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June, 1982 Differential Geometry of Curved Exponential Families-Curvatures and Information Loss
Shun-Ichi Amari
Ann. Statist. 10(2): 357-385 (June, 1982). DOI: 10.1214/aos/1176345779


The differential-geometrical framework is given for analyzing statistical problems related to multi-parameter families of distributions. The dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint. The duality connected by the Legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures. The second-order information loss is calculated for Fisher-efficient estimators, and is decomposed into the sum of two non-negative terms. One is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator. Only the latter term depends on the estimator, and vanishes for the maximum-likelihood estimator. A set of statistics which recover the second-order information loss are given. The second-order efficiency also is obtained. The differential geometry of the function space of distributions is discussed.


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Shun-Ichi Amari. "Differential Geometry of Curved Exponential Families-Curvatures and Information Loss." Ann. Statist. 10 (2) 357 - 385, June, 1982.


Published: June, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0507.62026
MathSciNet: MR653513
Digital Object Identifier: 10.1214/aos/1176345779

Primary: 62E20
Secondary: 62B10

Keywords: asymptotic estimation theory , Duality , information loss , Kullback-Leibler distance , recovery of information , second-order efficiency , statistical affine connections , Statistical curvatures

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • June, 1982
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