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Generating distributions by transformation of scale

Jones, M. C. (2014). Generating distributions by transformation of scale. Statistica Sinica, 24(2) pp. 749–771.

DOI (Digital Object Identifier) Link: https://doi.org/10.5705/ss.2011.304
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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-Schlömilch 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.

Item Type: Journal Item
ISSN: 1996-8507
Keywords: asymmetry function; Cauchy-Schlömilch transformation; invariant entropy; iterated distribution function; Khintchine theorem; normalising constant; self-inverse function; skew distributions; skewing function; two-piece distributions
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 40650
Depositing User: M. C. Jones
Date Deposited: 09 Feb 2015 10:44
Last Modified: 01 May 2019 13:22
URI: http://oro.open.ac.uk/id/eprint/40650
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