Asymptotically Stable Tests with Application to Robust Detection

  • Georgy Shevlyakov
Conference paper


To design highly robust and efficient tests, a new method based on the so-called variational optimization approach for robust estimation proposed by Shurygin (1994a, b) is developed. A new indicator of robustness of tests, the test stability, is introduced. The optimal decision rules maximizing test efficiency under the guaranteed level of test stability are written out. The proposed stable tests are based on redescending M-estimators as the corresponding test statistics. For hypothesis testing of location, one of those tests, namely a radical test, outperforms the conventional robust linear bounded Huber’s and redescending Hampel’s tests under heavy-tailed distributions although being slightly inferior to Huber’s test under the Gaussian and moderately contaminated Gaussian distributions.


Score Function Stable Estimation Asymptotic Variance Noise Distribution Monte Carlo Experiment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I would like to thank the referees for their important and helpful comments. Also I am so much grateful to Prof. Ayanendranath Basu for his patience and help, which allowed me to finalize this paper.


  1. Andrews DF, Bickel PJ, Hampel FR, Huber PJ, Rogers WH, Tukey JW (1972) Robust estimates of location. Princeton Univ. Press, PrincetonMATHGoogle Scholar
  2. Bentkus V, Bloznelis M, Götze F (1995) A Berry-Esseen bound for \(M\)-estimators. Preprint 95-068, Universität BielifeldGoogle Scholar
  3. Bohner M Guseinov GSH (2003) Improper integrals on time scales. Dyn Syst Appl 12:45–65Google Scholar
  4. Hampel FR (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69:383–393MathSciNetCrossRefMATHGoogle Scholar
  5. Hampel FR, Ronchetti E, Rousseeuw PJ, Stahel WA (1986) Robust statistics. The approach based on influence functions. Wiley, New YorkMATHGoogle Scholar
  6. Hossjer O, Mettiji M (1993) Robust multiple classification of known signals in additive noise - an asymptotic weak signal approach. IEEE Trans Inf Theory 39:594–608CrossRefMATHGoogle Scholar
  7. Huber PJ (1964) Robust estimation of a location parameter. Ann Math Statist 35:1–72MathSciNetCrossRefMATHGoogle Scholar
  8. Huber PJ (1981) Robust Stat. Wiley, New YorkCrossRefGoogle Scholar
  9. Kim K, Shevlyakov GL (2008) Why Gaussianity? IEEE Signal Process Mag 25:102–113CrossRefGoogle Scholar
  10. Michel R, Pfanzagle J (1971) The accuracy of the normal approximation for minimum contrast estimates. Z. Wahrsch Verw Geb 18:73–84MathSciNetCrossRefGoogle Scholar
  11. Noether GE (1967) Elements of nonparametric statistics. Wiley, New YorkMATHGoogle Scholar
  12. Rousseeuw PJ (1981) A new infinitesemal approach to robust estimation. Z Wahrsch Verw Geb 56:127–132MathSciNetCrossRefMATHGoogle Scholar
  13. Rousseeuw PJ, Croux C (1993) Alternatives to the median absolute deviation. J Amer Statist Assoc 88:1273–1283MathSciNetCrossRefMATHGoogle Scholar
  14. Shevlyakov GL, Vilchevski NO (2002) Robustness in data analysis: criteria and methods. VSP, UtrechtGoogle Scholar
  15. Shevlyakov GL, Morgenthaler S, Shurygin A (2008) Redescending \(M\)-estimators. J Stat Plann Infer 138:2906–2917MathSciNetCrossRefMATHGoogle Scholar
  16. Shevlyakov GL, Lee JW, Lee KM, Shin VI, Kim K (2010) Robust detection of a weak signal with redescending \(M\)-estimators: a comparative study. Int J Adapt Control Signal Proc 24:33–40MathSciNetMATHGoogle Scholar
  17. Shevlyakov GL, Shin VI, Lee S, Kim K (2014) Asymptotically stable detection of a weak signal. Int J Adapt Control Signal Proc 28:848–858CrossRefMATHGoogle Scholar
  18. Shurygin AM (1994a) New approach to optimization of stable estimation. In: Proceedings 1st US/Japan Conference on Frontiers of Statist. Modeling, Kluwer, Netherlands, pp 315–340Google Scholar
  19. Shurygin AM (1994b) Variational optimization of the estimator stability. Autom Remote Control 55:1611–1622MathSciNetMATHGoogle Scholar
  20. Shurygin AM (2009) Mathematical methods of forecasting. Text-book, Hot-line-Telecom, Moscow (in Russian)Google Scholar
  21. Tukey JW (1977) Exploratory data analysis. Addison-Wesley, ReadingMATHGoogle Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsPeter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia

Personalised recommendations