Abstract
An important classical problem is testing whether several centered multivariate normal distributions have the same covariance matrix, which is equivalent to testing that certain Wishart distributions have the same natural parameter. Wishart distributions, which are supported on sets of positive definite matrices, are a special case of generalized Riesz distributions, which are supported on sets of matrices related to the Markov properties of decomposable undirected graphs. This leads to the problem of testing whether several generalized Riesz distributions have the same natural parameter. In this paper, we derive the likelihood ratio statistic for this testing problem and find its moments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, T.W. (2003). An introduction to multivariate statistical analysis, (3rd ed.) John Wiley and Sons, New Jersey.
Andersson, S.A., Brøns, H.K., Jensen, S.T (1983). Distribution of eigenvalues in multivariate statistical analysis. Ann. Stat., 11, 2, 392–415.
Andersson, S.A. and Klein, T. (2010). On Riesz and Wishart distributions associated with decomposable undirected graphs. J. Multivar. Anal., 101, 3, 789–810.
Andersson, S.A., Madigan, D. and Perlman, M.D. (2001). Alternative Markov properties for chain graphs. Scand. J. Stat., 28, 1, 33–85.
Bartlett, M.S. (1937). Properties of sufficiency and statistical tests. Proc. R. Soc. Lond., 160, (A), 268–282.
Bartlett, M.S. (1938). Further aspects of the theory of multiple regression. Proc. Camb. Philos. Soc., 34, 33–40.
Box, G.E.P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36, 317–346.
Brown, G.W. (1939). On the power of the L 1 test for equality of several variances. Ann. Math. Statist., 10, 119–128.
Frydenberg, M. (1990). The chain graph Markov property. Scand. J. Stat., 17, 333–353.
Hassairi, A. and Lajmi, S. (2001). Riesz exponential families on symmetric cones. J. Theor. Probab., 14, (4), 927–948.
Lauritzen, S.L (1996). Graphical models. Clarendon Press, Oxford.
Lauritzen, S.L., Wermuth, N (1989). Graphical models for association between variables, some of which are qualitative and some quantitative. Ann. Stat., 17, 31–57.
Neyman, J. and Pearson, E.S. (1931). On the problem of k samples. Bull. Acad. Polonaise Sci. Let., A, 460–481.
Perlman, M. (1980). Unbiasedness of the likelihood ratio tests for equality of several covariance matrices and equality of several multivariate normal populations. Ann. Statist., 8, (2), 247–263.
Pitman, E.J.G. (1939). Tests of hypotheses concerning location and scale parameters. Biometrika, 31, 200–215.
Sugiura, N. and Nagao, N. (1968). Unbiasedness of some test criteria for the equality of one or two covariance matrices. Ann. Math. Statist., 39, 1686–1692.
Sugiura, N. and Nagao, N. (1969). On Bartletts test and Lehmanns test for homogeneity of variances. Ann. Math. Statist., 40, 2018–2032.
Wilks, S.S. (1932). Certain generalizations in the analysis of variance. Biometrika, 24, (3/4), 471–494.
Wishart, J. (1928). The generalised product moment distribution in samples from a normal multivariate population. Biometrika, 20A, (1/2), 32–52.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Andersson, S.A., Crawford, J.B. A Likelihood Ratio Test for Equality of Natural Parameters for Generalized Riesz Distributions. Sankhya A 77, 186–210 (2015). https://doi.org/10.1007/s13171-014-0052-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-014-0052-5
Keywords and phrases.
- Bartlett’s test
- Decomposable undirected graphs
- Graphical models
- Likelihood ratio test
- Riesz distribution
- Wishart distribution.