  In many statistical and econometric applications, we gather individual
samples from various interconnected populations that undeniably exhibit common
latent structures. Utilizing a model that incorporates these latent structures
for such data enhances the efficiency of inferences. Recently, many researchers
have been adopting the semiparametric density ratio model (DRM) to address the
presence of latent structures. The DRM enables estimation of each population
distribution using pooled data, resulting in statistically more efficient
estimations in contrast to nonparametric methods that analyze each sample in
isolation. In this article, we investigate the limit of the efficiency
improvement attainable through the DRM. We focus on situations where one
population's sample size significantly exceeds those of the other populations.
In such scenarios, we demonstrate that the DRM-based inferences for populations
with smaller sample sizes achieve the highest attainable asymptotic efficiency
as if a parametric model is assumed. The estimands we consider include the
model parameters, distribution functions, and quantiles. We use simulation
experiments to support the theoretical findings with a specific focus on
quantile estimation. Additionally, we provide an analysis of real revenue data
from U.S. collegiate sports to illustrate the efficacy of our contribution.
