*(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 | Statistical measures of associations |