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august 1999 The exponential statistical manifold: mean parameters, orthogonality and space transformations
Giovanni Pistone, Maria Piera Rogantin
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Bernoulli 5(4): 721-760 (august 1999).

Abstract

Let ( X,cal X,μ) be a measure space, and let cal M (X,cal X,μ) denote the set of the μ -almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on cal M (X,cal X,μ) can be realized by an affine atlas whose charts are defined locally by the mappings cal M (X,cal X,μ)cal U pqlog(q/p)+K(p,q)B p , where cal U p is a suitable open set containing p , K (p,q) is the Kullback--Leibler relative information and B p is the vector space of centred and exponentially ( pμ) -integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, i.e. measurable transformation of the sample space. A generalization of the mixed parametrization method for exponential models is also presented.

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Giovanni Pistone. Maria Piera Rogantin. "The exponential statistical manifold: mean parameters, orthogonality and space transformations." Bernoulli 5 (4) 721 - 760, august 1999.

Information

Published: august 1999
First available in Project Euclid: 19 February 2007

zbMATH: 0947.62003
MathSciNet: MR1704564

Keywords: exponential families , exponential statistical manifolds , Information , mean parameters , Orlicz spaces , orthogonality

Rights: Copyright © 1999 Bernoulli Society for Mathematical Statistics and Probability

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Vol.5 • No. 4 • august 1999
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