  A statistical model is said to be un-normalised when its likelihood function
involves an intractable normalising constant. Two popular methods for parameter
inference for these models are MC-MLE (Monte Carlo maximum likelihood
estimation), and NCE (noise contrastive estimation); both methods rely on
simulating artificial data-points to approximate the normalising constant.
While the asymptotics of MC-MLE have been established under general hypotheses
(Geyer, 1994), this is not so for NCE. We establish consistency and asymptotic
normality of NCE estimators under mild assumptions. We compare NCE and MC-MLE
under several asymptotic regimes. In particular, we show that, when m goes to
infinity while n is fixed (m and n being respectively the number of artificial
data-points, and actual data-points), the two estimators are asymptotically
equivalent. Conversely, we prove that, when the artificial data-points are IID,
and when n goes to infinity while m/n converges to a positive constant, the
asymptotic variance of a NCE estimator is always smaller than the asymptotic
variance of the corresponding MC-MLE estimator. We illustrate the variance
reduction brought by NCE through a numerical study.
