Bias-Reduced Estimation of Mean Absolute Deviation Around the Median

A bias-reduced estimator is proposed for the mean absolute deviation parameter of a median regression model. The proposal shares some similarities with the degrees of freedom correction for the variance estimator of a linear 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. An empirical approach is adopted, as a crucial quantity required for the task is generally unknown. Based on simulations, the proposal seems to work in high-dimensional settings.