Statistical Manifolds Admitting Torsion, Pre-contrast Functions and Estimating Functions

Abstract

It is well-known that a contrast function defined on a product manifold \(M \times M\) induces a Riemannian metric and a pair of dual torsion-free affine connections on the manifold M. This geometrical structure is called a statistical manifold and plays a central role in information geometry. Recently, the notion of pre-contrast function has been introduced and shown to induce a similar differential geometrical structure on M, but one of the two dual affine connections is not necessarily torsion-free. This structure is called a statistical manifold admitting torsion. This paper summarizes such previous results including the fact that an estimating function on a parametric statistical model naturally defines a pre-contrast function to induce a statistical manifold admitting torsion and provides some new insights on this geometrical structure. That is, we show that the canonical pre-contrast function can be defined on a partially flat space, which is a flat manifold with respect to only one of the dual connections, and discuss a generalized projection theorem in terms of the canonical pre-contrast function.