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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 $\hat{f}_n(n^{-\alpha})$, wherewe need to distinguish between different values of $0<\alpha<1$. Similarresults are obtained for the upper endpoint of the support, in the case it isfinite. This yields consistent estimators for the values of $f$ at theboundaries of the support. The limit distribution of these estimators isestablished and their performance is compared with the penalized maximumlikelihood estimator.

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