  The parameter estimation of unnormalized models is a challenging problem. The
maximum likelihood estimation (MLE) is computationally infeasible for these
models since normalizing constants are not explicitly calculated. Although some
consistent estimators have been proposed earlier, the problem of statistical
efficiency remains. In this study, we propose a unified, statistically
efficient estimation framework for unnormalized models and several efficient
estimators, whose asymptotic variance is the same as the MLE. The computational
cost of these estimators is also reasonable and they can be employed whether
the sample space is discrete or continuous. The loss functions of the proposed
estimators are derived by combining the following two methods: (1)
density-ratio matching using Bregman divergence, and (2) plugging-in
nonparametric estimators. We also analyze the properties of the proposed
estimators when the unnormalized models are misspecified. The experimental
results demonstrate the advantages of our method over existing approaches.
