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 ${\hat{f}}_n$ 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 maximum likelihood estimator.

Published at http://dx.doi.org/10.1214/009053606000000100 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)