  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.
