Elliptical distributions

TBD

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.