Optimal Estimation under a Semiparametric Density Ratio Model

TL;DR

This paper investigates the efficiency limits of semiparametric density ratio models in combining data from interconnected populations, showing they can achieve near-parametric efficiency for smaller samples when one population is large.

Contribution

It demonstrates that DRM-based inferences for smaller populations can attain the highest asymptotic efficiency, comparable to parametric models, especially when one population's sample size dominates.

Findings

DRM achieves near-parametric efficiency for small samples when a large population exists.

Simulation results support theoretical efficiency gains in quantile estimation.

Real data analysis confirms practical benefits of the proposed approach.

Abstract

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.