Statistical Methods & Applications

, Volume 19, Issue 3, pp 333–354 | Cite as

Infinitesimally Robust estimation in general smoothly parametrized models

Article

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.

Keywords

Exponential family Influence curves Asymptotically linear estimators Shrinking contamination, total variation, and Hellinger neighborhoods One-step construction Minmax MSE 

Mathematics Subject Classification (2000)

62-07 62F12 62F35 62G05 62G35 

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References

  1. Analytical Methods Committee: (1989) Robust statistics—how not to reject outliers. Analyst 114: 1693–1702CrossRefGoogle Scholar
  2. Andrews DF, Bickel PJ, Hampel FR, Huber PJ, Rogers WH, Tukey JW (1972) Robust estimates of location. Survey and advances. Princeton University Press, PrincetonMATHGoogle Scholar
  3. Bauer H (1990) Maß- und integrationstheorie. (Measure and integration theory). Walter de Gruyter, BerlinMATHGoogle Scholar
  4. Bickel PJ (1981) Quelques aspects de la statistique robuste. Ecole d’ete de probabilites de Saint-Flour IX-1979 876: 2–72MathSciNetGoogle Scholar
  5. Bickel PJ (1984) Robust regression based on infinitesimal neighbourhoods. Ann Stat 12: 1349–1368MATHCrossRefMathSciNetGoogle Scholar
  6. Bickel PJ, Klaassen CAJ, Ritov Y, Wellner JA (1998) Efficient and adaptive estimation for semiparametric models. Springer, New YorkMATHGoogle Scholar
  7. Chambers JM (2008) Software for data analysis. Programming with R. Springer, New YorkMATHCrossRefGoogle Scholar
  8. Chambers JM (1998) Programming with data: a guide to the S language. Springer, New YorkMATHGoogle Scholar
  9. Donoho DL, Liu RC (1988) Pathologies of some minimum distance estimators. Ann Stat 16(2): 587–608MATHCrossRefMathSciNetGoogle Scholar
  10. Fernholz LT (1983) Von Mises calculus for statistical functionals. Lecture notes in statistics #19. Springer, New YorkGoogle Scholar
  11. Fraiman R, Yohai VJ, Zamar RH (2001) Optimal robust M-estimates of location. Ann Stat 29(1): 194–223MATHCrossRefMathSciNetGoogle Scholar
  12. 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 175–194Google Scholar
  13. Hampel FR (1968) Contributions to the theory of robust estimation. Dissertation, University of California, Berkely, CAGoogle Scholar
  14. Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics. The approach based on influence functions. Wiley, New YorkMATHGoogle Scholar
  15. Huber PJ (1997) Robust statistical procedures, 2 edn. In: CBMS-NSF regional conference series in applied mathematics. 68. SIAM, Philadelphia, PAGoogle Scholar
  16. Huber PJ (1981) Robust statistics. Wiley, New YorkMATHCrossRefGoogle Scholar
  17. Huber-Carol C (1970) Étude asymptotique de tests robustes. Thèse de Doctorat, ETH ZürichGoogle Scholar
  18. 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
  19. 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
  20. Kohl M (2005) Numerical contributions to the asymptotic theory of robustness. Dissertation, University of Bayreuth, BayreuthGoogle Scholar
  21. Kohl M, Ruckdeschel P (2008a) RandVar: implementation of random variables. R package version 0.6.6, URL http://robast.r-forge.r-project.org
  22. Kohl M, Ruckdeschel P (2008b) RobAStBase: Robust asymptotic statistics. R package version 0.1.5, URL http://robast.r-forge.r-project.org
  23. Kohl M, Ruckdeschel P (2008c) ROptEst: optimally robust estimation. R package version 0.6.3, URL http://robast.r-forge.r-project.org
  24. Le Cam L (1969) Théorie asymptotique de la décision statistique. Les Presses de l’Université de Montréal, Montreal, CanadaGoogle Scholar
  25. 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
