A Comparison of the Estimators of the Scale Parameter of the Errors Distribution in the L<sub>1</sub> Regression

The L 1 regression is a robust alternative to the least squares regression whenever there are outliers in the values of the response variable, or the errors follow a long-tailed distribution. To calculate the standard errors of the L 1 estimators, construct confidence intervals and test hypotheses about the parameters of the model, or to calculate a robust coefficient of determination, it is necessary to have an estimate of a scale parameter τ. This parameter is such that τ 2 /n is the variance of the median of a sample of size n from the errors distribution. [1] proposed the use of ˆ τ , a consistent, and so, an asymptotically unbiased estimator of τ. However, this estimator is not stable in small samples, in the sense that it can increase with the introduction of new independent variables in the model. When the errors follow the Laplace distribution, the maximum likelihood estimator of τ, say ˆ τ ∗ , is the mean absolute error, that is, the mean of the absolute residuals. This estimator always de-creases when new independent variables are added to the model. Our objective is to develop asymptotic properties of ˆ τ ∗ under several errors distributions analytically. We also performed a simulation study to compare the distributions of both estimators in small samples with the objective to establish conditions in which ˆ τ ∗ is a good alternative to ˆ τ for such situations.