Abstract
The comparison of an estimated parameter to its standard error, the Wald test, is a well known procedure of classical statistics. Here we discuss its application to graphical Gaussian model selection. First we derive the Fisher information matrix and its inverse about the parameters of any graphical Gaussian model. Both the covariance matrix and its inverse are considered and a comparative analysis of the asymptotic behaviour of their maximum likelihood estimators (m.l.e.s) is carried out. Then we give an example of model selection based on the standard errors. The method is shown to produce almost identical inference to likelihood ratio methods in the example considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barndorff-Nielsen, O. E. (1978) Information and Exponential Families in Statistical Theory. Wiley: New York.
Cox, D. R. and Hinkley, D. V. (1974) Theoretical Statistics. Chapman and Hall: London.
Cox, D. R. and Wermuth, N. (1990) An approximation to maximum likelihood estimates in reduced models. Biometrika, 77, 747–61.
Darroch, J. N., Lauritzen, S. L. and Speed, T. P. (1980) Markov fields and log-linear interaction models for contingency tables. Annals of Statistics, 43, 1470–80.
Dawid, A. P. and Lauritzen, S. L. (1993) Hyper Markov laws in the statistical analysis of decomposable graphical models. Annals of Statistics, 21, 1272–1317.
Dempster, A. P. (1972) Covariance selection. Biometrics, 28, 157–75.
Edwards, D. (1995) Introduction to Graphical Modelling. Springer Verlag.
Graybill, F. A. (1983) Matrices with Applications in Statistics. 2nd Edition. Wadsworth: California.
Isserlis, L. (1918) On a formula for the product-moment correlation of any order of a normal frequency distribution in any number of variables. Biometrika, 12, 134–9.
Roverato, A. and Whittaker, J. (1996) A hyper Markov prior distribution for approximate Bayes factor calculations on non-decomposable graphical Gaussian models, Technical Report, Department of Mathematics and Statistics, Lancaster University.
Smith, P. W. F. (1990) Edge Exclusion Tests for Graphical Models. Unpublished Ph.D. thesis. Lancaster University.
Smith, P. W. F. and Whittaker, J. (1992) Edge Exclusion Tests for Graphical Gaussian Models. Technical Report. Department of Mathematics, Lancaster University.
Speed, T. P. and Kiiveri, H. (1986) Gaussian Markov distributions over finite graphs. Annals of Statistics, 14, 138–50.
Wermuth, N. (1976) Analogies between multiplicative models in contingency tables and covariance selection. Biometrics, 32, 95–108.
Whittaker, J. (1990) Graphical Models in Applied Multivariate Statistics. Wiley: Chichester.
Wright, S. (1954) The interpretation of multivariate systems. In: O. Kempthorne et ail (Eds) Statistics and Mathematics in Biology, Iowa State University Press: Ames 11–23.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Roverato, A., Whittaker, J. Standard errors for the parameters of graphical Gaussian models. Stat Comput 6, 297–302 (1996). https://doi.org/10.1007/BF00140874
Issue Date:
DOI: https://doi.org/10.1007/BF00140874