Deviations Regressions by Iterative Least Squares Regressions and by Linear Programming

This note considers some aspects of the computational problem of by minimizing the sum of absolute deviations Yt = t 1 x· t l3· l= l l fitting the regression model (t = 1, 2, ... , n) r!: 1 x· t l3·1 l= l l The iterative method recently proposed by Schlossmacher (1973) is shown to have undesirable features under certain conditions. The linear programming approach using the simplex method as suggested by Fisher (1961) requires that the simplex tableau contain a submatrix of order n by 2k + n which restricts the method to relatively small problems. We show that an n by k matrix and a 2k + n vector are sufficient to represent this submatrix. This representation improves the efficiency of the simplex method and allows its use in a large proportion of the problems which occur in applications.