# Elliptical distributions Generalising multivariate Gaussians to anything which has a density function of the form $f(x)\propto g((x-\mu )'\Sigma ^{-1}(x-\mu ))$ where $$\mu$$ is the mean vector, $$\Sigma$$ is a positive definite matrix, and $$g:\mathbb{R}^+\to\mathbb{R}^+$$. In fact, we do not need the density function to exist; it’s ok if $$\Sigma$$ is positive semi-definite or to allow $$g$$ to be a generalised function. Baby steps! If the mean of such an $$X\sim f$$ RV exists, it is $$\mu$$, and $$\Sigma$$ is proportional to the covariance matrix of $$X$$ if such a covariance matrix exists.

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.

## Elliptical processes

See Aste (2021);Bånkestad et al. (2020).

## References

Anderson, T. W. 2006. An introduction to multivariate statistical analysis. Hoboken, N.J.: Wiley-Interscience.
Aste, Tomaso. 2021. arXiv.
Bånkestad, Maria, Jens Sjölund, Jalil Taghia, and Thomas Schön. 2020. arXiv.
Cambanis, Stamatis, Steel Huang, and Gordon Simons. 1981. Journal of Multivariate Analysis 11 (3): 368–85.
Chamberlain, Gary. 1983. Journal of Economic Theory 29 (1): 185–201.
Culan, Christophe, and Claude Adnet. 2016. arXiv:1611.10266 [Math, Stat], November.
Davison, Andrew J., and Joseph Ortiz. 2019. arXiv:1910.14139 [Cs], October.
Fang, Kai-Tai, Samuel Kotz, and Kai Wang Ng. 2017. Symmetric Multivariate and Related Distributions. Boca Raton: Chapman and Hall/CRC.
Fang, Kaitai, and Yao-ting Zhang. 1990. Generalized Multivariate Analysis. Beijing: Science Press.
Gupta, A. K., T. Varga, and Taras Bodnar. 2013. Elliptically Contoured Models in Statistics and Portfolio Theory. Second edition. New York: Springer.
Landsman, Zinoviy, and Johanna Nešlehová. 2008. Journal of Multivariate Analysis 99 (5): 912–27.
Landsman, Zinoviy, Steven Vanduffel, and Jing Yao. 2013. Journal of Statistical Planning and Inference 143 (11): 2016–22.
Ley, Christophe, Slađana Babić, and Domien Craens. 2021. Annual Review of Statistics and Its Application 8 (1): 369–91.
Markatou, Marianthi, Dimitrios Karlis, and Yuxin Ding. 2021. Annual Review of Statistics and Its Application 8 (1): 301–27.
Ortiz, Joseph, Talfan Evans, and Andrew J. Davison. 2021. arXiv:2107.02308 [Cs], July.
Owen, Joel, and Ramon Rabinovitch. 1983. The Journal of Finance 38 (3): 745–52.

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