On the differentiability of $\phi$-projections in the discrete finite case

TL;DR

This paper establishes conditions for the continuous differentiability of -projections in finite discrete spaces, with applications to robust statistics and asymptotic analysis of estimators.

Contribution

It provides new conditions ensuring -projections are differentiable, facilitating their use in statistical estimation and asymptotic analysis in finite discrete settings.

Findings

-projections are differentiable under verifiable conditions for convex sets.

Derived influence functions for -projection estimators in robust statistics.

Established asymptotic properties of -projection estimators in various parametric and non-parametric contexts.

Abstract

In the case of finite measures on finite spaces, we state conditions under which {\phi}- projections are continuously differentiable. When the set on which one wishes to {\phi}- project is convex, we show that the required assumptions are implied by easily verifiable conditions. In particular, for input probability vectors and a rather large class of {\phi}-divergences, we obtain that {\phi}-projections are continuously differentiable when projecting on a set defined by linear equalities. The obtained results are applied to {\phi}- projection estimators (that is, minimum {\phi}-divergence estimators). A first application, rooted in robust statistics, concerns the computation of the influence functions of such estimators. In a second set of applications, we derive their asymptotics when projecting on parametric sets of probability vectors, on sets of probability vectors generated from distributions with certain moments fixed and on Fr\'echet classes of bivariate probability arrays. The resulting asymptotics hold whether the element to be {\phi}-projected belongs to the set on which one wishes to {\phi}-project or not.