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

The Annals of Statistics · 2006 · Volume 34 · Issue 2

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 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.

Keywords

62E20, 62G07, 62G20, asymptotic distribution, estimating a monotone density at zero, Grenander estimator, isotonic estimation

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