  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.
