Abstract.In this paper we discuss algorithms for Lp-methods, i.e. minimizers of the L~-norm of the residual vector. The statistical "goodness" of the different methods when applied to regression problems is compared in a Monte Carlo experiment. O. Introduction.The problem of robust regression, i.e. ignoring the influence of "outliers" or "wild points" in a discrete approximation model, is a field where the domains of numerical mathematics and mathematical statistics overlap to some extent. It seems as if the exchange of ideas across the border between these two disciplines has been very limited. This paper aims at presenting the main results of research in the two sciences and also at comparing some of the proposed solutions in a Monte Carlo experiment.

In this paper we are concerned with finding the L~-solution (i.e. minimizing the L~-norm of the residual vector) to a linear approximation problem or, equivalenty, to an overdetermined system of linear equations. An embedding method is described in which the damped Newton iteration is applied to a series of "perturbed problems" in order to guarantee convergence and also increase the convergence rate.

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Abstract.This paper considers algorithms for solving the linear robust regression problem by minimizing the Huber function. In the computational methods for this problem used so far, the scale estimate is adjusted separately. The new algorithm, based on Newton's method, treats both the scale and the location parameters as independent variables. The special form of the Hessian allows for an efficient updating scheme.Subject classification: AMS 6Z!05, 65U05.

Constrained M-estimators for regression were introduced by Mendes and Tyler in 1995 as an alternative class of robust regression estimators with high breakdown point and high asymptotic e‰ciency. To compute the CMestimate, the global minimum of an objective function with an inequality constraint has to be localized. To find the S-estimate for the same problem, we instead restrict ourselves to the boundary of the feasible region. The algorithm presented for computing CM-estimates can easily be modified to compute Sestimates as well. Testing is carried out with a comparison to the algorithm SURREAL by Ruppert.