An alternative data analytic approach to measure the univariate and multivariate skewness

Abstract

We introduce a new measure of univariate skewness of a distribution or data based on quantiles and by using the concepts of even and odd functions. Based on this new measure, we then suggest an approach to define the multivariate skewness for the multivariate distributions and multidimensional data and accordingly suggest a measure for it. Using numerous data sets, we illustrate that Mardia’s measure of multivariate skewness appears to be ambiguous in what it actually measures and show that our measure not only has an intuitive appeal, it also unambiguously quantifies what one would view as the multivariate skewness. Approach presented here is data analytic and can be implemented on a computer. Based on the idea of orthogonal transformation of the data, we also suggest another multivariate measure of skewness which may be simpler to compute.

Appendix

Proof of Equation (7)

The coefficient of skewness $\delta$ is defined in (6) as

$$\begin{aligned} \delta = \frac{\varvec{\int }_0^1\bigg ({\frac{F^{-1}(\alpha )+ F^{-1}(1 - \alpha )}{2}\bigg )^2 d\alpha }}{\varvec{\int }_0^1\bigg (\frac{F^{-1}(\alpha )+ F^{-1}(1 - \alpha )}{2}\bigg )^2 d\alpha + \varvec{\int }_0^1\bigg (\frac{F^{-1}(\alpha )- F^{-1}(1 - \alpha )}{2}\bigg )^2 d\alpha }. \end{aligned}$$

Note that $\int _0^1(F^{-1}(\alpha ))^2 d\alpha = \int _0^1(F^{-1}(1 - \alpha ))^2 d\alpha = \mu _2'$, the second raw moment of the underlying random variable. However, since the random variable is assumed to have been centered with zero mean, $\mu _2' = \mu _2 = \sigma ^2 > 0$, the variance. Thus, upon squaring and simplifying, we have

$$\begin{aligned} \delta =\frac{1}{2} {\bigg [ 1 + \frac{\int _0^1({F^{-1}(\alpha ) F^{-1}(1 - \alpha ))} d\alpha }{\mu _2}\bigg ]}. \end{aligned}$$

$\square$