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November, 1975 Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency)
Bradley Efron
Ann. Statist. 3(6): 1189-1242 (November, 1975). DOI: 10.1214/aos/1176343282


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.


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Bradley Efron. "Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency)." Ann. Statist. 3 (6) 1189 - 1242, November, 1975.


Published: November, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0321.62013
MathSciNet: MR428531
Digital Object Identifier: 10.1214/aos/1176343282

Primary: 62B10
Secondary: 62F20

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

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 6 • November, 1975
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