We investigate the behavior of the nonparametric maximum likelihood estimator f̂n for a decreasing density f near the boundaries of the support of f. We establish the limiting distribution of f̂n(n−α), where we need to distinguish between different values of 0<α<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.
"The behavior of the NPMLE of a decreasing density near the boundaries of the support." Ann. Statist. 34 (2) 742 - 768, April 2006. https://doi.org/10.1214/009053606000000100