Geometry of Pre-contrast Functions and Non-conservative Estimating Functions

Abstract

This paper introduces the notion of pre-contrast functions and studies their geometric properties. A contrast function is an asymmetric squared distance like function. A pre-contrast function corresponds to the differential of a contrast function, however, it is not integrable in general. While a contrast function induces a statistical manifold, a pre-contrast function induces a statistical manifold admitting torsion. As an application of pre-contrast functions, geometry of non-conservative estimating functions in statistics is also studied.

On some information geometric structures concerning Mercator projections

“…However, it is not well known that an affine connection with torsion plays a fundamental role to describe such a loxodrome as an auto-parallel path on the sphere. Based on this fact, it is worth to generalize the method of information geometry by incorporating an torsion such as the works [15,16,17,18] on SMAT. Further studies are needed in order to develop a generalization of the information geometry by using an affine connection with torsion.…”

“…To the best of my knowledge, it was Kurose [15] that considered torsions in statistical manifold for the first time. Matsuzoe [16,17] developed the statistical manifold admitting torsion (SMAT), and Henmi [18] applied it to the field of statistics. It is hence very interesting to further study SMAT from the view point of the results obtained in this paper for future research works.…”