Nonparametric maximum likelihood density estimation and simulation-based minimum distance estimators

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