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Adaptive choice of trimming proportions

  • Jana Jurečková
  • Roger Koenker
  • A. H. Welsh
Estimation

Abstract

We consider Jaeckel's (1971,Ann. Math. Statist.,42, 1540–1552) proposal for choosing the trimming proportion of the trimmed mean in the more general context of choosing a trimming proportion for a trimmedL-estimator of location. We obtain higher order expansions which enable us to evaluate the effect of the estimated trimming proportion on the adaptive estimator. We find thatL-estimators with smooth weight functions are to be preferred to those with discontinuous weight functions (such as the trimmed mean) because the effect of the estimated trimming proportion on the estimator is of ordern−1 rather thann−3/4. In particular, we find that valid inferences can be based on a particular “smooth” trimmed mean with its asymptotic standard error and the Studentt distribution with degrees of freedom given by the Tukey and McLaughlin (1963,Sankhyā Ser. A,25, 331–352) proposal.

Key words and phrases

Trimmed mean adaptive estimation L-statistics 

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References

  1. Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H. and Tukey, J. W. (1972)Robust Estimates of Location: Survey and Advances, Princeton University Press, New Jersey.Google Scholar
  2. Bickel, P. (1965). On some robust estimates of location,Ann. Math. Statist.,36, 847–858.Google Scholar
  3. Bickel, P. and Lehmann, E. L. (1975). Descriptive statistics for nonparametric models. II,Ann. Statist.,3, 1045–1069.Google Scholar
  4. Efron, B. and Olshen, R. A. (1978). How broad is the class of normal scale mixtures?,Ann. Statist.,6, 1159–1164.Google Scholar
  5. Gross, A. M. (1973). A Monte Carlo swindle for estimators of location,Applied Statistics,22, 347–353.Google Scholar
  6. Gross, A. M. (1977). Confidence intervals for bisquare regression estimates,J. Amer. Statist. Assoc.,72, 341–354.Google Scholar
  7. Hall, P. (1981). Large sample properties of Jaeckel's adaptive trimmed mean,Ann. Inst. Statist. Math.,33, 449–462.Google Scholar
  8. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986).Robust Statistics: The Approach Based in Influence Functions, Wiley, New York.Google Scholar
  9. Hill, M. A. and Dixon, W. J. (1982). Robustness in real life: a study of clinical laboratory data,Biometricsi,38, 377–396.Google Scholar
  10. Huber, P. J. (1972). Robust statistics: a review,Ann. Math. Statist.,43, 1041–1067.Google Scholar
  11. Jaeckel, L. A. (1971). Some flexible estimates of location,Ann. Math. Statist.,42, 1540–1552.Google Scholar
  12. Jurečková, J. (1986). Asymptotic representations ofL-estimators and their relations toM-estimators,Sequential Anal.,5, 317–338.Google Scholar
  13. Relles, D. A. (1970). Variance reduction techniques for Monte Carlo sampling from Student distributions,Technometrics,12, 499–515.Google Scholar
  14. Rocke, D. M., Downs, G. W. and Rocke, A. J. (1982). Are robust estimators really necessary?,Technometrics,24, 95–101.Google Scholar
  15. Siddiqui, M. M. (1960). Distribution of quantiles in samples from a bivariate population,J. Res. Nat. Bur. Standards,64B, 145–150.Google Scholar
  16. Spjotvoll, E. and Aastreit, A. H. (1980). Comparison of robust estimators on data from field experiments,Scand. J. Statist.,7, 1–13.Google Scholar
  17. Stigler, S. M. (1973). The asymptotic distribution of the trimmed mean,Ann. Statist.,1, 472–477.Google Scholar
  18. Stigler, S. M. (1977). Do robust estimators work with real data?,Ann. Statist.,5, 1055–1077.Google Scholar
  19. Tukey, J. W. and McLaughlin, D. H. (1963). Less vulnerable confidence and significance procedures for location based on a single sample (Trimming/Winsorisation 1),Sankhyā Ser. A,25, 331–352.Google Scholar
  20. Welsh, A. H. (1988). Asymptotically efficient estimation of the sparsity function at a point,Statist. Probab. Lett.,6, 427–432.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1994

Authors and Affiliations

  • Jana Jurečková
    • 1
  • Roger Koenker
    • 2
  • A. H. Welsh
    • 3
  1. 1.Department of Probability and StatisticsCharles UniversityPragueCzech Republic
  2. 2.Department of EconomicsUniversity of IllinoisChampaignUSA
  3. 3.Department of StatisticsThe Australian National UniversityCanberraAustralia

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