Bootstrapping the Grenander estimator

By Michael R. Kosorok. 2008

68 citations recorded by the source corpus.

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 $L_1$ 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 $L_1$ and uniform confidence bands and discuss some open questions.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d9818e9eb4dd34ab9f3e4e47b58f0a2063d4d51c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Adityanand Guntuboyina
2. Bodhisattva Sen
3. Cécile Durot
4. Hendrik P. Lopuhaä
5. P. Groeneboom

In Statistics and Probability

1. Cécile Durot
2. Adityanand Guntuboyina
3. P. Groeneboom
4. Bodhisattva Sen
5. Hendrik P. Lopuhaä
6. Pierre C. Bellec
7. Richard J. Samworth
8. Fadoua Balabdaoui

Citations in the corpus

• Nonparametric Shape-Restricted Regression (2017, background): “It is well-known that both the above bootstrap schemes — bootstrapping pairs and bootstrapping residuals with f̌n = f̂n — yield inconsistent estimators of Hn; see [1, 56, 80, 119, 122].”
• Testing equality of functions under monotonicity constraints (2012, methodology): “From the results in Sen et al. (2010) and Kosorok (2008), it appears that bootstrapping from the Grenander does not work when the statistic of interest is f̂n0 at a fixed point.”