Generating distributions by transformation of scale

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-Schlomilch 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.

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