On Resampling Methods for Variance and Bias Estimation in Linear Models

Shao, Jun

Ann. Statist., Tome 16 (1988) no. 1, p. 986-1008 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1988-09-14
Classification:  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,  62J05,  62F35