Groups related to Gaussian graphical models

Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. We describe the maximal group acting on the space of covariance matrices, which leaves this model invariant for any fixed graph. This group furnishes a representation of Gaussian graphical models as composite transformation families and enables to analyze properties of parameter estimators. We use these results in the robustness analysis to compute upper bounds on finite sample breakdown points of equivariant estimators of the covariance matrix. In addition we provide conditions on the sample size so that an equivariant estimator exists with probability 1.