Bahadur Representations for Robust Scale Estimators Based on Regression Residuals

We investigate the asymptotic behaviour of the median deviation and the semi-interquartile range based on the residuals from a linear regression model by deriving weak asymptotic representations for the estimators. These representations may be used to obtain a variety of central limit theorems and yield conditions under which the median deviation and the semi-interquartile range are asymptotically equivalent. The results justify the use of the estimators as concommitant scale estimators in the general scale equivariant M-estimation of a regression parameter problem. Finally, the results contain as a special case those obtained by Hall and Welsh (1985) for independent and identically distributed random variables.