Abstract
Amari's ±1-divergences and geometries provide an important description of statistical inference. The ±1-divergences are constructed so that they are compatible with a metric that is defined by the Fisher information. In many cases, the ±1-divergences are but two in a family of divergences, called the f-divergences, that are compatible with the metric. We study the geometries induced by these divergences. Minimizing the f-divergence provides geometric estimators that are naturally described using certain curvatures. These curvatures are related to asymptotic bias and efficiency loss. Under special but important restrictions, the geometry of f-divergence is closely related to the α-geometry, Amari's extension of the ±1-geometries. One application of these results is illustrated in an example.
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 Statist., 28, Springer, New York.
Amari, S. (1987). Differential geometrical theory of statistics, Differential Geometry in Statistical Inference, 19–94, IMS Lecture Notes-Monograph Series, Vol. 10, Hayward, California.
Barndorff-Nielsen, O. E. (1987). Differential geometry and statistics: some mathematical aspects, Indian J. Math., 29, 335–350.
Bates, D. and Watts, D. (1988). Nonlinear Regression Analysis and Its Applications, Wiley, New York.
Blæsild, P. (1987). Yokes: elemental properties with statistical applications, Geometrization of Statistical Theory, Proceedings of the GST Workshop, University of Lancaster (ed. C. T. J. Dodson), 193–198, ULDM Publications, University of Lancaster.
Csiszar, I. (1967). On topological properties of f-divergence, Studia Sci. Math. Hungar., 2, 329–339.
Eguchi, S. (1983). Second-order efficiency of minimum contrast estimators in a curved exponential family, 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.
Hoaglin, D., Mosteller, F. and Tukey, J. (1983). Understanding Robust and Exploratory Data Analysis, Wiley, New York.
Lauritzen, S. L. (1987). Statistical manifolds, Differential Geometry in Statistical Inference, 163–216, ISM Lecture Notes-Monograph Series, Vol. 10, Hayward, California.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed., Chapman and Hall, New York.
Pfanzagl, J. (1973). Asymptotic expansions related to minimum contrast estimators, Ann. Statist., 1, 993–1026.
Rao, C. R. (1987). Differential metrics in probability spaces, Differential Geometry in Statistical Inference, 217–240, ISM Lecture Notes-Monograph Series, Vol. 10, Hayward, California.
Read, N. and Cressie, T. R. C. (1988). Goodness-of-fit statistics for discrete multivariate data, Springer Ser. Statist., Springer, New York.
Ruppert, D. and Aldershof, B. (1989). Transformations to symmetry and homoscedasticity, J. Amer. Statist. Assoc., 84, 437–446.
Vos, P. W. (1989). Fundamental equations for statistical submanifolds with applications to the Bartlett correction, Ann. Inst. Statist. Math., 41, 429–450.
Vos, P. W. (1991). Minimum f-divergence estimators and quasi-likelihood functions, to appear in Ann. Inst. Statist. Math.
Author information
Authors and Affiliations
Additional information
This work was supported by National Science Foundation grant DMS 88-03584.
About this article
Cite this article
Vos, P.W. Geometry of f-divergence. Ann Inst Stat Math 43, 515–537 (1991). https://doi.org/10.1007/BF00053370
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00053370