Annals of Statistics

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

Bradley Efron

Full-text: Open access

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


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