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.