Abstract
The aim of the paper is to give a coherent account of the robustness approach based on shrinking neighborhoods in the case of i.i.d. observations, and add some theoretical complements. An important aspect of the approach is that it does not require any particular model structure but covers arbitrary parametric models if only smoothly parametrized. In the meantime, equal generality has been achieved by object-oriented implementation of the optimally robust estimators. Exponential families constitute the main examples in this article. Not pretending a complete data analysis, we evaluate the robust estimates on real datasets from literature by means of our R packages ROptEst and RobLox.
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References
Analytical Methods Committee: (1989) Robust statistics—how not to reject outliers. Analyst 114: 1693–1702
Andrews DF, Bickel PJ, Hampel FR, Huber PJ, Rogers WH, Tukey JW (1972) Robust estimates of location. Survey and advances. Princeton University Press, Princeton
Bauer H (1990) Maß- und integrationstheorie. (Measure and integration theory). Walter de Gruyter, Berlin
Bickel PJ (1981) Quelques aspects de la statistique robuste. Ecole d’ete de probabilites de Saint-Flour IX-1979 876: 2–72
Bickel PJ (1984) Robust regression based on infinitesimal neighbourhoods. Ann Stat 12: 1349–1368
Bickel PJ, Klaassen CAJ, Ritov Y, Wellner JA (1998) Efficient and adaptive estimation for semiparametric models. Springer, New York
Chambers JM (2008) Software for data analysis. Programming with R. Springer, New York
Chambers JM (1998) Programming with data: a guide to the S language. Springer, New York
Donoho DL, Liu RC (1988) Pathologies of some minimum distance estimators. Ann Stat 16(2): 587–608
Fernholz LT (1983) Von Mises calculus for statistical functionals. Lecture notes in statistics #19. Springer, New York
Fraiman R, Yohai VJ, Zamar RH (2001) Optimal robust M-estimates of location. Ann Stat 29(1): 194–223
Hájek J (1972) Local asymptotic minimax and admissibility in estimation. In: Proceedings of 6th Berkeley symposium mathematics statistics probability, vol 1. University of California 1970, pp 175–194
Hampel FR (1968) Contributions to the theory of robust estimation. Dissertation, University of California, Berkely, CA
Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics. The approach based on influence functions. Wiley, New York
Huber PJ (1997) Robust statistical procedures, 2 edn. In: CBMS-NSF regional conference series in applied mathematics. 68. SIAM, Philadelphia, PA
Huber PJ (1981) Robust statistics. Wiley, New York
Huber-Carol C (1970) Étude asymptotique de tests robustes. Thèse de Doctorat, ETH Zürich
Hubert M, Vandervieren E (2006) An adjusted boxplot for skewed distributions. Technical report TR-06-11, KU Leuven, Section of Statistics, Leuven, URL http://wis.kuleuven.be/stat/robust/Papers/TR0611.pdf
Kohl M (2008) RobLox: optimally robust influence curves for location and scale. R package version 0.6.1, URL http://robast.r-forge.r-project.org
Kohl M (2005) Numerical contributions to the asymptotic theory of robustness. Dissertation, University of Bayreuth, Bayreuth
Kohl M, Ruckdeschel P (2008a) RandVar: implementation of random variables. R package version 0.6.6, URL http://robast.r-forge.r-project.org
Kohl M, Ruckdeschel P (2008b) RobAStBase: Robust asymptotic statistics. R package version 0.1.5, URL http://robast.r-forge.r-project.org
Kohl M, Ruckdeschel P (2008c) ROptEst: optimally robust estimation. R package version 0.6.3, URL http://robast.r-forge.r-project.org
Le Cam L (1969) Théorie asymptotique de la décision statistique. Les Presses de l’Université de Montréal, Montreal, Canada
Marazzi A (1993) Algorithms, routines, and S functions for robust statistics. The FORTRAN library ROBETH with an interface to S-PLUS. With the collaboration of Johann Joss and Alex Randriamiharisoa. Brooks/Cole Statistics/Probability Series, Wadsworth, URL http://www.iumsp.ch/Unites/us/Alfio/msp_programmes.htm
