SUMM/ENH: TModel use t-distribution for estimation

Start by reviewing the existing TModel and TArma in miscmodels, then inspect sandbox.distribution.MVT and MVElliptical. Determine the intended multivariate t estimation and terminology before defining the scope. Done would include a usable t-distribution fit method and its integration with TModel, with tests covering estimation behavior.

(coming back to t-distribution MLE after robust readings)

motivation: heavy-tailed distribution but not clear outliers

current: TModel, TArma in miscmodels, t-based covariance in outlier robust cov PR
sandbox.distribution has MVT as basic distribution class, including rvs for simulation, but no fit method.
sandbox.distribution also has a class MVElliptical, which was supposed to work as a superclass for various elliptically symmetric distributions, but AFAIR I didn't do much or anything with it.

(terminology: I'm not sure what to call the multivariate t scatter/cov/shape matrix Sigma.
cov_y = Sigma * df / (df - 2) if df > 2
same terminology problem for all elliptical distributions. Taylor uses "scatter" where scaling factor is indeterminate. robust literature uses "shape" for cov scaled so that det(cov) = 1
)

extension:

• heterogeneous kurtosis or degrees of freedom (varying by variable), copulas
• skew-t (skew-elliptical distribution ?)

technical notes:

Property 4 in Lange, Little, Taylor 1989: Mahalanobis distance is F-distributed
(I couldn't find that property while searching for it with google)
Appendix B has score for general elliptically symmetric models and expected information matrix for multivariate t model.
Another important property: estimates for mean parameter and cov, df are asymptotically uncorrelated, i.e. block-diagonal EIM. This holds for all elliptically symmetric models. (As a consequence we can concentrate on inference for mean parameters separately from cov, df as in GLM/LEF.)
They also mention generalized power exponential family as another elliptically symmetric distribution family that includes also short tails, but is computationally and analytically less tractable.

Lange, Kenneth L., Roderick J. A. Little, and Jeremy M. G. Taylor. 1989. “Robust Statistical Modeling Using the T Distribution.” Journal of the American Statistical Association 84 (408): 881–96. doi:10.2307/2290063.