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Connecting Distributions with Power Tails on the Real Line, the Half Line and the Interval

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  • M.C. Jones
Abstract
Univariate continuous distributions have three possible types of support exemplified by: the whole real line, , the semi‐finite intervaland the bounded interval (0,1). This paper is about connecting distributions on these supports via ‘natural’ simple transformations in such a way that tail properties are preserved. In particular, this work is focussed on the case where the tails (at ±∞) of densities are heavy, decreasing as a (negative) power of their argument; connections are then especially elegant. At boundaries (0 and 1), densities behave conformably with a directly related dependence on power of argument. The transformation from (0,1) tois the standard odds transformation. The transformation fromtois a novel identity‐minus‐reciprocal transformation. The main points of contact with existing distributions are with the transformations involved in the Birnbaum–Saunders distribution and, especially, the Johnson family of distributions. Relationships between various other existing and newly proposed distributions are explored. Une loi de probabilité univariée peut être définie sur trois types de support, représentés par la droite des réels , la demi‐droite , et l'intervalle borné (0, 1). Ce papier met en rapport ces trois types de lois par des transformations simples et ‘naturelles’ de façon à préserver les propriétés de leurs queues. On considère plus particulièrement le cas où les queues aux limites ±∞ sont épaisses, leur taille décroissant selon une puissance négative de leur argument. Les interconnexions sont dans ce cas particulièrement élégantes. Ces types de loi sont conformes aux bornes 0 et 1, la probabilité de la queue variant selon une fonction de puissance. La transformation de l'intervalle (0, 1) en l'intervalle est la transformation odds usuelle. La transformation de en est une transformation nouvelle de type ‘identité moins réciproque’. Les principaux liens avec les lois de probabilité existantes sont mis en évidence par le biais des transformations relatives à la loi de Birnbaum–Saunders et, surtout, avec la famille des lois de Johnson. Quelques liens avec d'autres lois de probabilité, connues et nouvellement proposées, sont explorés.

Suggested Citation

  • M.C. Jones, 2007. "Connecting Distributions with Power Tails on the Real Line, the Half Line and the Interval," International Statistical Review, International Statistical Institute, vol. 75(1), pages 58-69, April.
    • Handle: RePEc:bla:istatr:v:75:y:2007:i:1:p:58-69
    DOI: 10.1111/j.1751-5823.2007.00006.x
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    Cited by:

    1. M. C. Jones, 2015. "On Families of Distributions with Shape Parameters," International Statistical Review, International Statistical Institute, vol. 83(2), pages 175-192, August.
    2. J. Rosco & M. Jones & Arthur Pewsey, 2011. "Skew t distributions via the sinh-arcsinh transformation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 630-652, November.
    3. Jones, M.C., 2012. "Relationships between distributions with certain symmetries," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1737-1744.

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