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.

## 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.

## 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.

## No comments yet. Why not leave one?