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Jones, M. C.
(2015).
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 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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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