A note on a measure of asymmetry

A recently proposed measure of asymmetry (Patil et al. in Stat Papers 53: 971–985, 2012) is analyzed in detail. Several examples illustrate the peculiar behavior of this measure η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} as a measure of asymmetry or skewness. These findings are supported by theoretical considerations. Specifically, η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} is revealed to be a measure of similarity with the exponential distribution rather than an asymmetry measure. To illustrate this, we consider a related goodness of fit test for exponentiality. Moreover, we show that the partly erratic behavior of η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} also has a negative impact on the estimation of the measure.