Elliptical distributions


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.


Anderson, T. W. 2006. An introduction to multivariate statistical analysis. Hoboken, N.J.: Wiley-Interscience.
Cambanis, Stamatis, Steel Huang, and Gordon Simons. 1981. On the Theory of Elliptically Contoured Distributions.” Journal of Multivariate Analysis 11 (3): 368–85.
Chamberlain, Gary. 1983. A Characterization of the Distributions That Imply Mean—Variance Utility Functions.” Journal of Economic Theory 29 (1): 185–201.
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, 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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