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September, 1988 On Resampling Methods for Variance and Bias Estimation in Linear Models
Jun Shao
Ann. Statist. 16(3): 986-1008 (September, 1988). DOI: 10.1214/aos/1176350945


Let $g$ be a nonlinear function of the regression parameters $\beta$ in a heteroscedastic linear model and $\hat{\beta}$ be the least squares estimator of $\beta.$ We consider the estimation of the variance and bias of $g(\hat{\beta})$ [as an estimator of $g(\beta)$] by using three resampling methods: the weighted jackknife, the unweighted jackknife and the bootstrap. The asymptotic orders of the mean squared errors and biases of the resampling variance and bias estimators are given in terms of an imbalance measure of the model. Consistency of the resampling estimators is also studied. The results indicate that the weighted jackknife variance and bias estimators are asymptotically unbiased and consistent and their mean squared errors are of order $o(n^{-2})$ if the imbalance measure converges to zero as the sample size $n \rightarrow \infty$. Furthermore, based on large sample properties, the weighted jackknife is better than the unweighted jackknife. The bootstrap method is shown to be asymptotically correct only under a homoscedastic error model. Bias reduction, a closely related problem, is also discussed.


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Jun Shao. "On Resampling Methods for Variance and Bias Estimation in Linear Models." Ann. Statist. 16 (3) 986 - 1008, September, 1988.


Published: September, 1988
First available in Project Euclid: 12 April 2007

zbMATH: 0651.62063
MathSciNet: MR959186
Digital Object Identifier: 10.1214/aos/1176350945

Primary: 62J05
Secondary: 62F35

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 3 • September, 1988
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