  We propose elliptical graphical models based on conditional uncorrelatedness
as a general- ization of Gaussian graphical models by letting the population
distribution be elliptical instead of normal, allowing the fitting of data with
arbitrarily heavy tails. We study the class of propor- tionally affine
equivariant scatter estimators and show how they can be used to perform
elliptical graphical modelling, leading to a new class of partial correlation
estimators and analogues of the classical deviance test. General expressions
for the asymptotic variance of partial correla- tion estimators, unconstrained
and under decomposable models, are given, and the asymptotic chi square
approximation of the pseudo-deviance test statistic is proved. The feasibility
of our approach is demonstrated by a simulation study, using, among others,
Tyler's scatter estimator, which is distribution-free within the elliptical
model. Our approach provides a robustification of Gaussian graphical modelling.
The latter is likelihood-based and known to be very sensitive to model
misspecification and outlying observations.
