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Efficiency and robustness in resampling. (English) Zbl 0755.62038

The goal of this paper is to provide a unified framework for classifying different jackknife and bootstrap procedures for linear regression models into two types, type \(E\) (the efficient ones) and type \(R\) (the robust ones). After an introductory first section, section 2 addresses the simple linear regression model \(Y_ i=x_ i\beta+\varepsilon_ i\) with independent errors \(\varepsilon_ i\approx (0,\sigma^ 2_ 1)\) where least squares estimation of \(\beta\) is of interest.
It is shown that the resampling variances of the regression estimators satisfy exactly one of two possible representations. This depends mainly on \(\sum x^ 2_ i\). The two representations characterize the efficient and robust groups.
Section 3 extends the result to general linear regression. The case of the classical bootstrap in the \(E\)-type and the external bootstrap in the \(R\)-type are presented in detail, other resampling procedures are outlined.

MSC:

62G09 Nonparametric statistical resampling methods
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
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