Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras

Abstract

In this paper we generalize the major results of Andersson and Perlman on LCI models to the setting of symmetric cones and give an explicit closed form formula for the estimate of the covariance matrix in the generalized LCI models that we define. To this end, we replace the cone $H_I^+(\mathbb{R})$ sitting inside the Jordan algebra of symmetric real $I \times I$-matrices by the symmetric cone $\Omega$ of an Euclidean Jordan algebra $V$. We introduce the Andersson-Perlman cone $\Omega(\mathscr{K}\subseteq\Omega$ which generalizes $\mathscr{P}(\mathscr{K})\subseteq H_I^+(\mathscr{R})$. We prove several characterizations and properties of $\Omega(\mathscr{K})$ which allows us to recover, though with different proofs, the main results of Andersson and Perlman regarding $\mathscr{P}(\mathscr{K})$. The new lattice conditional independence models are defined, assuming that the Euclidean Jordan algebra $V$ has a symmetric representation. Using standard results from the theory of Jordan algebras, we can reduce the general model to the case where $V$ is the Jordan algebra of Hermitian matrices over the real, complex or quaternionic numbers, and $\Omega$ is the corresponding cone of positive-definite matrices. Our main statistical result is a closed-form formula for the estimate of the covariance matrix in the generalized LCI model. We also give the likelihood ratio test for testing a given model versus another one, nested within the first.