Elliptical graphical modelling

TL;DR

This paper introduces elliptical graphical models that extend Gaussian graphical models to elliptical distributions, enabling robust analysis of data with heavy tails and outliers through new estimators and tests.

Contribution

It develops a new class of elliptical graphical models using affine equivariant scatter estimators, generalizing Gaussian models for heavy-tailed data, with theoretical properties and practical demonstrations.

Findings

Robust elliptical graphical models can be fitted using affine equivariant estimators.

Asymptotic properties of partial correlation estimators are derived.

Simulation studies confirm the robustness and effectiveness of the proposed methods.

Abstract

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.