Abstract
Many applications of Amari's dual geometries involve one or more submanifolds imbedded in a supermanifold. In the differential geometry literature, there is a set of equations that describe relationships between invariant quantities on the submanifold and supermanifold when the Riemannian connection is used. We extend these equations to statistical manifolds, manifolds on which a pair of dual connections is defined. The invariant quantities found in these equations include the mean curvature and the statistical curvature which are used in statistical calculations involving such topics as information loss and efficiency. As an application of one of these equations, the Bartlett correction is interpreted in terms of curvatures and other invariant quantities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amari, S. (1985).Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics,28, Springer, New York.
Amari, S. (1986). Geometrical theory on manifolds of linear systems,METR,86-1, University of Tokyo.
Amari, S. (1987). Differential geometrical theory of statistics, referred in Amariet al. (1987).
Amari, S., Barndorff-Nielsen, O. E., Kass, R. E., Lauritzen, S. L. and Rao, C. R. (1987).Differential Geometry in Statistical Inference, Institute of Mathematical Statistics, Monograph Series,10, Hayward, California.
Barndorff-Nielsen, O. E. (1978).Information and Exponential Families in Statistical Theory, Wiley, New York.
Barndorff-Nielsen, O. E. (1986). Likelihood and observed geometries,Ann. Statist.,14, 856–873.
Barndorff-Nielsen, O. E. and Blæsild, P. (1986). A note on the calculation of Bartlett adjustments,J. Roy. Statist. Soc. Ser. B,48, 353–358.
Barndorff-Nielsen, O. E. and Cox, D. R. (1984). Bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator,J. Roy. Statist. Soc. Ser. B,46, 483–495.
Barndorff-Nielsen, O. E. and Hall, P. (1988). On the level-error after Bartlett adjustment of the likelihood ratio statistic,Biometrika,75, 374–378.
Chen, B.-Y. (1984).Total Mean Curvature and Manifolds of Finite Type, World Scientific, Singapore.
Efron, B. (1975). Defining the curvature of a statistical problem,Ann. Statist.,6, 1189–1242.
Lauritzen, S. L. (1987). Statistical manifolds, referred in Amariet al. (1987).
Lawley, D. N. (1956). A general method for approximating to the distribution of likelihood-ratio criteria,Biometrika,43, 295–303.
McCullagh, P. (1987).Tensor Methods in Statistics, Chapman & Hall, New York.
McCullagh, P. and Cox, D. R. (1986). Invariants and likelihood ratio statistics,Ann. Statist.,14, 1419–1430.
Møller, J. (1986). Bartlett adjustments for structured covariances,Scand. J. Statist.,13, 1–15.
Spivak, M. (1975).A Comprehensive Introduction to Differential Geometry, Volume IV, Publish or Perish, Boston.
Vos, P. W. (1987). Dual geometries and their applications to generalized linear models, Ph. D. Dissertation, University of Chicago.
Author information
Authors and Affiliations
About this article
Cite this article
Vos, P.W. Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Ann Inst Stat Math 41, 429–450 (1989). https://doi.org/10.1007/BF00050660
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00050660