On a Vector-Valued Measure of Multivariate Skewness

Nicola Loperfido

doi:10.3390/sym13101817

Abstract

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the canonical skewness vector with respect to representations, transformations, and norm. In particular, the paper shows its connections with tensor contraction, scalar measures of multivariate kurtosis and Mardia’s skewness, the best-known scalar measure of multivariate skewness. A simulation study empirically compares the powers of tests for multivariate normality based on the squared norm of the canonical skewness vector and on Mardia’s skewness. An example with financial data illustrates the statistical applications of the canonical skewness vector.

Highlights

This section uses the financial data in [17] to illustrate a statistical application of the canonical skewness vector and the partial skewness
For the data at hand, the canonical skewness vector and the partial skewness nicely generalize to the multivariate case and the univariate features of financial returns
This paper investigated some theoretical and empirical properties of the canonical skewness vector

Summary

Introduction

That is, when the only component of x is the random variable X, the skewness vector coincides with the skewness of X, that is, its third standardized moment, as follows:. The canonical skewness vector is a null vector when x is centrally symmetric, that is, if x − μ and if μ − x are identically distributed [4]. The canonical skewness vector might be a null vector, even if the underlying distribution is skewed with finite third moments and a positive definite covariance matrix. Total skewness is nonnegative, equals zero if the underlying distribution is centrally symmetric, and is invariant with respect to nonsingular affine transformations. The total skewness is always positive when some third-order cumulants of the underlying distribution are non-null and the covariance matrix is positive definite.

Theory

E Z1 Z22

Simulations

Example

Conclusions