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Bahadur representations for robust scale estimators based on regression residuals. (English) Zbl 0604.62028
Let \(\theta_ n\) be an initial estimate in the regression model \(y_ j=x_ j'\theta +e_ j\) and denote by \(e_ j(\theta_ n)\) the residuals \(y_ j-x_ j'\theta_ n\). The paper derives asymptotic Bahadur representations for the interquartile range and the median absolute deviation of the \(e_ j(\theta_ n)'s\) under weak conditions on the design and the distribution of the \(e_ j's\). These representations are the same as for known \(\theta\). As a corollary the asymptotic behaviour of these two scale estimates follows and their use as concomitant scale estimates in robust regression M-estimators is justified.
Reviewer: H.R.K√ľnsch

MSC:
62F35 Robustness and adaptive procedures (parametric inference)
62F12 Asymptotic properties of parametric estimators
62J05 Linear regression; mixed models
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