Abstract
A unified treatment of all currently available cumulant-based indexes of multivariate skewness and kurtosis is provided here, expressing them in terms of the third and fourth-order cumulant vectors respectively. Such a treatment helps reveal many subtle features and inter-connections among the existing indexes as well as some deficiencies, which are hitherto unknown. Computational formulae for obtaining these measures are provided for spherical and elliptically-symmetric, as well as skew-symmetric families of multivariate distributions, yielding several new results and a systematic exposition of many known results.
Highlights
Using the standard normal distribution as the yardstick, statisticians have defined notions of skewness and kurtosis in the univariate case
The analysis presented here is based on the cumulant vectors of the third and fourth order, defined below
We believe that using only the tensor products of vectors leads to an intuitive and natural way to deal with higher order moments and cumulants for multivariate distributions, as it will be demonstrated in the paper
Summary
Using the standard normal distribution as the yardstick, statisticians have defined notions of skewness (asymmetry) and kurtosis (peakedness) in the univariate case. In a broad discussion and analysis of multivariate cumulants, their properties and their use in inference, Jammalamadaka et al (2006) proposed using the full vector of third and fourth order cumulants, as vectorial measures for multivariate skewness and kurtosis respectively. We believe that using only the tensor products of vectors leads to an intuitive and natural way to deal with higher order moments and cumulants for multivariate distributions, as it will be demonstrated in the paper. Another comprehensive reference on matrix derivatives is the book by Mathai (1997). A word about the notations: bold uppercase letters are used for random vectors and matrices while bold lowercase letters denote their specific values
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