Multivariate skewness and kurtosis measures with an application in ICA

In this paper skewness and kurtosis characteristics of a multivariate p-dimensional distribution are introduced. The skewness measure is defined as a p-vector while the kurtosis is characterized by a pxp-matrix. The introduced notions are extensions of the corresponding measures of Mardia [K.V. Mardia, Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (1970) 519-530] and Mori, Rohatgi & Szekely [T.F. Mori, V.K. Rohatgi, G.J. Szekely, On multivariate skewness and kurtosis, Theory Probab. Appl. 38 (1993) 547-551]. Basic properties of the characteristics are examined and compared with both the above-mentioned results in the literature. Expressions for the measures of skewness and kurtosis are derived for the multivariate Laplace distribution. The kurtosis matrix is used in Independent Component Analysis (ICA) where the solution of an eigenvalue problem of the kurtosis matrix determines the transformation matrix of interest [A. Hyvärinen, J. Karhunen, E. Oja, Independent Component Analysis, Wiley, New York, 2001].