Abstract
The Huber criterion for data fitting is a combination of thel 1 and thel 2 criteria which is robust in the sense that the influence of “wild” data points can be reduced. We present a trust region and a Marquardt algorithm for Huber estimation in the case where the functions used in the fit are non-linear. It is demonstrated that the algorithms converge under the usual conditions.
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References
J. E. Dennis,Non-linear Least Squares and Equations, in:State of the Art in Numerical Analysis, (ed. D. A. H. Jacobs), Academic Press, 269–312, 1977.
R. Dutter and P. J. Huber,Numerical methods for the nonlinear robust regression problem, J. Stat. Comp. Sim. (1981).
H. Ekblom,Programs for the Huber estimator in linear models, Dept. of Math. 1985-5, Luleå University, Sweden (1985).
H. Ekblom,A new algorithm for the Huber estimator in linear models, BIT 28 (1988), 123–132.
Li Gao and K. Madsen,Robust Non-linear Parameter Estimation, in: in:Numerical Analysis 1987, D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Mathematics Series 170, Longman, UK, 176–191, 1988.
F. R. Hampel, E. Ronchetti, P. Rousseeuw, W. Stahel,Robust Statistics: The Infinitestimal Approach, John Wiley, New York, 1986.
P. Huber,Robust Statistics, John Wiley, New York, 1981.
K. Levenberg,A method for the solution of certain problems in least squares, Quart. Appl. Math. 2 (1944).
K. Madsen,Minimization of Non-linear Approximation Functions, Dr. techn. thesis, Institute for Numerical Analysis, Technical University of Denmark, DK2800 Lyngby, (1985).
A. Marazzi,On the Numerical Solutions of Bounded Influence Regression Problems, in F. de Antonio, N. Lauro and A. Rizzi (eds.),COMPST AT 86:Proceedings in Computational Statistics, Physica Verlag, Vienna, 1986.
D. Marquardt,An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. (1963).
J. Moré,Recent Developments in Algorithms and Software for Trust Region Methods, inMathematical Programming, the state of the art, (Bonn 1982), Springer Verlag, 1983.
G. Nagel and H. Wolff,Ein Verfahren zur Minimierung einer Quadratsumme nichtlinearer Funktionen, Biometrische Zeitschrift 16 (1974).
M. R. Osborne,Some aspects of non-linear least squares calculations, in:Numerical Methods for Non-linear Optimization, (Ed. F. Lootsma). Academic Press, New York and London, 1972.
D. F. Shanno and K. H. Phua,Research on Algorithm 500:Minimization of unconstrained multivariate functions, ACM Trans. Math. Software 6 (1980), 618–622.
D. F. Shanno and D. M. Rocke,Numerical methods for robust regression: Linear models, SIAM J. Scient. Stat. Comp. 7 (1986), 86–97.
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Ekblom, H., Madsen, K. Algorithms for non-linear huber estimation. BIT 29, 60–76 (1989). https://doi.org/10.1007/BF01932706
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DOI: https://doi.org/10.1007/BF01932706