  26. Marazzi A, Paccaud F, Ruffieux C, Beguin C (1998) Fitting the distributions of length of stay by parametric models. Med Care 36: 915–927CrossRefGoogle Scholar
  27. Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics: theory and methods. Wiley, New YorkMATHCrossRefGoogle Scholar
  28. Meyer PA (1966) Probabilités et potential. Hermann (Editions Scientifiques), ParisGoogle Scholar
  29. Pfanzagl J (1994) Parametric statistical theory. De Gruyter Textbook, BerlinMATHGoogle Scholar
  30. Pfanzagl J (1990) Estimation in semiparametric models. Some recent developments. In: Lecture notes in statistics, 63, Springer, New YorkGoogle Scholar
  31. 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
  32. Reeds JA (1976) On the definition of von Mises functionals. Ph.D. Thesis, Harvard University, CambridgeGoogle Scholar
  33. 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
  34. Rieder H (1994) Robust asymptotic statistics. Springer, New YorkMATHGoogle Scholar
  35. Rieder H (1980) Estimates derived from robust tests. Ann Stat 8: 106–115MATHCrossRefMathSciNetGoogle Scholar
  36. Rieder H (1978) A robust asymptotic testing model. Ann Stat 6: 1080–1094MATHCrossRefMathSciNetGoogle Scholar
  37. Rieder H, Kohl M, Ruckdeschel P (2008) The cost of not knowing the radius. Stat Meth Appl 17: 13–40MATHCrossRefMathSciNetGoogle Scholar
  38. Rieder H, Ruckdeschel P (2001) Short proofs on L r—differentiability. Stat Decis 19: 419–425MATHMathSciNetGoogle Scholar
  39. Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New YorkMATHCrossRefGoogle Scholar
  40. Ruckdeschel P (2009b) Uniform higher order asymptotics for risks on neighborhoods. In preparation. A preliminary version is available on requestGoogle Scholar
  41. Ruckdeschel P (2009a) Uniform integrability on neighborhoods. In preparation. A preliminary version is available on requestGoogle Scholar
  42. Ruckdeschel P (2006) A Motivation for \({1/\sqrt{n}}\)-Shrinking-Neighborhoods. Metrika 63(3): 295–307MATHCrossRefMathSciNetGoogle Scholar
  43. 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, GermanyGoogle Scholar
  44. Ruckdeschel P, Kohl M, Stabla T, Camphausen F (2006) S4 classes for distributions. R News 6(2): 2–6Google Scholar
  45. Ruckdeschel P, Rieder H (2004) Optimal influence curves for general loss functions. Stat Decis 22: 201–223MATHCrossRefMathSciNetGoogle Scholar
  46. Rutherford E, Geiger H (1910) The probability variations in the distribution of alpha particles. Philos Mag 20: 698–704Google Scholar
  47. Shevlyakov G, Morgenthaler S, Shurygin A (2008) Redescending M-estimators. J Stat Plan Inference 138(10): 2906–2917MATHCrossRefMathSciNetGoogle Scholar
  48. 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
  49. 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/ Google Scholar
  50. van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, CambridgeMATHGoogle Scholar
  51. Venables WN, Ripley BD (2002) Modern applied statistics with S, 4 edn. Springer, New YorkMATHGoogle Scholar
  52. 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
  53. Witting H (1985) Mathematische statistik I: parametrische verfahren bei festem stichprobenumfang. B.G. Teubner, StuttgartMATHGoogle Scholar
  54. Yohai VJ (1987) High breakdown-point and high efficiency robust estimates for regression. Ann Stat 15(2): 642–656MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Matthias Kohl
    • 1
  • Peter Ruckdeschel
    • 2
  • Helmut Rieder
    • 3
  1. 1.Jena University Hospital, Department of Anesthesiology and Intensive Care MedicineFriedrich-Schiller-University JenaJenaGermany
  2. 2.Fraunhofer-InstitutTechno-und WirtschaftsmathematikKaiserslauternGermany
  3. 3.Department of MathematicsUniversity of BayreuthBayreuthGermany

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