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Model selection for Gaussian concentration graphs. (English) Zbl 1108.62098
Summary: A multivariate Gaussian graphical Markov model for an undirected graph $$G$$, also called a covariance selection model or concentration graph model, is defined in terms of the Markov properties, i.e., conditional independences associated with $$G$$, which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients. By means of Fisher’s $$z$$-transformation and Z. Šidák’s correlation inequality [J. Am. Stat. Assoc. 62, 626–633 (1967; Zbl 0158.17705)], conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained, leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. The simultaneous $$p$$-values corresponding to the partial correlations are partitioned into three disjoint sets, a significant set $$S$$, an indeterminate set $$I$$ and a nonsignificant set $$N$$.
Our model selection method selects two graphs, a graph $$\widehat{G}_{SI}$$ whose edges correspond to the set $$S\cup I$$, and a more conservative graph $$\widehat{G}_S$$ whose edges correspond to $$S$$ only. Similar considerations apply to covariance graph models, which are defined in terms of marginal independence rather than conditional independence. The method is applied to some well-known examples and to simulated data.

##### MSC:
 62M99 Inference from stochastic processes 05C90 Applications of graph theory 62H20 Measures of association (correlation, canonical correlation, etc.)
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