## The Annals of Statistics

- Ann. Statist.
- Volume 10, Number 2 (1982), 357-385.

### Differential Geometry of Curved Exponential Families-Curvatures and Information Loss

#### Abstract

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.

#### Article information

**Source**

Ann. Statist. Volume 10, Number 2 (1982), 357-385.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176345779

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aos/1176345779

**Mathematical Reviews number (MathSciNet)**

MR653513

**Zentralblatt MATH identifier**

0507.62026

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62B10: Information-theoretic topics [See also 94A17]

**Keywords**

Statistical curvatures statistical affine connections information loss recovery of information duality second-order efficiency Kullback-Leibler distance asymptotic estimation theory

#### Citation

Amari, Shun-Ichi. Differential Geometry of Curved Exponential Families-Curvatures and Information Loss. Ann. Statist. 10 (1982), no. 2, 357--385. doi:10.1214/aos/1176345779. http://projecteuclid.org/euclid.aos/1176345779.