Univariate continuous distributions: symmetries and transformations

Jones, M. C. (2015). Univariate continuous distributions: symmetries and transformations. Journal of Statistical Planning and Inference, 161 pp. 119–124.

DOI: https://doi.org/10.1016/j.jspi.2014.12.011


If the univariate random variable X follows the distribution with distribution function F, then so does Y=F−1(1−F(X)). This known result defines the type of (generalised) symmetry of F, which is here referred to as T-symmetry; for example, ordinary symmetry about θ corresponds to Y=2θ−X. Some distributions, with density fS, display a density-level symmetry of the form fS(x)=fS(s(x)), for some decreasing transformation function s(x); I call this S-symmetry. The main aim of this article is to introduce the S-symmetric dual of any (necessarily T-symmetric) F, and to explore the consequences thereof. Chief amongst these are the connections between the random variables following F and fS, and relationships between measures of ordinary symmetry based on quantiles and on density values.

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