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

Univariate continuous distributions have three possible types of support exemplified by: the whole real line, R, the semi-finite interval R+= (0,∞) 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 ±∞) 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.