Abstract
We present, compare and classify popular families of flexible multivariate distributions. Our classification is based on the type of symmetry (spherical, elliptical, central symmetry or asymmetry) and the tail behaviour (a single tail weight parameter or multiple tail weight parameters). We compare the families both theoretically (relevant properties and distinctive features) and with a Monte Carlo study (comparing the fitting abilities in finite samples).
Highlights
Probability distributions are the building blocks of statistical modelling and inference
Given the appealing properties of the univariate SAS distribution, we suggest to use them as marginal distributions in combination with the t-copula described in the previous section
Two further remarks should be made: (1) we focus on the multivariate SAS distributions within the family of multivariate distributions obtained via the transformation approach, and (2) we omit copulas in general and rather focus on meta-elliptical distributions
Summary
Probability distributions are the building blocks of statistical modelling and inference. One needs multivariate distributions that are flexible, in the sense that they can incorporate skewness and heavy tails. Their parameters should bear clear interpretations, and parameter estimation ought to be feasible. We decided to present those popular flexible families of multivariate distributions that we deem the most suitable for fitting skew and heavy-tailed data, and discuss their advantages and drawbacks. Our comparison goes crescendo in the sense that we start from the simplest families and every new section extends the previous one These extensions are based on the tail behavior, moving from a single tail weight parameter to multiple tail weight parameters, and/or the type of symmetry, moving from spherical, elliptical, central symmetry to skewness. We use =d for equality in distribution between random quantities
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