Edge Exclusion Tests for Improper Complex Gaussian Graphical Model Selection

Abstract

We consider the problem of inferring the conditional independence graph (CIG) of improper complex-valued Gaussian vectors. A $p$ -variate improper complex Gaussian graphical model associated with an undirected graph with $p$ vertices is defined as the family of improper complex Gaussian distributions that obey the conditional independence restrictions implied by the edge set of the graph. For real random vectors, considerable body of paper exists where one first tests for exclusion of each edge from the saturated model, and then infers the CIG. Prior work on proper complex Gaussian graphical models is sparse, while that on improper complex Gaussian graphical models is non-existent. In this paper, we propose and analyze a generalized likelihood ratio test (GLRT) based edge exclusion test statistic for improper complex Gaussian graphical models. The null distribution of the test statistic is specified explicitly to allow analytical calculation of the test threshold. An alternative computationally fast version of the GLRT statistic is also derived. Simulation examples are presented to illustrate the proposed statistic.