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There are many models, often called unnormalized models, whose normalizingconstants are not calculated in closed form. Maximum likelihood estimation isnot directly applicable to unnormalized models. Score matching, contrastivedivergence method, pseudo-likelihood, Monte Carlo maximum likelihood, and noisecontrastive estimation (NCE) are popular methods for estimating parameters ofsuch models. In this paper, we focus on NCE. The estimator derived from NCE isconsistent and asymptotically normal because it is an M-estimator. NCEcharacteristically uses an auxiliary distribution to calculate the normalizingconstant in the same spirit of the importance sampling. In addition, there areseveral candidates as objective functions of NCE.  We focus on how to reduce asymptotic variance. First, we propose a method forreducing asymptotic variance by estimating the parameters of the auxiliarydistribution. Then, we determine the form of the objective functions, where theasymptotic variance takes the smallest values in the original estimator classand the proposed estimator classes. We further analyze the robustness of theestimator.

Analysis of Noise Contrastive Estimation from the Perspective of Asymptotic Variance