  A bias-reduced estimator is proposed for the mean absolute deviation
parameter of a median regression model. A workaround is devised for the lack of
smoothness in the sense conventionally required in general bias-reduced
estimation. A local asymptotic normality property and a Bahadur--Kiefer
representation suffice in proving the validity of the bias correction. The
proposal is developed under a classical asymptotic regime but, based on
simulations, it seems to work also in high-dimensional settings.
