Abstract
Graphical models are used to examine conditional independence among random variables. In this chapter we take graphical models for the multivariate complex normal distribution w.r.t. simple undirected graphs into consideration. This is the first published presentation of these models. Graphical models for the multivariate real normal distribution, also called covariance selection models, have already been studied in the literature. The initial work on covariance selection models is done by Dempster (1972) and Wermuth (1976) and a presentation of these models is given in Eriksen (1992). Graphical models for contingency tables are introduced in statistics by Darroch et al. (1980) and further these are well-described in Lauritzen (1989). Graphical association models are treated in general in Whittaker (1990) and Lauritzen (1993). The complex normal graphical models are quite similar to the covariance selection models. We have chosen to develop this chapter without use of exponential families. We study definition of the model, maximum likelihood estimation and hypothesis testing. To verify some of the results in the chapter we use results from mathematical analysis. These can be found in e.g. Rudin (1987). In graphical models one uses the concentration matrix instead of the variance matrix as it is more advantageous. Therefore we define this matrix and derive a relation which is basic for complex normal graphical models. Afterwards we formally define a complex normal graphical model w.r.t. a simple undirected graph. As these models are used to examine conditional independence of selected pairs of variables given the remaining ones we are mainly interested in inference on the concentration matrix. It is possible to base the maximum likelihood estimation of the concentration matrix on a complex random matrix with mean zero.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Andersen, H.H., Højbjerre, M., Sørensen, D., Eriksen, P.S. (1995). Complex Normal Graphical Models. In: Linear and Graphical Models. Lecture Notes in Statistics, vol 101. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4240-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4240-6_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94521-7
Online ISBN: 978-1-4612-4240-6
eBook Packages: Springer Book Archive