Noise-contrastive estimation of dynamic mixture distributions

Maximum likelihood estimation of unnormalized distributions is notoriously challenging, given the intractability of the normalizing constant. We propose an unconditional and a conditional version of the noise-contrastive estimation method, based on two different noise distributions. Our analysis suggests that the two methods are approximately equivalent in terms of statistical efficiency. Compared to maximum likelihood, the noise-contrastive-based estimates have a slightly smaller RMSE for larger sample sizes, especially for the tail distribution parameters. On the other hand, their computing times are about one order of magnitude shorter than maximum likelihood. Finally, in an empirical application to operational risk measurement data, the noise-contrastive-based parameters and risk measures estimates are more precise than the corresponding maximum likelihood-based quantities.