Geometry of minimum contrast
type: Item
instance of: scholarly article
title: Geometry of minimum contrast
zbMATH Open document ID: 0780.53015
author: Shinto Eguchi
published in: Hiroshima Mathematical Journal
publication date: 1993-05-16T00:00:00Z
review text: The paper is concerned with a differential geometric approach to some concepts of mathematical statistics [cf. also the author's paper, ibid. 15, 341-391 (1985; Zbl 0625.62004)]. A contrast function \(\rho\) on a manifold \(M\) is a differentiable function on \(M \times M\) with \(\rho(x,y) \geq 0\) and equality if and only if \(x=y\). For any two vector fields \(X,Y\) on \(M\), the partial derivative \(\rho(X | Y)(x)=((X,Y) \rho)\) \((x,x)\) defines a Riemannian metric \(g=-\rho(X | Y)\) on \(M\), if, as it is further assumed, \((XX \rho(\;,x))(x)>0\) for every nonvanishing vector field \(X\). If on a Riemannian manifold one takes as \(\rho\) the squared arc-length of a geodesic curve connecting \(x\) and \(y\), then the metric constructed in this way is the original one. The metric \(g\) is accompanied by two torsionfree connections, half the sum of which is the Levi-Civita connection of \(g\), and by two curvature- like tensor fields, the antisymmetric part of which are the Riemannian curvature tensors. Both of these tensor fields and the Riemannian curvature tensors vanish for one of the above connections if and only if they do for both of them. Some deformations of contrast functions and so- called minimum contrast leafs are discussed.
Mathematics Subject Classification ID: 53B15
Mathematics Subject Classification ID: 62J12
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 169023
zbMATH Keywords: contrast function
zbMATH Keywords: Riemannian metric
zbMATH Keywords: squared arc-length
zbMATH Keywords: connections
zbMATH Keywords: curvature-like tensor fields
zbMATH Keywords: curvature tensors
zbMATH Keywords: minimum contrast leafs
reviewed by: Klaus Horneffer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 169023