文档介绍：Journal of Multivariate Analysis 66, 133187 (1998)
Article No. MV981745
Normal Linear Regression Models With Recursive
Graphical Markov Structure*
Steen A. Andersson
Indiana University
and
Michael D. Perlman-
University of Washington
Received April 17, 1996; revised January 25, 1998
A multivariate normal statistical model defined by the Markov properties deter-
mined by an acyclic digraph admits a recursive factorization of its likelihood
function (LF) into the product of conditional LFs, each factor having the form of
a classical multivariate linear regression model (#MANOVA model). Here these
models are extended in a natural way to normal linear regression models whose
LFs continue to admit such recursive factorizations, from which maximum
likelihood estimators and likelihood ratio (LR) test statistics can be derived by
classical linear methods. The central distribution of the LR test statistic for testing
one such multivariate normal linear regression model against another is derived, and
the relation of these regression models to block-recursive normal linear systems is
established. It is shown how a collection of nonnested dependent normal linear regres-
sion models (#seemingly unrelated regressions) can bined into a single multi-
variate normal linear regression model by imposing a parsimonious set of graphical
Markov (#conditional independence) restrictions. 1998 Academic Press
AMS 1991 subject classifications: 62H12, 62H15, 62J05, 62J10.
Key words and phrases: multivariate normal distribution; multivariate analysis of
variance (MANOVA); linear regression; recursive linear models; directed graph;
graphical Markov model; conditional independence; maximum likelihood estimate;
likelihood ratio test; seemingly unrelated regressions.
1. INTRODUCTION
Graphical Markov models use graphs, either undirected, directed, or
mixed, to represent multivariate statistical dependencies. Statistical variables
* Th