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 NPMLE of \mycite{M.Woodroofe and J.Sun}.