Unbiasedness of the Likelihood Ratio Test for Lattice Conditional Independence Models

The lattice conditional independence (LCI) model N(K) is defined to be the set of all normal distributions N(0, Σ) on RI such that for every pair L, M ∈ K, xL and xM are conditionally independent given xL ∩ M. Here K is a ring of subsets (hence a distributive lattice) of the finite index set I such that ∅ I ∈ K, while for K ∈ K, xK is the coordinate projection of x ∈ RI onto RK. Andersson and Perlman in the preceding paper derived the likelihood ratio (LR) statistic λ for testing one LCI model against another, i.e., for testing N(K) vs N(M) based on a random sample from N(0, Σ), where M is a subring of K. In the present paper the strict unbiasedness of the LR test is established, and related results regarding the distribution of the maximum likelihood estimator of Σ under the LCI model N(K) are presented.