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

TL;DR

This paper analyzes the asymptotic behavior of the NPMLE for decreasing densities near support boundaries, providing limiting distributions and boundary estimators, and comparing them with penalized MLE.

Contribution

It establishes the limiting distribution of the NPMLE at the boundaries of the support for decreasing densities, including finite endpoints, and compares these with penalized MLE.

Findings

Derived the limiting distribution of the NPMLE near boundaries.

Provided consistent boundary estimators for the density.

Compared performance of NPMLE with penalized MLE.

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.