Symmetry and lattice conditional independence in a multivariate normal distribution
A class of multivariate normal models which combines the lattice conditional independence (LCI) restrictions with the group symmetry (GS) restrictions is introduced as follows. Suppose that the observation space is \(\mathbb{R}^I\) where \(I\) is a finite index set. Let an orthogonal group representation \(\rho\) on a finite group \(G\) on \(\mathbb{R}^I\) and a ring \(K\) of subsets of \(I\) be given. The GS-LCI models are defined by imposing symmetry conditions given by \(\rho\) and conditional independence conditions given by \(K\). The models generalize the LCI models introduced by \textit{S.A. Andersson} and \textit{M.D. Perlman} [ibid. 21. No. 3, 1318-1358 (1993; Zbl 0803.62042)], and the GS models investigated in numerous Danish unpublished manuscripts in 1969-1983. In the Appendix of the present paper a summary of the basic theory of GS is presented. In the paper the following statistical properties of GS-LCI models are given: the conditions for the existence and uniqueness of the maximum likelihood (ML) estimator, an explicit expression for the ML estimator, an explicit expression for the likelihood ratio statistic \(Q\) for testing one GS-LCI model against another, and the central distribution of \(Q\) in terms of the moments \(E (Q^{\alpha})\), \(\alpha > 0\). Numerous examples illustrate an importance of the presented very general and abstract theory.
scientific article; zbMATH DE number 1359530