Abstract
A yoke on a differentiable manifold M gives rise to a whole family of derivative strings. Various elemental properties of a yoke are discussed in terms of these strings. In particular, using the concept of intertwining from the theory of derivative strings it is shown that a yoke induces a family of tensors on M. Finally, the expected and observed α-geometries of a statistical model and related tensors are shown to be derivable from particular yokes.
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 geometric methods in statistics, Lecture Notes in Statist., 28, Springer, Heidelberg.
Amari, S. (1987). Differential geometrical theory of statistics—towards new developments, Differential Geometry in Statistical Inference, 19–94, IMS Lecture Notes-Monograph Series, Hayward, California.
Amari, S. and Kumon, M. (1983). Differential geometry of Edgeworth expansion in curved exponential family, Ann. Inst. Statist. Math., 35, 1–24.
Barndorff-Nielsen, O. E. (1978). Information and Exponential Families, Wiley, Chichester.
Barndorff-Nielsen, O. E. (1986a). Likelihood and observed geometries, Ann. Statist., 14, 856–873.
Barndorff-Nielsen, O. E. (1986b). Strings, tensorial combinants, and Bartlett adjustments, Proc. Roy. Soc. London Ser. A, 406, 127–137.
Barndorff-Nielsen, O. E. (1987a). On some differential geometric concepts and their relations to statistics, Geometrization of Statistical Theory (ed. C. T. J. Dodson), 53–90, ULDM Publications, Dept. Math., University of Lancaster, England.
Barndorff-Nielsen, O. E. (1987b). Differential and integral geometry in statistical inference, Differential Geometry in Statistical Inference, 95–162, IMS Lecture Notes-Monograph Series, Hayward, California.
Barndorff-Nielsen, O. E. (1988a). Differential geometry and statistics: Some mathematical aspects, Indian J. Math., 29, Ramanujan Centenary Volume.
Barndorff-Nielsen, O. E. (1988b). Parametric statistical models and likelihood, Lecture Notes in Statist., 50, Springer, Heidelberg.
Barndorff-Nielsen, O. E. (1989). In the discussion of Kass, R. E.: The geometry of asymptotic inference, Statist. Sci., 4, 188–234.
Barndorff-Nielsen, O. E. and Blæsild, P. (1987a). Strings: Mathematical theory and statistical examples, Proc. Roy. Soc. London Ser. A, 411, 155–176.
Barndorff-Nielsen, O. E. and Blæsild, P. (1987b). Strings: Contravariant aspect, Proc. Roy. Soc. London Ser. A, 411, 421–444.
Barndorff-Nielsen, O. E. and Blæsild, P. (1988). Coordinate-free definition of structurally symmetric derivative strings, Adv. in Appl. Math., 9, 1–6.
Barndorff-Nielsen, O. E., Blæsild, P. and Mora, M. (1988). Derivative strings and higher order differentiation, Memoir 11, Dept. Theor. Statist., Aarhus University, Aarhus, Denmark.
Blæsild, P. (1987a). Yokes: Elemental properties with statistical applications, Geometrization of Statistical Theory (ed. C. T. J. Dodson), 193–196, ULDM Publications, Dept. Math., University of Lancaster, Lancaster, England.
Blæsild, P. (1987b). Further analogies between expected and observed geometries of statistical models, Research Report 160, Dept. Theor. Statist., Aarhus University, Aarhus, Denmark.
Blæsild, P. (1988). Yokes and tensors derived from yokes, Research Report 173, Dept. Theor. Statist., Aarhus University, Aarhus, Denmark.
Dodson, C. T. J., Jupp, P. E., Kendall, W. S. and Lauritzen, S. L. (1987). A summary of points raised in the Discussion Sessions, Geometrization of Statistical Theory (ed. C. T. J. Dodson), 235–250, ULDM Publications, Dept. Math., University of Lancaster, Lancaster, England.
Eguchi, S. (1983). Second order efficiency of minimum contrast estimators in curved exponential families, Ann. Statist., 11, 793–803.
Eguchi, S. (1985). A differential geometric approach to statistical inference on the basis of contrast functionals, Hiroshima Math. J., 15, 341–391.
Lauritzen, S. L. (1987). Statistical manifolds, Differential Geometry in Statistical Inference, 163–216, IMS Lecture Notes-Monograph Series, Hayward, California.
McCullagh, P. (1987). Tensor Methods in Statistics, Chapman and Hall, London.
McCullagh, P. and Cox, D. R. (1986). Invariants and likelihood ratio statistics, Ann. Statist., 14, 1419–1430.
Mora, M. (1988). Geometrical expansions for the distributions of the score vector and the maximum likelihood estimator, Research Report 172, Dept. Theor. Statist., Aarhus University, Aarhus, Denmark.
Murray, M. K. and Rice, J. W. (1987). On differential geometry in statistics, Research Report, School of Mathematical Sciences, The Flinders University of South Australia, Adelaide, Australia.
Author information
Authors and Affiliations
About this article
Cite this article
Blæsild, P. Yokes and tensors derived from yokes. Ann Inst Stat Math 43, 95–113 (1991). https://doi.org/10.1007/BF00116471
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00116471