Asymptotics of maximum likelihood estimators based on Markov chain Monte Carlo methods

In complex statistical models, in which exact computation of the likelihood is intractable, Monte Carlo methods can be applied to approximate maximum likelihood estimates. In this paper we consider approximation obtained via Markov chain Monte Carlo. We prove consistency and asymptotic normality of the resulting estimator, when both sample sizes (the initial and Monte Carlo one) tend to infinity. Our results can be applied to models with intractable normalizing constants and missing data models. We also investigate properties of estimators in numerical experiments.