Marazzi A, Paccaud F, Ruffieux C, Beguin C (1998) Fitting the distributions of length of stay by parametric models. Med Care 36: 915–927
Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics: theory and methods. Wiley, New York
Meyer PA (1966) Probabilités et potential. Hermann (Editions Scientifiques), Paris
Pfanzagl J (1994) Parametric statistical theory. De Gruyter Textbook, Berlin
Pfanzagl J (1990) Estimation in semiparametric models. Some recent developments. In: Lecture notes in statistics, 63, Springer, New York
R Development Core Team (2009) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0, URL http://www.R-project.org
Reeds JA (1976) On the definition of von Mises functionals. Ph.D. Thesis, Harvard University, Cambridge
Rieder H (2003) Robust estimation for time series models based on infinitesimal neighborhoods. Talk presented at EPF Lausanne. Slides available under http://www.stoch.uni-bayreuth.de/de/pdfFiles/zzk12Jun03.pdf
Rieder H (1994) Robust asymptotic statistics. Springer, New York
Rieder H (1980) Estimates derived from robust tests. Ann Stat 8: 106–115
Rieder H (1978) A robust asymptotic testing model. Ann Stat 6: 1080–1094
Rieder H, Kohl M, Ruckdeschel P (2008) The cost of not knowing the radius. Stat Meth Appl 17: 13–40
Rieder H, Ruckdeschel P (2001) Short proofs on L r —differentiability. Stat Decis 19: 419–425
Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New York
Ruckdeschel P (2009b) Uniform higher order asymptotics for risks on neighborhoods. In preparation. A preliminary version is available on request
Ruckdeschel P (2009a) Uniform integrability on neighborhoods. In preparation. A preliminary version is available on request
Ruckdeschel P (2006) A Motivation for \({1/\sqrt{n}}\)-Shrinking-Neighborhoods. Metrika 63(3): 295–307
Ruckdeschel P, Kohl M, Stabla T, Camphausen F (2008) S4 classes for distributions—a manual for packages distr, distrSim, distrTEst, distrEx, distrMod, and distrTeach. Technical report, Fraunhofer ITWM, Kaiserslautern, Germany
Ruckdeschel P, Kohl M, Stabla T, Camphausen F (2006) S4 classes for distributions. R News 6(2): 2–6
Ruckdeschel P, Rieder H (2004) Optimal influence curves for general loss functions. Stat Decis 22: 201–223
Rutherford E, Geiger H (1910) The probability variations in the distribution of alpha particles. Philos Mag 20: 698–704
Shevlyakov G, Morgenthaler S, Shurygin A (2008) Redescending M-estimators. J Stat Plan Inference 138(10): 2906–2917
Todorov V, Ruckstuhl A, Salibian-Barrera M, Verbeke T, Maechler M (2009) Robustbase: basic Robust statistics. Original code by many authors, notably Rousseeuw P, Croux C, see file ‘Copyrights’, R package version 0.5-0-1, URL http://CRAN.R-project.org/package=robustbase
Todorov V, Filzmoser P (2009) An object-oriented framework for robust multivariate analysis. J Stat Softw 32(3):1–47, URL http://www.jstatsoft.org/v32/i03/
van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge
Venables WN, Ripley BD (2002) Modern applied statistics with S, 4 edn. Springer, New York
Wang J, Zamar R, Marazzi A, Yohai V, Salibian-Barrera M, Maronna R, Zivot E, Rocke D, Martin D, Maechler M, Konis K (2009) Robust: insightful Robust library. R package version 0.3–9, URL http://CRAN.R-project.org/package=robust
Witting H (1985) Mathematische statistik I: parametrische verfahren bei festem stichprobenumfang. B.G. Teubner, Stuttgart
Yohai VJ (1987) High breakdown-point and high efficiency robust estimates for regression. Ann Stat 15(2): 642–656
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Kohl, M., Ruckdeschel, P. & Rieder, H. Infinitesimally Robust estimation in general smoothly parametrized models. Stat Methods Appl 19, 333–354 (2010). https://doi.org/10.1007/s10260-010-0133-0
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DOI: https://doi.org/10.1007/s10260-010-0133-0
Keywords
- Exponential family
- Influence curves
- Asymptotically linear estimators
- Shrinking contamination, total variation, and Hellinger neighborhoods
- One-step construction
- Minmax MSE