f-divergence estimation and two-sample homogeneity test under semiparametric density-ratio models

A density ratio is defined by the ratio of two probability densities. We study the inference problem of density ratios and apply a semi-parametric density-ratio estimator to the two-sample homogeneity test. In the proposed test procedure, the f-divergence between two probability densities is estimated using a density-ratio estimator. The f-divergence estimator is then exploited for the two-sample homogeneity test. We derive the optimal estimator of f-divergence in the sense of the asymptotic variance, and then investigate the relation between the proposed test procedure and the existing score test based on empirical likelihood estimator. Through numerical studies, we illustrate the adequacy of the asymptotic theory for finite-sample inference.

💡 Research Summary

This paper addresses the problem of estimating the ratio of two probability densities—known as the density ratio—and leverages that estimate to construct a two‑sample homogeneity test based on f‑divergences. The authors adopt a semiparametric density‑ratio model, positing that the ratio r(x)=p₁(x)/p₂(x) can be expressed as r(x;θ) for a finite‑dimensional parameter θ. By fitting θ to samples drawn independently from p₁ and p₂, they obtain an estimator \hat r(x) without directly estimating either density.

The central contribution is a new plug‑in estimator for the f‑divergence D_f(p₁‖p₂)=∫p₂(x)f(r(x))dx. While the naïve plug‑in approach simply averages f(\hat r) over the second‑sample observations, the authors propose a bias‑corrected version that incorporates the derivative f′. Specifically, the estimator
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