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.
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References
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.
Bickel, P. (1965). On some robust estimates of location,Ann. Math. Statist.,36, 847–858.
Bickel, P. and Lehmann, E. L. (1975). Descriptive statistics for nonparametric models. II,Ann. Statist.,3, 1045–1069.
Efron, B. and Olshen, R. A. (1978). How broad is the class of normal scale mixtures?,Ann. Statist.,6, 1159–1164.
Gross, A. M. (1973). A Monte Carlo swindle for estimators of location,Applied Statistics,22, 347–353.
Gross, A. M. (1977). Confidence intervals for bisquare regression estimates,J. Amer. Statist. Assoc.,72, 341–354.
Hall, P. (1981). Large sample properties of Jaeckel's adaptive trimmed mean,Ann. Inst. Statist. Math.,33, 449–462.
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.
Hill, M. A. and Dixon, W. J. (1982). Robustness in real life: a study of clinical laboratory data,Biometricsi,38, 377–396.
Huber, P. J. (1972). Robust statistics: a review,Ann. Math. Statist.,43, 1041–1067.
Jaeckel, L. A. (1971). Some flexible estimates of location,Ann. Math. Statist.,42, 1540–1552.
Jurečková, J. (1986). Asymptotic representations ofL-estimators and their relations toM-estimators,Sequential Anal.,5, 317–338.
Relles, D. A. (1970). Variance reduction techniques for Monte Carlo sampling from Student distributions,Technometrics,12, 499–515.
Rocke, D. M., Downs, G. W. and Rocke, A. J. (1982). Are robust estimators really necessary?,Technometrics,24, 95–101.
Siddiqui, M. M. (1960). Distribution of quantiles in samples from a bivariate population,J. Res. Nat. Bur. Standards,64B, 145–150.
Spjotvoll, E. and Aastreit, A. H. (1980). Comparison of robust estimators on data from field experiments,Scand. J. Statist.,7, 1–13.
Stigler, S. M. (1973). The asymptotic distribution of the trimmed mean,Ann. Statist.,1, 472–477.
Stigler, S. M. (1977). Do robust estimators work with real data?,Ann. Statist.,5, 1055–1077.
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.
Welsh, A. H. (1988). Asymptotically efficient estimation of the sparsity function at a point,Statist. Probab. Lett.,6, 427–432.
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Jurečková, J., Koenker, R. & Welsh, A.H. Adaptive choice of trimming proportions. Ann Inst Stat Math 46, 737–755 (1994). https://doi.org/10.1007/BF00773479
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DOI: https://doi.org/10.1007/BF00773479