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Bootstrapping the Grenander estimator

Michael R. Kosorok

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Abstract

The goal of this paper is to study the bootstrap for the Grenander estimator. The first result is a proof of the inconsistency of the nonparametric bootstrap for the Grenander estimator at a given point. The second result is the development and verification of a bootstrap for the L1 confidence band for the Grenander estimator. As part of this work, kernel estimators are studied as alternatives to the Grenander estimator. We show that when the second derivative of the true density is assumed to be uniformly bounded, there exist kernel estimators with faster convergence rates than the Grenander estimator. We study the implications of this in developing L1 and uniform confidence bands and discuss some open questions.

Chapter information

Source
N. Balakrishnan, Edsel A. Peña and Mervyn J. Silvapulle, eds., Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 282-292

Dates
First available in Project Euclid: 1 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1207058280

Digital Object Identifier
doi:10.1214/193940307000000202

Subjects
Primary: 62G09: Resampling methods 62G07: Density estimation
Secondary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes

Keywords
Chernoff’s distribution confidence bands kernel estimators L_1 error Monte Carlo methods pointwise error uniform error

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

Kosorok, Michael R. Bootstrapping the Grenander estimator. Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 282--292, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000202. https://projecteuclid.org/euclid.imsc/1207058280


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