The Annals of Statistics

New methods for bias correction at endpoints and boundaries

Peter Hall and Byeong U. Park

Full-text: Open access

Abstract

We suggest two new, translation-based methods for estimating and correcting for bias when estimating the edge of a distribution. The first uses an empirical translation applied to the argument of the kernel, in order to remove the main effects of the asymmetries that are inherent when constructing estimators at boundaries. Placing the translation inside the kernel is in marked contrast to traditional approaches, such as the use of high-order kernels, which are related to the jackknife and, in effect, apply the translation outside the kernel. Our approach has the advantage of producing bias estimators that, while enjoying a high order of accuracy, are guaranteed to respect the sign of bias. Our second method is a new bootstrap technique. It involves translating an initial boundary estimate toward the body of the dataset, constructing repeated boundary estimates from data that lie below the respective translations, and employing averages of the resulting empirical bias approximations to estimate the bias of the original estimator. The first of the two methods is most appropriate in univariate cases, and is studied there; the second approach may be used to bias-correct estimates of boundaries of multivariate distributions, and is explored in the bivariate case.

Article information

Source
Ann. Statist., Volume 30, Number 5 (2002), 1460-1479.

Dates
First available in Project Euclid: 28 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1035844983

Digital Object Identifier
doi:10.1214/aos/1035844983

Mathematical Reviews number (MathSciNet)
MR1936326

Zentralblatt MATH identifier
1014.62041

Subjects
Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Keywords
Bias estimation bootstrap curve estimation free disposal hull estimator frontier estimation kernel methods nonparametric density estimation productivity analysis translation

Citation

Hall, Peter; Park, Byeong U. New methods for bias correction at endpoints and boundaries. Ann. Statist. 30 (2002), no. 5, 1460--1479. doi:10.1214/aos/1035844983. https://projecteuclid.org/euclid.aos/1035844983


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  • CANBERRA, ACT 0200 AUSTRALIA DEPARTMENT OF STATISTICS SEOUL NATIONAL UNIVERSITY SEOUL 151-747 KOREA E-MAIL: bupark@stats.snu.ac.kr