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.

## Recommended reading

OG paper introduction Cambanis, Huang, and Simons (1981) is basically a textbook on the bits that are important to me at least, and it is not a bad textbook at that. K.-T. Fang, Kotz, and Ng (2017) is an actual textbook.

## 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.

*Symmetric Multivariate and Related Distributions*. Boca Raton: Chapman and Hall/CRC.

*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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