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It is well known that the least absolute value (l(infinity)) and the least sum of absolute deviations (l(1)) algorithms produce estimators that are not necessarily unique. In this paper it is shown how the set of all solutions of the l(1) and l(infinity) regression problems for moderately large sample sizes can be obtained. In addition, if the multiplicity of solutions wants to be avoided, two new methods giving the same optimal l(1) and l(infinity) values, but supplying unique solutions, are proposed. The idea consists of using two steps: in the first step the optimal values of the l(1) and l(infinity) errors are calculated, and in the second step, in case of non-uniqueness of solutions, one of the multiple solutions is selected according to a different criterion. For the l(infinity) the procedure is used sequentially but removing, in each iteration, the data points with maximum absolute residual and adding the corresponding constraints for keeping these residuals, and this process is repeated until no change in the solution is obtained. In this way not only the maximum absolute residual values are minimized in the modified method, but also the maximum absolute residual values of the remaining points sequentially, until no further improvement is possible. In the l(1) case a least squares criterion is used but restricted to the, residual condition. Thus, in the modified l(1) method not only the, residual is minimized, but also the sum of squared residuals subject to the, residual. The methods are illustrated by their application to some well known examples and their performances are tested by some simulations, which show that the lack of uniqueness problem cannot be corrected for some experimental designs by increasing the sample size.
least squares; l(1)-norml(infinity)-norm; parameter estimation; set of all solutions; asymptotic properties