The exponential statistical manifold: mean parameters, orthogonality and space transformations. (English) Zbl 0947.62003

Pistone, Giovanni; Rogantin, Maria Piera

Bernoulli 5, No. 4, 721-760 (1999).

The present paper is dedicated to further developments of the following ideas. The statistical object that induces the geometry is the exponential model with its particular form of the Fisher information; so the starting point has to be a nonparametric definition of the exponential model. This definition in turn is related to the class of exponentially integrable random variables whose natural topology is given by the notion of Orlicz spaces for the exponential functions.
The authors study the transformation of an affine atlas and the related manifold structure under measurable transformations of the sample space. A nonparametric version of the concept of mixed parametrization in exponential models, and the effect of space transformation on the exponential manifold are given. An example aimed at showing how each of the obtained abstract results corresponds to the usual objects of the statistical analysis of statistical dependence is given.

Reviewer: Serguey M.Pokas (Odessa)

MSC:

62A01	Foundations and philosophical topics in statistics
53A15	Affine differential geometry
53A99	Classical differential geometry

Keywords:

exponential families; exponential statistical manifolds; Fisher information operator; mean parameters; Orlicz spaces; splitting; orthogonality; affine atlas; tangent space; regular parametrization; submanifolds