Non-Parametric Maximum Likelihood Density Estimation and Simulation-Based Minimum Distance Estimators

Indirect inference estimators (i.e., simulation-based minimum distance estimators) in a parametric model that are based on auxiliary non-parametric maximum likelihood density estimators are shown to be asymptotically normal. If the parametric model is correctly specified, it is furthermore shown that the asymptotic variance-covariance matrix equals the Cramér-Rao bound. These results are based on uniform-in-parameters convergence rates and a uniform-in-parameters Donsker-type theorem for non-parametric maximum likelihood density estimators.