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A density ratio is defined by the ratio of two probability densities. We study the inference problem of density ratios and apply a semiparametric density-ratio estimator to the two-sample homogeneity test. In the proposed test procedure, the <formula formulatype="inline"> <tex Notation="TeX">$f$</tex></formula>-divergence between two probability densities is estimated using a density-ratio estimator. The <formula formulatype="inline"><tex Notation="TeX">$f$</tex> </formula>-divergence estimator is then exploited for the two-sample homogeneity test. We derive an optimal estimator of <formula formulatype="inline"><tex Notation="TeX">$f$</tex></formula>-divergence in the sense of the asymptotic variance in a semiparametric setting, and provide a statistic for two-sample homogeneity test based on the optimal estimator. We prove that the proposed test dominates the existing empirical likelihood score test. Through numerical studies, we illustrate the adequacy of the asymptotic theory for finite-sample inference.

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<formula formulatype="inline"><tex Notation="TeX">$f$</tex> </formula>-Divergence Estimation and Two-Sample Homogeneity Test Under Semiparametric Density-Ratio Models

$f$ -Divergence Estimation and Two-Sample Homogeneity Test Under Semiparametric Density-Ratio Models