Some measures for asymmetry of distributions

Abstract

We propose several measures, functional and scalar, for asymmetry of distributions by comparing the behavior of probability densities to the right and left of the mode(s) and show how to generate classes of equivalent distributions from a given distribution, allowing for varying asymmetry but retaining some information theoretic properties of the original distribution, such as the entropy. r

On reciprocal symmetry

“…The respective quantile and mode emphases of the different types of symmetry extend to positivity of skewness measures based on quantiles and modes, respectively; the latter is a new observation utilising the mode-based skewness measure of Arnold and Groeneveld (1995) given in the table. Indeed, Mudholkar and Wang's R-symmetry lends itself to a deeper analysis of skewness via the density-based skewness functions of Averous, Fougéres and Meste (1996), Critchley and Jones (2005) and Boshnakov (2007). Suppose f is unimodal with, for convenience, f (0) = 0 and let y L (p) be the unique value in (0, θ) such that f (y L (p)) = pf (θ), 0 < p < 1.…”

Skew t distributions via the sinh-arcsinh transformation

“…2.6) and Critchley and Jones (2008) for unimodal distributions; see also Avérous et al (1996) and Boshnakov (2007). Figure 3 shows several examples of their asymmetry function…”

On Families of Distributions with Shape Parameters

“…This is a consequence of f S taking the same values as g in the same order as x increases; W acts as a ‘transformation of scale’ (only) moving through the values of g ( x ) at a varying rate. Skewness properties of f S can be readily interpreted in terms of density‐based skewness ( Avérous et al , ; Boshnakov , ; Critchley & Jones , ), as opposed to the quantile‐based skewness of van Zwet . See also Fujisawa & Abe for more on skewness of f S .…”