Least absolute deviations analysis of variance

Abstract

Asymptotic methods for testing linear hypotheses based on the L₁-norm regression estimator have been recently discussed by a number of authors. The suggested tests are similar to those based on the least squares theory. Reduction in sums of squares is simply replaced by reduction in sums of absolute deviations. The appropriate distribution theory in such a case has been developed by a number of authors. The object of the present paper is to provide a rigorous proof of the asymptotic distribution of the reduction in sum of absolute deviations, the statistic used in testing a linear hypothesis. The asymptotic distribution is not directly useful as it involves a nuisance parameter. A new method of adjusting for the unknown parameter is suggested.