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March, 1978 The Geometry of Exponential Families
Bradley Efron
Ann. Statist. 6(2): 362-376 (March, 1978). DOI: 10.1214/aos/1176344130


There are two important spaces connected with every multivariate exponential family, the natural parameter space and the expectation parameter space. We describe some geometric results relating the two. (In the simplest case, that of a normal translation family, the two spaces coincide and the geometry is the familiar Euclidean one.) Maximum likelihood estimation, within one-parameter curved subfamilies of the multivariate family, has two simple and useful geometric interpretations. The geometry also relates to the Fisherian question: to what extent can the Fisher information be replaced by $-\partial^2/\partial\theta^2\lbrack\log f_\theta(x)\rbrack\mid_{\theta=\hat{\theta}}$ in the variance bound for $\hat{\theta}$, the maximum likelihood estimator?


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Bradley Efron. "The Geometry of Exponential Families." Ann. Statist. 6 (2) 362 - 376, March, 1978.


Published: March, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0436.62027
MathSciNet: MR471152
Digital Object Identifier: 10.1214/aos/1176344130

Primary: 62F10

Keywords: curvature , Duality , Kullback-Leibler distance , maximum likelihood estimation

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 2 • March, 1978
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