# Elliptical distributions

June 24, 2015 — January 3, 2023

algebra
density
geometry
high d
linear algebra
measure
probability
risk
signal processing
spheres
statistics

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, though let us have densities for now. 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.

## 2 Elliptical processes

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

## 4 References

Anderson. 2006. An introduction to multivariate statistical analysis.
Bånkestad, Sjölund, Taghia, et al. 2020.
Cambanis, Huang, and Simons. 1981. Journal of Multivariate Analysis.
Chamberlain. 1983. Journal of Economic Theory.
Culan, and Adnet. 2016. arXiv:1611.10266 [Math, Stat].
Davison, and Ortiz. 2019. arXiv:1910.14139 [Cs].
Fang, Kai-Tai, Kotz, and Ng. 2017. Symmetric Multivariate and Related Distributions.
Fang, Kaitai, and Zhang. 1990. Generalized Multivariate Analysis.
Gupta, Varga, and Bodnar. 2013. Elliptically Contoured Models in Statistics and Portfolio Theory.
Landsman, and Nešlehová. 2008. Journal of Multivariate Analysis.
Landsman, Vanduffel, and Yao. 2013. Journal of Statistical Planning and Inference.
Ley, Babić, and Craens. 2021. Annual Review of Statistics and Its Application.
Markatou, Karlis, and Ding. 2021. Annual Review of Statistics and Its Application.
Ortiz, Evans, and Davison. 2021. arXiv:2107.02308 [Cs].
Owen, and Rabinovitch. 1983. The Journal of Finance.