Uniform central limit theorems for the Grenander estimator

We consider the Grenander estimator that is the maximum likelihood estimator for non-increasing densities. We prove uniform central limit theorems for certain subclasses of bounded variation functions and for H\"older balls of smoothness s>1/2. We do not assume that the density is differentiable or continuous. The proof can be seen as an adaptation of the method for the parametric maximum likelihood estimator to the nonparametric setting. Since nonparametric maximum likelihood estimators lie on the boundary, the derivative of the likelihood cannot be expected to equal zero as in the parametric case. Nevertheless, our proofs rely on the fact that the derivative of the likelihood can be shown to be small at the maximum likelihood estimator.