The Annals of Statistics

On Resampling Methods for Variance and Bias Estimation in Linear Models

Jun Shao

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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.

Article information

Ann. Statist., Volume 16, Number 3 (1988), 986-1008.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62J05: Linear regression
Secondary: 62F35: Robustness and adaptive procedures

Resampling variance and bias estimators jackknife weighted jackknife bootstrap bias reduction homoscedastic and heteroscedastic linear models asymptotic unbiasedness consistency mean squared error imbalance measure of a linear model


Shao, Jun. On Resampling Methods for Variance and Bias Estimation in Linear Models. Ann. Statist. 16 (1988), no. 3, 986--1008. doi:10.1214/aos/1176350945.

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