On resampling methods for variance and bias estimation in linear models
type: Item
instance of: scholarly article
title: On resampling methods for variance and bias estimation in linear models
zbMATH Open document ID: 0651.62063
author: Jun Shao
published in: The Annals of Statistics
publication date: 1988-01-01T00:00:00Z
review text: 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 \(n\to \infty.\) 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.
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G99
zbMATH DE Number: 4062396
zbMATH Keywords: heteroscedastic linear model
zbMATH Keywords: least squares estimator
zbMATH Keywords: resampling methods
zbMATH Keywords: weighted jackknife
zbMATH Keywords: bootstrap
zbMATH Keywords: resampling variance
zbMATH Keywords: imbalance measure
zbMATH Keywords: Consistency
zbMATH Keywords: bias estimators
zbMATH Keywords: asymptotically unbiased
zbMATH Keywords: mean squared errors
zbMATH Keywords: large sample properties
zbMATH Keywords: homoscedastic error model.
zbMATH Keywords: Bias reduction
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350945
OpenAlex ID: W2009586519
DOI: 10.1214/AOS/1176350945
description: scientific article; zbMATH DE number 4062396