  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.
