On a New Measure of Skewness for Unimodal Distributions

Abstract

In this paper we introduce a concept of skewness and suggest its measure among a class of unimodal distributions. The measure is built on the lack of symmetry of the density function around the the mode of the distribution. It is shown to satisfy all standard properties expected from a measure of skewness, including location and scale invariance. The extreme values of the measure are characterized. Although the measure is defined for a relatively narrow class of distributions, its utility is established by showing that it is applicable for most popularly used continuous distribution families. The introduced measure is compared with the other established measures like Pearson's skewness and standardized third moment and it is shown to be more strict. Two alternative ways of partial ordering among the distributions based on this skewness are also described. The utility of the proposed measure is examined in other cases including discrete distributions.