# Uniform minimum risk equivariant estimates for moment condition models

## TL;DR

This paper studies invariant semiparametric models and shows that minimum empirical divergence estimates, including empirical likelihood, are equivariant, providing a way to identify the minimum risk equivariant estimate using conditional expectations.

## Contribution

It introduces a novel approach to identify minimum risk equivariant estimates in moment condition models using empirical divergence methods.

## Key findings

- Minimum empirical divergence estimates are equivariant.
- The minimum risk equivariant estimate can be characterized via conditional expectations.
- Asymptotic approximation of the conditional expectation is derived.

## Abstract

We consider semiparametric moment condition models invariant to transformation groups. The parameter of interest is estimated by minimum empirical divergence approach, introduced by Broniatowski and Keziou (2012). It is shown that the minimum empirical divergence estimates, including the empirical likelihood one, are equivariants. The minimum risk equivariant estimate is then identied to be any one of the minimum empirical divergence estimates minus its expectation conditionally to maximal invariant statistic of the considered group of transformations. An asymptotic approximation to the conditional expectation, is obtained, using the result of Jureckov{\'a} and Picek (2009).