On l₁ regression coeficients

Abstract

We propose an alternative method to find the well-known L₁ estimators for general regression models. We find that the L₁ regression coefficients can be defined in terms of the convergent weighted medians of the slopes from each data point to the point that is assumed to be on a predicted regression line. Since the L₁ estimators are obtained by the convergent weighted medians, the L₁ estimation method could be regarded as finding the point on a predicted regression line. Moreover, we extend the above method to the multiple regression models. In these cases, the regression coefficients cannot be mentioned as the medians of some slopes, but can be defined in terms of me convergent weighted medians of some slopes. With these situations, we do the analogous argument to find the point on the Lx regression line so that the sum of absolute residuals would be minimized.

Keywords:

• convergent weighted median
• least absolute deviation
• L1 estimator