Geometry of minimum contrast
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.
scientific article; zbMATH DE number 169023