The behavior of the NPMLE of a decreasing density near the boundaries of the support

We investigate the behavior of the nonparametric maximum likelihood estimator $\hat{f}_n$ for a decreasing density $f$ near the boundaries of the support of $f$. We establish the limiting distribution of $\hat{f}_n(n^{-\alpha})$, where we need to distinguish between different values of $0<\alpha<1$. Similar results are obtained for the upper endpoint of the support, in the case it is finite. This yields consistent estimators for the values of $f$ at the boundaries of the support. The limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator.