The Annals of Statistics

Estimation of means in graphical Gaussian models with symmetries

Helene Gehrmann and Steffen L. Lauritzen

Full-text: Open access


We study the problem of estimability of means in undirected graphical Gaussian models with symmetry restrictions represented by a colored graph. Following on from previous studies, we partition the variables into sets of vertices whose corresponding means are restricted to being identical. We find a necessary and sufficient condition on the partition to ensure equality between the maximum likelihood and least-squares estimators of the mean.

Article information

Ann. Statist., Volume 40, Number 2 (2012), 1061-1073.

First available in Project Euclid: 18 July 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62F99: None of the above, but in this section

Conditional independence invariance maximum likelihood estimation patterned mean vector symmetry


Gehrmann, Helene; Lauritzen, Steffen L. Estimation of means in graphical Gaussian models with symmetries. Ann. Statist. 40 (2012), no. 2, 1061--1073. doi:10.1214/12-AOS991.

Export citation


  • Andersen, H. H., Højbjerre, M., Sørensen, D. and Eriksen, P. S. (1995). Linear and Graphical Models for the Multivariate Complex Normal Distribution. Lecture Notes in Statistics 101. Springer, New York.
  • Andersson, S. (1975). Invariant normal models. Ann. Statist. 3 132–154.
  • Andersson, S. A., Brøns, H. K. and Jensen, S. T. (1983). Distribution of eigenvalues in multivariate statistical analysis. Ann. Statist. 11 392–415.
  • Bollobás, B. (1998). Modern Graph Theory. Graduate Texts in Mathematics 184. Springer, New York.
  • Chan, A. and Godsil, C. D. (1997). Symmetry and eigenvectors. In Graph Symmetry (Montreal, PQ, 1996). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 497 75–106. Kluwer Academic, Dordrecht.
  • Dawid, A. P. (1988). Symmetry models and hypotheses for structured data layouts. J. Roy. Statist. Soc. Ser. B 50 1–34.
  • Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 11. IMS, Hayward, CA.
  • Drton, M. (2008). Multiple solutions to the likelihood equations in the Behrens–Fisher problem. Statist. Probab. Lett. 78 3288–3293.
  • Eaton, M. L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley, New York.
  • Eaton, M. L. (1989). Group Invariance Applications in Statistics. Regional Conference Series in Probability and Statistics 1. IMS and American Statistical Association, Hayward, CA and Alexandria, VA.
  • Frets, G. P. (1921). Heredity of head form in man. Genetica 3 193–400.
  • Gehrmann, H. (2011). Lattices of graphical Gaussian models with symmetries. Symmetry 3 653–679.
  • Haberman, S. J. (1975). How much do Gauss–Markov and least square estimates differ? A coordinate-free approach. Ann. Statist. 3 982–990.
  • Højsgaard, S. and Lauritzen, S. L. (2008). Graphical Gaussian models with edge and vertex symmetries. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 1005–1027.
  • Højsgaard, S. and Lauritzen, S. L. (2011). gRc: Inference in graphical Gaussian models with edge and vertex symmetries. R package version 0.3.0.
  • Hylleberg, B., Jensen, M. and Ørnbøl, E. (1993). Graphical symmetry models. M.Sc. thesis, Aalborg Univ., Aalborg.
  • Jensen, S. T. (1988). Covariance hypotheses which are linear in both the covariance and the inverse covariance. Ann. Statist. 16 302–322.
  • Kruskal, W. (1968). When are Gauss–Markov and least squares estimators identical? A coordinate-free approach. Ann. Math. Statist 39 70–75.
  • Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Clarendon Press, New York.
  • Madsen, J. (2000). Invariant normal models with recursive graphical Markov structure. Ann. Statist. 28 1150–1178.
  • Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, New York, NY.
  • Olkin, I. (1972). Testing and estimation for structures which are circularly symmetric in blocks. Technical report, Educational Testing Service, Princeton, NJ.
  • Olkin, I. and Press, S. J. (1969). Testing and estimation for a circular stationary model. Ann. Math. Statist. 40 1358–1373.
  • Sachs, H. (1966). Über Teiler, Faktoren und charakteristische Polynome von Graphen. Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau 12 7–12.
  • Scheffé, H. (1944). A note on the Behrens–Fisher problem. Ann. Math. Statist. 15 430–432.
  • Siemons, J. (1983). Automorphism groups of graphs. Arch. Math. (Basel) 41 379–384.
  • Viana, M. A. G. (2008). Symmetry Studies: An Introduction to the Analysis of Structured Data in Applications. Cambridge Univ. Press, Cambridge.
  • Votaw, D. F. Jr. (1948). Testing compound symmetry in a normal multivariate distribution. Ann. Math. Statist. 19 447–473.
  • Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley, Chichester.
  • Wilks, S. S. (1946). Sample criteria for testing equality of means, equality of variances, and equality of covariances in a normal multivariate distribution. Ann. Math. Statist. 17 257–281.