Relationships between distributions with certain symmetries

Jones, M. C. (2012). Relationships between distributions with certain symmetries. Statistics and Probability Letters, 82(9) pp. 1737–1744.

DOI: https://doi.org/10.1016/j.spl.2012.05.014

Abstract

The genesis of two-way links between the inverse Gaussian and Birnbaum–Saunders distributions is explored and extended. The most general results apply to pairs of distributions with a general ‘S-symmetry’ structure involving a self-inverse function closely related to a transformation function with certain properties. These general results arise by transformation from very simple properties of the familiar Azzalinitype skew-symmetric distributions. They specialise again to relationships between R-symmetric and log-symmetric distributions, between various models related to the inverse Gaussian and Birnbaum–Saunders distributions, relationships involving the sinh–arcsinh transformation, and others. Simple random variate generation is a practical consequence of these relationships.

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