TBD

In the easy non-degenerate case this is intuitive to state: An elliptical distribution with a density function \(f\) has the form \[ f(x)\propto g((x-\mu )'\Sigma ^{-1}(x-\mu ))\]. where \(x\in\mathbb{R}^p\) and the RV of \(X\sim f\) has mean \(\mu\) (which is also the mean vector if the latter exists), and \(\Sigma\) is a positive definite matrix which is proportional to the covariance matrix of \(X\) if such a covariance matrix exists.

These are useful in generalising multivariate Gaussians.

I assume they did not invent this idea, but Davison and Ortiz (2019) point out that if you have a least-squares-compatible model, usually it can generalise to any elliptical density, which includes many M-estimator-style robust losses.

## References

*An introduction to multivariate statistical analysis*. Hoboken, N.J.: Wiley-Interscience.

*Journal of Multivariate Analysis*11 (3): 368–85.

*Journal of Economic Theory*29 (1): 185–201.

*arXiv:1611.10266 [Math, Stat]*, November.

*arXiv:1910.14139 [Cs]*, October.

*Generalized Multivariate Analysis*. Beijing: Science Press.

*Elliptically Contoured Models in Statistics and Portfolio Theory*. Second edition. New York: Springer.

*Journal of Multivariate Analysis*99 (5): 912–27.

*Journal of Statistical Planning and Inference*143 (11): 2016–22.

*Annual Review of Statistics and Its Application*8 (1): 369–91.

*Annual Review of Statistics and Its Application*8 (1): 301–27.

*arXiv:2107.02308 [Cs]*, July.

*The Journal of Finance*38 (3): 745–52.

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