The Annals of Statistics

On Resampling Methods for Variance and Bias Estimation in Linear Models

Jun Shao

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350945

Digital Object Identifier
doi:10.1214/aos/1176350945

Mathematical Reviews number (MathSciNet)
MR959186

Zentralblatt MATH identifier
0651.62063

JSTOR
links.jstor.org

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

Keywords
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

Citation

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. https://projecteuclid.org/euclid.aos/1176350945


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