Abstract: We develop directional tests to compare incomplete undirected graphs in the general context of covariance selection for Gaussian graphical models. The exactness of the underlying saddlepoint approximation is proved for chordal graphs, and leads to exact control of the size of the tests, given that the only approximation error involved is from the numerical calculation of two scalar integrals. Although exactness is not guaranteed for non-chordal graphs, the ability of the saddlepoint approximation to control the relative error means the proposed method outperforms its competitors even in these cases. The accuracy of our proposal is verified using simulation experiments under challenging scenarios in which inference via standard asymptotic approximations to the likelihood ratio test and some of its higher- order modifications fails. The directional approach is used to illustrate the assessment of Markovian dependencies in a data set from a veterinary trial on cattle. A second example with microarray data shows how to select the graph structure related to genetic anomalies due to acute lymphocytic leukemia.

Key words and phrases: Covariance selection, exponential family, higher-order asymptotics, likelihood ratio test, saddlepoint approximation, undirected graph.