Unbiasedness of the Likelihood Ratio Test for Lattice Conditional Independence Models

Abstract

The lattice conditional independence (LCI) model N( K ) is defined to be the set of all normal distributions N(0, Σ) on R I such that for every pair L, M ∈ K , x L and x M are conditionally independent given x L ∩ 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 , x K is the coordinate projection of x ∈ R I onto R K . 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.