Bahadur representations for robust scale estimators based on regression residuals (Q1083144)

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.