Divergences and Duality for Estimation and Test under Moment Condition Models

TL;DR

This paper develops divergence-based estimation and testing methods for models with linear constraints, extending empirical likelihood techniques and analyzing their duality, limiting distributions, and power properties.

Contribution

It introduces a unified divergence minimization framework for estimation and testing under moment conditions, generalizing empirical likelihood methods with duality analysis.

Findings

Derived limiting distributions for the proposed estimators and tests.

Provided an approximation to the power function and sample size calculations.

Characterized the duality and divergence projections in the model context.

Abstract

We introduce estimation and test procedures through divergence minimiza- tion for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood approach. We treat the problems of existence and characterization of the divergence projections of probability distributions on sets of signed finite measures. We give a precise characterization of duality, for the proposed class of estimates and test statistics, which is used to derive their limiting distributions (including the EL estimate and the EL ratio statistic) both under the null hypotheses and under alterna- tives or misspecification. An approximation to the power function is deduced as well as the sample size which ensures a desired power for a given alternative.