Model selection for Gaussian concentration graphs.

*(English)*Zbl 1108.62098Summary: 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.

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.) |