There are many models, often called unnormalized models, whose normalizing constants are not calculated in closed form. Maximum likelihood estimation is not directly applicable to unnormalized models. Score matching, contrastive divergence method, pseudo-likelihood, Monte Carlo maximum likelihood, and noise contrastive estimation (NCE) are popular methods for estimating parameters of such models. In this paper, we focus on NCE. The estimator derived from NCE is consistent and asymptotically normal because it is an M-estimator. NCE characteristically uses an auxiliary distribution to calculate the normalizing constant in the same spirit of the importance sampling. In addition, there are several candidates as objective functions of NCE. We focus on how to reduce asymptotic variance. First, we propose a method for reducing asymptotic variance by estimating the parameters of the auxiliary distribution. Then, we determine the form of the objective functions, where the asymptotic variance takes the smallest values in the original estimator class and the proposed estimator classes. We further analyze the robustness of the estimator.
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