×

Trimmed and Winsorized means based on a scaled deviation. (English) Zbl 1149.62047

Summary: Trimmed (and Winsorized) means based on a scaled deviation are introduced and studied. The influence functions of the estimators are derived and their limiting distributions are established via asymptotic representations. As a main focus of the paper, the performance of the estimators with respect to various robustness and efficiency criteria is evaluated and compared with leading competitors including the ordinary Tukey trimmed (and Winsorized) means. Unlike the Tukey trimming which always trims a fixed fraction of sample points at each end of data, the trimming scheme here only trims points at one or both ends that have a scaled deviation beyond some threshold. The resulting trimmed (and Winsorized) means are much more robust than their predecessors.
Indeed they can share the best breakdown point robustness of the sample median for any common trimming thresholds. Furthermore, for appropriate trimming thresholds they are highly efficient at light-tailed symmetric models and more efficient than their predecessors at heavy-tailed or contaminated symmetric models. Detailed comparisons with leading competitors on various robustness and efficiency aspects reveal that the scaled deviation trimmed (Winsorized) means behave very well overall and consequently represent very favorable alternatives to the ordinary trimmed (Winsorized) means.

MSC:

62H12 Estimation in multivariate analysis
62G35 Nonparametric robustness
62E20 Asymptotic distribution theory in statistics
62G05 Nonparametric estimation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bickel, P. J., On some robust estimates of location, Ann. Math. Statist., 36, 847-858 (1965) · Zbl 0192.25802
[2] Donoho, D. L.; Huber, P. J., The notion of breakdown point, (Bickel, P. J.; Doksum, K. A.; Hodges, J. L., A Festschrift for Erich L. Lehmann (1983), Wadsworth: Wadsworth Belmont, CA), 157-184
[3] Hampel, F. R.; Ronchetti, E. Z.; Rousseeuw, P. J.; Stahel, W. A., Robust Statistics: The Approach Based on Influence Function (1986), Wiley: Wiley New York
[4] Hogg, R. V., Adaptive robust procedures: a partial review and some suggestions for future applications and theory, J. Amer. Statist. Assoc., 69, 909-923 (1974) · Zbl 0305.62030
[5] Jaeckel, L. A., Some flexible estimates of location, Ann. Math. Statist., 42, 1540-1552 (1971) · Zbl 0232.62008
[6] Jurečková, J.; Koenker, R.; Welsh, A. H., Adaptive choice of trimming proportions, Ann. Inst. Statist. Math., 46, 737-755 (1994) · Zbl 0822.62019
[7] Kim, S., The metrically trimmed mean as a robust estimator of location, Ann. Statist., 20, 1534-1547 (1992) · Zbl 0782.62039
[8] Pollard, D., Convergence of Stochastic Processes (1984), Springer: Springer New York · Zbl 0544.60045
[9] Serfling, R., Approximation Theorems of Mathematical Statistics (1980), Wiley: Wiley New York · Zbl 0538.62002
[10] Shorack, G. R., Random means, Ann. Statist., 2, 661-675 (1974) · Zbl 0286.62029
[11] Simpson, D. G., M-estimation for discrete data: asymptotic distribution theory and implications, Ann. Statist., 15, 657-669 (1987) · Zbl 0652.62011
[12] Stigler, S. M., Do robust estimators work with real data?, Ann. Statist., 5, 1055-1077 (1977) · Zbl 0374.62050
[13] Tukey, J. W., Some elementary problems of importance to small sample practice, Hum. Biol., 20, 205-214 (1948)
[14] van der Vaart, A. W.; Wellner, J. A., Weak Convergence and Empirical Processes With Applications to Statistics (1996), Springer: Springer New York · Zbl 0862.60002
[15] Zuo, Y., Multivariate trimmed means based on data depth, (Dodge, Y., Statistical Data Analysis based on the L1-Norm and Related Methods (2002), Birkhäuser: Birkhäuser Basel), 313-322 · Zbl 1145.62337
[16] Zuo, Y., Multi-dimensional trimming based on projection depth, Ann. Statist., 34, 2211-2251 (2006) · Zbl 1106.62057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.