normalising constant, self-inverse function, skew distributions, skewing function, two-piece distributions.