Imputation estimators for unnormalized models with missing data

Several statistical models are given in the form of unnormalized densities and calculation of the normalization constant is intractable. We propose estimation methods for such unnormalized models with missing data. The key concept is to combine imputation techniques with estimators for unnormalized models including noise contrastive estimation and score matching. Further, we derive asymptotic distributions of the proposed estimators and construct confidence intervals. Simulation results with truncated Gaussian graphical models and the application to real data of wind direction demonstrate that the proposed methods enable statistical inference from missing data properly.