Unbiasedness of the Likelihood Ratio Testfor Lattice Conditional Independence Models

The lattice conditional independence (LCI) model N(C)<) is defined to be the set of all normal distributions N(O, E) on IR I such that for every pair L, M E 1<, XL and xM are conditionally independent given xLnM (ct , [AP] (1993a)). Here C)< is a ring of subsets (hence a distributive lattice) of the finite index set I such that 0, IE ".K, while for K E 1<, XK is the coordinate projection of x E IRI onto IRK. [AP] (1993b) derived the likelihood ratio (LR) statistic A for testing one LCI model against another, i.e.. for testing N(C)<) vs. N(1I\) based on a random sample from N(O, E), where 11\ is a subring of C)<. 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 E under the LCI model N(1<) are presented.