Abstract
M-estimators and M-kernel estimators with a redescending ψ-function are not in general consistent. This is often handled by means of coupling the estimator to a consistent one. Coupling the estimator to the (inconsistent) starting point improves the jump preserving properties. However, the consistency depends heavily on the shape of the density of the residuals. This paper shows inconsistency under convenient conditions as well as consistency – even at jump points – under somewhat stronger conditions.
Similar content being viewed by others
References
Andrews DF, Bickel PJ, Hampel FR, Huber PJ, Rogers WH, Tukey JW (1972) Robust estimates of location. Survey and advances. Princeton University Press, Princeton
Candès EJ, Donoho DL (1999) Ridgelets: a key to higher-dimensional intermittency. In: Wavelets, Silvermann B, Vassilicos J (eds) Oxford University Press, pp 111–127
Chu CK, Glad IK, Godtliebsen F, Marron JS (1998) Edge-preserving smoothers for image processing. J Am Stat Assoc 93:526–541
Clarke BR (1983) Uniqueness and Frechét differentiability of functional solutions to maximum likelihood type equations. Ann Stat 4:1196–1205
Clarke BR (1986) Asymptotic theory for description of regions in which Newton-Raphson iterations converge to location M-estimators. J Stat Plann Inference 15:71–85
Collins JR (1976) Robust estimation of a location parameter in the presence of asymmetry. Ann Stat 4:68–85
Copas JB (1975) On the unimodality of the likelihood for the Cauchy distribution. Biometrika 62:701–704
Donoho DL (1999) Wedgelets: nearly minimax estimation of edges. Ann Stat 27:859–897
Donoho DL, Johnstone IM, Kerkyacharian G, Picard D (1995) Wavelet shrinkage: asymptopia? J R Stat Soc B 57:301–369
Eubank RL (1988) Spline smoothing and nonparametric regression. Marcel Dekker, New York
Freedman DA, Diaconis P (1982) On inconsistent M-estimators. Ann Stat 10:454–461
Härdle W, Gasser T (1984) Robust nonparametric function fitting. J R Stat Soc B 46:42–51
Hillebrand M (2003) On robust corner-preserving smoothing in image processing. PhD thesis, University of Oldenburg, Germany (http://docserver.bis.uni-oldenburg.de/publikationen/dissertation/2003/hilonr03/hilonr03.html)
Huber P (1964) Robust estimator of a location parameter. Ann Math Stat 36:73–101
Huber P (1981) Robust statistics. Wiley, New York
Jurečková J, Sen PK (1996) Robust statistical procedures Asymptotics and interrelations. Wiley, New York
Kent JT, Tyler DE (1991) Redescending M-estimates of multivariate location and scatter. Ann Stat 19:2102–2119
Koch I (1996) On the asymptotic performance of median smoothers in image analysis and nonparametric regression. Ann Stat 24:1648–1666
Mizera I (1994) On consistent M-estimators: Tuning constants, unimodality and breakdown. Kybernetika 30:289–300
Mizera I (1996) Weak continuity of redescending M-estimators of location with an unbounded objective function. Tatra Mountains Math Publ 7:343–347
Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33:1065–1076
Polzehl J, Spokoiny VG (2000) Adaptive weights smoothing with applications to image restoration. J R Stat Soc B 62:335–354
Polzehl J, Spokoiny V (2003) Image denoising: pointwise adaptive approach. Ann Stat 31:30–57
Portnoy SL (1977) Robust estimation in dependent situations. Ann Stat 5:22–43
Serfling R (1980) Approximation theorems of mathematical statistics. Wiley, New York
Smith S, Brady J (1997) SUSAN – a new approach to low level image processing. Int J Comput Vis 23:45–78
Tsybakov AB (1986) Robust reconstruction of functions by the local-approximation method. Probl Inf Transm 22:133–146
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by the Friedrich Ebert Foundation and by grant Mu 1031/4-1/2 of the Deutsche Forschungsgemeinschaft
Rights and permissions
About this article
Cite this article
Hillebrand, M., Müller, C.H. On consistency of redescending M-kernel smoothers. Metrika 63, 71–90 (2006). https://doi.org/10.1007/s00184-005-0007-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-005-0007-x