Skewness-Invariant Measures of Kurtosis

Measures of kurtosis, when applied to asymmetric distributions, are typically much affected by the asymmetry which muddies their already murky interpretation yet further. Certain kurtosis measures, however, when applied to certain wide families of skew-symmetric distributions display the attractive property of skewness-invariance. In this article, we concentrate mainly on quantile-based measures of kurtosis and their interaction with skewness-inducing transformations, identifying classes of transformations that leave kurtosis measures invariant. Further miscellaneous aspects of skewness-invariant kurtosis measures are briefly considered, these not being quantile-based and/or not involving transformations. While our treatment is as unified as we are able to make it, we do not claim anything like a complete characterization of skewness-invariant kurtosis measures but hope that our results will stimulate further research into the issue.