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

TL;DR

This paper introduces a semi-parametric method for estimating f-divergences between two probability densities using density ratios, and applies it to develop a two-sample homogeneity test with proven asymptotic properties.

Contribution

It proposes an optimal semi-parametric estimator of f-divergence and connects it with existing score tests, enhancing two-sample testing under density-ratio models.

Findings

The estimator achieves minimal asymptotic variance.

The proposed test aligns with score tests based on empirical likelihood.

Numerical studies confirm the finite-sample effectiveness.

Abstract

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.