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

Abstract

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 f_S, display a density-level symmetry of the form f_S(x)=f_S(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 f_S, and relationships between measures of ordinary symmetry based on quantiles and on density values.

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About

• Item ORO ID
• 44630
• Item Type
• Journal Item
• ISSN
• 0378-3758
• Keywords
• density-based asymmetry; probability integral transformation; quantile-based skewness; R-symmetry; S-symmetry
• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
  Faculty of Science, Technology, Engineering and Mathematics (STEM)
• Copyright Holders
• © 2015 Elsevier B.V.
• Depositing User
• M. C. Jones