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A density ratio is defined by the ratio of two probability densities. Westudy the inference problem of density ratios and apply a semi-parametricdensity-ratio estimator to the two-sample homogeneity test. In the proposedtest procedure, the f-divergence between two probability densities is estimatedusing a density-ratio estimator. The f-divergence estimator is then exploitedfor the two-sample homogeneity test. We derive the optimal estimator off-divergence in the sense of the asymptotic variance, and then investigate therelation between the proposed test procedure and the existing score test basedon empirical likelihood estimator. Through numerical studies, we illustrate theadequacy of the asymptotic theory for finite-sample inference.

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