Semiparametric empirical likelihood inference with estimating equations under density ratio models

TL;DR

This paper develops semiparametric empirical likelihood methods for inference under density ratio models, incorporating auxiliary information via estimating equations, and demonstrates improved efficiency and valid testing through asymptotic analysis and simulations.

Contribution

It introduces a novel empirical likelihood framework for two-sample density ratio models with auxiliary estimating equations, analyzing asymptotic properties and proposing tests for auxiliary information validity.

Findings

Asymptotic normality of estimators established

Dropping an estimating equation does not reduce estimator efficiency

Simulations show improved efficiency with correct auxiliary info

Abstract

The density ratio model (DRM) provides a flexible and useful platform for combining information from multiple sources. In this paper, we consider statistical inference under two-sample DRMs with additional parameters defined through and/or additional auxiliary information expressed as estimating equations. We examine the asymptotic properties of the maximum empirical likelihood estimators (MELEs) of the unknown parameters in the DRMs and/or defined through estimating equations, and establish the chi-square limiting distributions for the empirical likelihood ratio (ELR) statistics. We show that the asymptotic variance of the MELEs of the unknown parameters does not decrease if one estimating equation is dropped. Similar properties are obtained for inferences on the cumulative distribution function and quantiles of each of the populations involved. We also propose an ELR test for the validity and usefulness of the auxiliary information. Simulation studies show that correctly specified estimating equations for the auxiliary information result in more efficient estimators and shorter confidence intervals. Two real-data examples are used for illustrations.