## 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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