## Annals of Statistics

- Ann. Statist.
- Volume 3, Number 6 (1975), 1189-1242.

### Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency)

#### Abstract

Statisticians know that one-parameter exponential families have very nice properties for estimation, testing, and other inference problems. Fundamentally this is because they can be considered to be "straight lines" through the space of all possible probability distributions on the sample space. We consider arbitrary one-parameter families $\mathscr{F}$ and try to quantify how nearly "exponential" they are. A quantity called "the statistical curvature of $\mathscr{F}$" is introduced. Statistical curvature is identically zero for exponential families, positive for nonexponential families. Our purpose is to show that families with small curvature enjoy the good properties of exponential families. Large curvature indicates a breakdown of these properties. Statistical curvature turns out to be closely related to Fisher and Rao's theory of second order efficiency.

#### Article information

**Source**

Ann. Statist., Volume 3, Number 6 (1975), 1189-1242.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343282

**Digital Object Identifier**

doi:10.1214/aos/1176343282

**Mathematical Reviews number (MathSciNet)**

MR428531

**Zentralblatt MATH identifier**

0321.62013

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 62F20

**Keywords**

Curvature exponential families Cramer-Rao lower bound locally most powerful tests Fisher information second order efficiency deficiency maximum likelihood estimation

#### Citation

Efron, Bradley. Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency). Ann. Statist. 3 (1975), no. 6, 1189--1242. doi:10.1214/aos/1176343282. https://projecteuclid.org/euclid.aos/1176343282