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 f̂n 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. © Institute of Mathematical Statistics, 2006.