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Distributions generated by transformation of scale using an extended Cauchy-Schlömilch transformation

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Abstract

Baker (2008) shows how more flexible densities on ℝ+ can be generated from others by applying the Cauchy-Schlömilch transformation to the abscissa. Such “transformation of scale” is not even guaranteed to provide integrable functions in general. The appeal of the Cauchy-Schlömilch transformation is that it automatically does so; moreover, the normalising constant is unaffected and hence immediately available. In this paper, we fit the original Cauchy-Schlömilch transformation into a broader framework of novel extended Cauchy-Schlömilch transformations based on self-inverse functions, and propose the corresponding newly generated densities which also retain the same normalising constant. As well as providing parallels with, and extensions of, the many properties of the new densities developed by Baker, we investigate the skewness properties of both original and extended Cauchy- Schlömilch-based distributions via application of a recently proposed densitybased approach to quantifying asymmetry.

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Authors and Affiliations

  1. The Open University, Milton Keynes, UK

    M. C. Jones

  2. Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

    M. C. Jones

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  1. M. C. Jones
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Correspondence to M. C. Jones.

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Jones, M.C. Distributions generated by transformation of scale using an extended Cauchy-Schlömilch transformation. Sankhya A 72, 359–375 (2010). https://doi.org/10.1007/s13171-010-0021-6

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  • DOI: https://doi.org/10.1007/s13171-010-0021-6

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