Connecting distributions with power tails on the real line, the half line and the interval

Jones, M.C. (2007). Connecting distributions with power tails on the real line, the half line and the interval. International Statistical Review, 75(1) pp. 58–69.

Abstract

Univariate continuous distributions have three possible types of support exemplified by: the whole real line, R, the semi-finite interval R+ = (0, infinity) and 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 +/-infinity) 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) to R+ is the standard odds transformation. The transformation from R+ to R is 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.

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