Discussiones Mathematicae Probability and Statistics 24(1) (2004) 59-75
Christine H. Müller
Carl von Ossietzky University Oldenburg
Institute for Mathematics
Postfach 2503, D-26111 Oldenburg, Germany
Keywords: redescending M-estimator, regression, breakdown point, optimality, cluster analysis, image analysis, kernel estimator.
2000 Mathematics Subject Classification: 62J05, 62G35, 62H30, 62G07, 62M40, 62P99.
[1] | D.F. Andrews, P.J. Bickel, F.R. Hampel, P.J. Huber, W.H. Rogers and J.W. Tukey, Robust Estimates of Location. Survey and Advances, Princeton University Press, Princeton 1972. |
[2] | O. Arslan, A simple test to identify good solutions of redescending M estimating equations for regression, in: Developments in Robust Statistics, Proceedings of ICORS 2001, R. Dutter, U. Gather, P.J. Rousseeuw and P. Filzmoser, (Eds.), (2003), 50-61. |
[3] | T. Bednarski and Ch.H. Müller, Optimal bounded influence regression and scale M-estimators, Statistics 35 (2001), 349-369. |
[4] | P.J. Bickel, Quelque aspects de la statistique robuste, In École d'Été de Probabilités de St. Flour. Springer Lecture Notes in Math. 876 (1981), 1-72. |
[5] | P.J. Bickel, Robust regression based on infinitesimal neighbourhoods, Ann. Statist. 12 (1984), 1349-1368. |
[6] | H. Chen and P. Meer, Robust computer vision through kernel density estimation, in: ECCV 2002, LNCS 2350, A. Heyden et al. (Eds.), Springer, Berlin (2002), 236-250. |
[7] | H. Chen, P. Meer and D.E. Tyler, Robust regression for data with multiple structures, in: 2001 IEEE Conference on Computer Vision and Pattern Recognition, vol. I, Kauai, HI, (2001), 1069-1075. |
[8] | C.K. Chu, I.K. Glad, F. Godtliebsen and J.S. Marron, Edge-preserving smoothers for image processing, J. Amer. Statist. Assoc. 93, (1998), 526-541. |
[9] | B.R. Clarke, Uniqueness and Frechét differentiability of functional solutions to maximum likelihood type equations, Ann. Statist. 4 (1983), 1196-1205. |
[10] | B.R. Clarke, Asymptotic theory for description of regions in which Newton-Raphson iterations converge to location M-estimators, J. Statist. Plann. Inference 15 (1986), 71-85. |
[11] | J.R. Collins, Robust estimation of a location parameter in the presence of asymmetry, Ann. Statist. 4 (1976), 68-85. |
[12] | J.B. Copas, On the unimodality of the likelihood for the Cauchy distribution, Biometrika 62 (1975), 701-704. |
[13] | D.L. Donoho. and J.P. Huber, The notion of breakdown point, in: P.J. Bickel, K.A. Doksum and J.L. Hodges, Jr., Eds., A Festschrift for Erich L. Lehmann, Wadsworth, Belmont, CA, (1983), 157-184. |
[14] | S.P. Ellis and S. Morgenthaler, Leverage and breakdown in L_{1} regression, J. Amer. Statist. Assoc. 87 (1992), 143-148. |
[15] | D.A. Freedman and P. Diaconis, On inconsistent M-estimators, Ann. Statist. 10 (1982), 454-461. |
[16] | G. Gabrielsen, On the unimodality of the likelihood for the Cauchy distribution: Some comments, Biometrika 69 (1982), 677-678. |
[17] | F.R. Hampel, Optimally bounding the gross-error-sensitivity and the influence of position in factor space, Proceedings of the ASA Statistical Computing Section, ASA, Washington, D.C., (1978), 59-64. |
[18] | F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics - The Approach Based on Influence Functions, John Wiley, New York 1986. |
[19] | W. Härdle and T. Gasser, Robust nonparametric function fitting, J. R. Statist. Soc. B 46 (1984), 42-51. |
[20] | X. He, J. Jurecková, R. Koenker and S. Portnoy, Tail behavior of regression estimators and their breakdown points, Econometrica 58 (1990), 1195-1214. |
[21] | X. He, D.G. Simpson and G. Wang, Breakdown points of t-type regression estimators, Biometrika 87 (2000), 675-687. |
[22] | C. Hennig, Regression fixed point clusters: motivation, consistency and simulations, Preprint 2000-02, Fachbereich Mathematik, Universität Hamburg 2000. |
