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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.Künsch

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