The Annals of Statistics

Resampling: consistency of substitution estimators

Hein Putter and Willem R. van Zwet

Full-text: Open access

Abstract

On the basis of N i.i.d. random variables with a common unknown distribution P we wish to estimate a functional $\tau_N(P)$. An obvious and very general approach to this problem is to find an estimator $\hat{P}_N$ of P first, and then construct a so-called substitution estimator $\tau_N (\hat{P}_N)$ of $\tau_N(P)$. In this paper we investigate how to choose the estimator $\hat{P}_N$ so that the substitution estimator $\tau_N (\hat{P}_N)$ will be consistent.

Although our setup covers a broad class of estimation problems, the main substitution estimator we have in mind is a general version of the bootstrap where resampling is done from an estimated distribution $\hat{P}_N$. We do not focus in advance on a particular estimator $\hat{P}_N$, such as, for example, the empirical distribution, but try to indicate which resampling distribution should be used in a particular situation. The conclusion that we draw from the results and the examples in this paper is that the bootstrap is an exceptionally flexible method which comes into its own when full use is made of its flexibility. However, the choice of a good bootstrap method in a particular case requires rather precise information about the structure of the problem at hand. Unfortunately, this may not always be available.

Article information

Source
Ann. Statist., Volume 24, Number 6 (1996), 2297-2318.

Dates
First available in Project Euclid: 16 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032181156

Digital Object Identifier
doi:10.1214/aos/1032181156

Mathematical Reviews number (MathSciNet)
MR1425955

Zentralblatt MATH identifier
0867.62036

Subjects
Primary: 62G09: Resampling methods
Secondary: 62F12: Asymptotic properties of estimators

Keywords
Resampling bootstrap substitution estimator consistency set of first category equicontinuity local uniform convergence

Citation

Putter, Hein; van Zwet, Willem R. Resampling: consistency of substitution estimators. Ann. Statist. 24 (1996), no. 6, 2297--2318. doi:10.1214/aos/1032181156. https://projecteuclid.org/euclid.aos/1032181156


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