[23] | C. Hennig, Clusters, outliers, and regression: Fixed point clusters, Journal of Multivariate Analysis. 86/1 (2003), 183-212. |
[24] | M. Hillebrand, On robust corner-preserving smoothing in image processing, Ph.D. thesis at the Carl von Ossietzky University Oldenburg, Germany 2002. |
[25] | M. Hillebrand and Ch.H. Müller, On consistency of redescending M-kernel smoothers, Submitted 2002. |
[26] | P.J. Huber, Minimax aspects of bounded-influence regression (with discussion), J. Amer. Statist. Assoc. 78 (1983), 66-80. |
[27] | J. Jurecková and P.K. Sen, Robust Statistical Procedures. Asymptotics and Interrelations, Wiley, New York 1996. |
[28] | W.S. Krasker, Estimation in linear regression models with disparate data points, Econometrica 48 (1980), 1333-1346. |
[29] | V. Kurotschka and Ch.H. Müller, Optimum robust estimation of linear aspects in conditionally contaminated linear models, Ann. Statist. 20 (1992), 331-350. |
[30] | K.L. Lange, R.J.A. Little and J.M.G. Taylor, Robust statistical modeling using the t distribution J. Amer. Statist. Assoc. 84 (1989), 881-896. |
[31] | R.A. Maronna, O.H. Bustos and V.J. Yohai, Bias- and efficiency-robustness of general M-estimators for regression with random carriers, in: Smoothing Techniques for Curve Estimation (T. Gasser and M. Rosenblatt, eds.) Springer, Berlin, Lecture Notes in Mathematics 757 (1979), 91-116. |
[32] | I. Mizera, On consistent M-estimators: tuning constants, unimodality and breakdown, Kybernetika 30 (1994), 289-300. |
[33] | I. Mizera, Weak continuity of redescending M-estimators of location with an unbounded objective function, Tatra Mountains Math. Publ. 7 (1996),343-347. |
[34] | I. Mizera and Ch.H. Müller, Breakdown points and variation exponents of robust M-estimators in linear models, Ann. Statist. 27 (1999), 1164-1177. |
[35] | I. Mizera and Ch.H. Müller, Breakdown points of Cauchy regression-scaleestimators, Stat. & Prob. Letters 57 (2002), 79-89. |
[36] | S. Morgenthaler, Fitting redescending M-estimators in regression, in: Robust Regression, H.D. Lawrence and S. Arthur, (Eds.), Dekker, New York (1990), 105-128. |
[37] | Ch.H. Müller, Optimal designs for robust estimation in conditionally contaminated linear models, J. Statist. Plann. Inference. 38 (1994), 125-140. |
[38] | Ch.H. Müller, Breakdown points for designed experiments, J. Statist. Plann. Inference, 45 (1995), 413-427. |
[39] | Ch.H. Müller, Optimal breakdown point maximizing designs, Tatra Mountains Math. Publ. 7, (1996), 79-85. |
[40] | Ch.H. Müller, Robust Planning and Analysis of Experiments, Springer, New York, Lecture Notes in Statistics 124 (1997). |
[41] | Ch.H. Müller, On the use of high breakdown point estimators in the image analysis, Tatra Mountains Math. Publ. 17 (1999), 283-293. |
[42] | Ch.H. Müller, Robust estimators for estimating discontinuous functions, Metrika 55 (2002a), 99-109. |
[43] | Ch.H. Müller, Comparison of high-breakdown-point estimators for image denoising, Allg. Stat. Archiv 86 (2002b), 307-321. |
[44] | Ch.H. Müller and T. Garlipp, Simple consistent cluster methods based on redescending M-estimators with an application to edge identification in images, To appear in Journal of Multivariate Analysis, (2002). |
[45] | P. Qiu, Nonparametric estimation of jump surface, The Indian Journal of Statistics 59, Series A, (1997) 268-294. |
[46] | H. Rieder, Robust regression estimators and their least favorable contamination curves, Stat. Decis. 5 (1987), 307-336. |
[47] | H. Rieder, Robust Asymptotic Statistics, Springer, New York 1994. |
[48] | B.W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, London 1986. |
[49] | S. Smith and J. Brady, SUSAN - a new approach to low level image processing, International Journal of Computer Vision 23 (1997), 45-78. |
[50] | R.H. Zamar, Robust estimation in the errors-in-variables model, Biometrika 76 (1989), 149-160. |
Received 4 October 2003