The behavior of the NPMLE of a decreasing density near the boundaries of the support
Vladimir Kulikov and Hendrik Lopuhaa
Annals of Statistics Volume Vol. 34, Number No. 2, 2006.

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 NPMLE of \mycite{M.Woodroofe and J.Sun}.

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