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, 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.

Elliptical processes

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


Anderson, T. W. 2006. An introduction to multivariate statistical analysis. Hoboken, N.J.: Wiley-Interscience.
Aste, Tomaso. 2021. Stress Testing and Systemic Risk Measures Using Multivariate Conditional Probability.” arXiv.
Bånkestad, Maria, Jens Sjölund, Jalil Taghia, and Thomas Schön. 2020. The Elliptical Processes: A Family of Fat-Tailed Stochastic Processes.” arXiv.
Cambanis, Stamatis, Steel Huang, and Gordon Simons. 1981. On the Theory of Elliptically Contoured Distributions.” Journal of Multivariate Analysis 11 (3): 368–85.
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Culan, Christophe, and Claude Adnet. 2016. Regularized Maximum Likelihood Estimation of Covariance Matrices of Elliptical Distributions.” arXiv:1611.10266 [Math, Stat], November.
Davison, Andrew J., and Joseph Ortiz. 2019. FutureMapping 2: Gaussian Belief Propagation for Spatial AI.” 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. Stein’s Lemma for Elliptical Random Vectors.” Journal of Multivariate Analysis 99 (5): 912–27.
Landsman, Zinoviy, Steven Vanduffel, and Jing Yao. 2013. A Note on Stein’s Lemma for Multivariate Elliptical Distributions.” Journal of Statistical Planning and Inference 143 (11): 2016–22.
Ley, Christophe, Slađana Babić, and Domien Craens. 2021. Flexible Models for Complex Data with Applications.” Annual Review of Statistics and Its Application 8 (1): 369–91.
Markatou, Marianthi, Dimitrios Karlis, and Yuxin Ding. 2021. Distance-Based Statistical Inference.” Annual Review of Statistics and Its Application 8 (1): 301–27.
Ortiz, Joseph, Talfan Evans, and Andrew J. Davison. 2021. A Visual Introduction to Gaussian Belief Propagation.” arXiv:2107.02308 [Cs], July.
Owen, Joel, and Ramon Rabinovitch. 1983. On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice.” The Journal of Finance 38 (3): 745–52.

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