BIRS Workshop Lecture Videos
Moment-Free Measures for the Multivariate Skew-t Distribution Giorgi, Emanuele
The main features that are used to describe a multivariate probability distribution are its location, dispersion, structure of dependence, asymmetry and kurtosis. These are generally associated to the moments of the distribution if these are defined. In the case of the multivariate skew-t distribution, the moments of order greater than or equal to the respective degrees of freedom do not exist. To overcome such a problem, we propose a set of measures that are defined regardless of the value of the degrees of freedom. We use measures based on the halfspace depth function in order to summarize location and kurtosis. The halfspace depth function can be viewed as a multivariate generalization of the rank method in the univariate case and it leads to the definition of a multivariate median with desirable properties, such as affine invariance and robustness. We investigate the properties of the halfspace depth function for the skew-t distribution and its relation to radial symmetry. For the particular case of one degree of freedom, we show that the skew-t distribution is radially symmetric with center of radial symmetry given by the halfspace median. Moreover, in this case, the elliptical shape of the halfspace depth contours eases the computation of a multivariate measure of kurtosis. As for dispersion and structure of dependence, we use measures based on notions of concordance (e.g. Kendall's tau and Spearman's rho) and quantile. Finally, asymmetry is quantified by computing the Bowley index for each of the marginal distributions.
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