Annals of Statistics

Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis

C. F. J. Wu

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Motivated by a representation for the least squares estimator, we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time. They are unbiased for homoscedastic errors and a special case, the delete-one jackknife, is almost unbiased for heteroscedastic errors. The method is extended to cover nonlinear parameters, regression $M$-estimators, nonlinear regression and generalized linear models. Interval estimators can be constructed from the jackknife histogram. Three bootstrap methods are considered. Two are shown to give biased variance estimators and one does not have the bias-robustness property enjoyed by the weighted delete-one jackknife. A general method for resampling residuals is proposed. It gives variance estimators that are bias-robust. Several bias-reducing estimators are proposed. Some simulation results are reported.

Article information

Ann. Statist., Volume 14, Number 4 (1986), 1261-1295.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62J05: Linear regression
Secondary: 62J02: General nonlinear regression 62G05: Estimation

Weighted jackknife bootstrap linear regression variable jackknife jackknife percentile bias-robustness bias reduction Fieller's linterval representation of the least squares estimator $M$-regression nonlinear regression generalized linear models balanced residuals


Wu, C. F. J. Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis. Ann. Statist. 14 (1986), no. 4, 1261--1295. doi:10.1214/aos/1176350142.

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