Measuring Skewness with Respect to the Mode

Abstract

There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or medians of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained the convex or c-ordering of distributions that he introduced. This ordering has been widely accepted as appropriate for ordering two distributions in relation to skewness. More recently, measures based on the medians have been shown to honor the convex ordering. The measure of skewness (μ – M) / [sgrave] where μ, [sgrave], and M are, respectively, the expectation, standard deviation, and mode of the distribution was initially proposed by Karl Pearson. It unfortunately does not maintain the convex ordering. Here we introduce a measure based on the mode of a distribution that maintains the c-ordering. For many classes of right-skewed distributions, it is easily computed as a function of the shape parameter of the family and the distribution function of the distribution. The measure γ _M satisfies −1 ≤ γ _M ≤ 1, with 1(−1) indicating extreme right (left) skewness. As γ _M can be found explicitly in the gamma, log-logistic, lognormal, and Weibull cases, and its influence function suggests appropriate properties as a skewness measure, it may be considered as an attractive competitor to other measures based on the mean or median.

Key Words:

• Asymmetry
• Convex ordering
• Log-logistic distribution
• Lognormal distribution