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

Abstract

We investigate the behavior of the nonparametric maximum likelihood estimator f̂_n for a decreasing density f near the boundaries of the support of f. We establish the limiting distribution of f̂_n(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.