Distributions that are both log-symmetric and R-symmetric

Jones, M. C. and Arnold, Barry C. (2008). Distributions that are both log-symmetric and R-symmetric. Electronic Journal of Statistics, 2 pp. 1300–1308.

DOI: https://doi.org/10.1214/08-EJS301

Abstract

Two concepts of symmetry for the distributions of positive random variables Y are log-symmetry (symmetry of the distribution of logY) and R-symmetry. In this paper, we characterise the distributions that have both properties, which we call doubly symmetric. It turns out that doubly symmetric distributions constitute a subset of those distributions that are moment-equivalent to the lognormal distribution. They include the lognormal, some members of the Berg/Askey class of distributions, and a number of others for which we give an explicit construction (based on work of A.J. Pakes) and note some properties; Stieltjes classes, however, are not doubly symmetric.

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