Affine Gateaux Differentials and the von Mises Statistical Calculus

TL;DR

This paper explores a generalized form of differentiability for functionals on convex domains using affine functionals, providing a unified framework relevant to applications in statistics.

Contribution

It introduces and studies affine Gateaux differentials, extending traditional differentiability concepts to non-open convex domains with applications in statistical calculus.

Findings

Develops a unified theory of affine Gateaux differentials

Connects affine differentiability to statistical applications

Provides a comprehensive perspective on non-open domain differentiability

Abstract

This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear functionals typically employed in standard Gateaux differentiability. This affine notion of differentiability naturally arises in certain applications and has been utilized by some authors in the statistics literature. We aim to offer a unified and comprehensive perspective on this concept.