# Projection Theorems of Divergences and Likelihood Maximization Methods

## TL;DR

This paper explores projection theorems for various divergences used in robust statistics, linking them to likelihood maximization and sufficiency principles, and deriving new theorems for density power divergences.

## Contribution

It establishes the equivalence between divergence projection methods and likelihood-based estimation, and derives a new projection theorem for density power divergences.

## Key findings

- Projection theorems relate reverse and forward divergence projections.
- Equivalence shown between divergence projections and estimating equations.
- New projection theorem derived for density power divergences.

## Abstract

Projection theorems of divergences enable us to find reverse projection of a divergence on a specific statistical model as a forward projection of the divergence on a different but rather "simpler" statistical model, which, in turn, results in solving a system of linear equations. Reverse projection of divergences are closely related to various estimation methods such as the maximum likelihood estimation or its variants in robust statistics. We consider projection theorems of three parametric families of divergences that are widely used in robust statistics, namely the R\'enyi divergences (or the Cressie-Reed power divergences), density power divergences, and the relative $\alpha$-entropy (or the logarithmic density power divergences). We explore these projection theorems from the usual likelihood maximization approach and from the principle of sufficiency. In particular, we show the equivalence of solving the estimation problems by the projection theorems of the respective divergences and by directly solving the corresponding estimating equations. We also derive the projection theorem for the density power divergences.