LATTICE MODELS FOR CONDITIONAL-INDEPENDENCE IN A MULTIVARIATE NORMAL-DISTRIBUTION

The lattice conditional independence model N(K) is defined to be the set of all normal distributions on R(I) such that for every pair L, M is-an-element-of K, x(L) and x(M) are conditionally independent given x(L and M). Here K is a ring of subsets of the finite index set I and, for K is-an-element-of K, x(K) is the coordinate projection of x is-an-element-of R(I) onto R(K). Statistical properties of N(K) may be studied, for example, maximum likelihood inference, invariance and the problem of testing H-0: N(K) vs. H: N(M) when M is a subring of K. The set J(K) of join-in-educible elements of K plays a central role in the analysis of N(K). This class of statistical models occurs in the analysis of nonnested multivariate missing data patterns and nonnested dependent linear regression models.