## The Annals of Statistics

- Ann. Statist.
- Volume 21, Number 3 (1993), 1318-1358.

### Lattice Models for Conditional Independence in a Multivariate Normal Distribution

Steen Arne Andersson and Michael D. Perlman

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 21, Number 3 (1993), 1318-1358.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349261

**Digital Object Identifier**

doi:10.1214/aos/1176349261

**Mathematical Reviews number (MathSciNet)**

MR1241268

**Zentralblatt MATH identifier**

0803.62042

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H12: Estimation

Secondary: 62H15: Hypothesis testing 62H20: Measures of association (correlation, canonical correlation, etc.) 62H25: Factor analysis and principal components; correspondence analysis

**Keywords**

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

#### Citation

Andersson, Steen Arne; Perlman, Michael D. Lattice Models for Conditional Independence in a Multivariate Normal Distribution. Ann. Statist. 21 (1993), no. 3, 1318--1358. doi:10.1214/aos/1176349261. https://projecteuclid.org/euclid.aos/1176349261