Least Absolute Deviations Estimator
The practical importance of the least absolute deviations (LAD) estimator as a robust estimator was noted in Section 2.3. Besides its practical importance, the LAD estimator poses an interesting theoretical problem because the general results of Section 4.1 can be used to prove only the consistency ofthe LAD estimator but not its asymptotic normality, even though the LAD estimator is an extremum estimator. In Section 4.6.1 we shall prove the asymptotic normality of the median, which is the LAD estimator in the i. i.d. sample case, using a method different from the method of Section 4.1.2 and shall point out why the latter method fails in this case. In Section 4.6.2 we shall use the general results of Section 4.1.1 to prove the consistency of the LAD estimator in a regression model. Finally, in Section 4.6.3 we shall indicate what lines of proof of asymptotic normality may be used for the LAD estimator in a regression model.
The cases where the general asymptotic results of Section 4.1 cannot be wholly used may be referred to as nonregular cases. Besides the LAD estimator, we have already noted some nonregular cases in Section 4.2.3 and will encounter more in Sections 9.S and 9.6. It is hoped that the methods outlined in the present section may prove useful for dealing with unsolved problems in other nonregular cases.