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August 2013 Groups acting on Gaussian graphical models
Jan Draisma, Sonja Kuhnt, Piotr Zwiernik
Ann. Statist. 41(4): 1944-1969 (August 2013). DOI: 10.1214/13-AOS1130

Abstract

Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. From an inferential point of view, it is important to realize that they are composite exponential transformation families. We reveal this structure by explicitly describing, for any undirected graph, the (maximal) matrix group acting on the space of concentration matrices in the model. The continuous part of this group is captured by a poset naturally associated to the graph, while automorphisms of the graph account for the discrete part of the group. We compute the dimension of the space of orbits of this group on concentration matrices, in terms of the combinatorics of the graph; and for dimension zero we recover the characterization by Letac and Massam of models that are transformation families. Furthermore, we describe the maximal invariant of this group on the sample space, and we give a sharp lower bound on the sample size needed for the existence of equivariant estimators of the concentration matrix. Finally, we address the issue of robustness of these estimators by computing upper bounds on finite sample breakdown points.

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Jan Draisma. Sonja Kuhnt. Piotr Zwiernik. "Groups acting on Gaussian graphical models." Ann. Statist. 41 (4) 1944 - 1969, August 2013. https://doi.org/10.1214/13-AOS1130

Information

Published: August 2013
First available in Project Euclid: 23 October 2013

zbMATH: 1292.62098
MathSciNet: MR3127854
Digital Object Identifier: 10.1214/13-AOS1130

Subjects:
Primary: 62F35 , 62H99
Secondary: 54H15

Keywords: Breakdown point , concentration matrix , Covariance matrix , equivariant estimator , Gaussian graphical models , robust estimator , transformation families

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 4 • August 2013
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