Lattice Models for Conditional Independence in a Multivariate Normal Distribution

Andersson, Steen Arne ; Perlman, Michael D.

Ann. Statist., Tome 21 (1993) no. 1, p. 1318-1358 / Harvested from Project Euclid

Résumé

The lattice conditional independence model $\mathbf{N}(\mathscr{K})$ is defined to be the set of all normal distributions on $\mathbb{R}^I$ such that for every pair $L, M \in \mathscr{K}, x_L$ and $x_M$ are conditionally independent given $x_{L \cap M}$. Here $\mathscr{K}$ is a ring of subsets of the finite index set $I$ and, for $K \in \mathscr{K}, x_K$ is the coordinate projection of $x \in \mathbb{R}^I$ onto $\mathbb{R}^K$. Statistical properties of $\mathbf{N}(\mathscr{K})$ may be studied, for example, maximum likelihood inference, invariance and the problem of testing $H_0: \mathbf{N}(\mathscr{K})$ vs. $H: \mathbf{N}(\mathscr{M})$ when $\mathscr{M}$ is a subring of $\mathscr{K}$. The set $J(\mathscr{K})$ of join-irreducible elements of $\mathscr{K}$ plays a central role in the analysis of $\mathbf{N}(\mathscr{K})$. This class of statistical models occurs in the analysis of nonnested multivariate missing data patterns and nonnested dependent linear regression models.

Détail

Publié le : 1993-09-14
Classification:  Distributive lattice,  join-irreducible elements,  pairwise conditional independence,  multivariate normal distribution,  generalized block-triangular matrices,  maximum likelihood estimator,  quotient space,  nonested missing data,  nonnested linear regressions,  62H12,  62H15,  62H20,  62H25