×

A note on the uniform asymptotic normality of location M-estimates. (English) Zbl 1095.62027

Summary: In the robustness framework, the parametric model underlying the data is usually embedded in a neighborhood of other plausible distributions. Accordingly, the asymptotic properties of robust estimates should be uniform over the whole set of possible models. We study location M-estimates calculated with a previous generalized S-scale and show that, under some regularity conditions, they are uniformly asymptotically normal over contamination neighborhoods of known size.
There is a trade off between the size of the neighborhood and the breakdown point of the GS-scale, but it is possible to adjust the estimates so that they have 50% breakdown point whereas the uniform asymptotic normality is ensured over neighborhoods that contain up to 25% of contamination. Alternatively, both the breakdown point and the size of the neighborhood could be chosen to be 38%. These results represent an improvement over those obtained recently by M. Salibian-Barrera and R. H. Zamar [Ann. Stat. 32, No. 4, 1434–1447 (2004; Zbl 1047.62022)].

MSC:

62F12 Asymptotic properties of parametric estimators
62F35 Robustness and adaptive procedures (parametric inference)

Citations:

Zbl 1047.62022
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bednarski T., Müller C. (2001). Optimal bounded influence regression and scale M-estimators in the context of experimental design. Statistics 35:349–369 · Zbl 0995.62028
[2] Bednarski T., Zontek S. (1996). Robust estimation of parameters in a mixed unbalanced model. Ann Stat 24:1493–1510 · Zbl 0878.62024
[3] Berrendero JR. (2003). Uniform strong consistency of robust estimators. Stat Probab Lett 64:159–168 · Zbl 1113.62310
[4] Clarke BR. (1986). Nonsmooth analysis and Fréchet differentiability of M-functionals. Prob Theory Relat Fields 73:197–209 · Zbl 0595.60005
[5] Croux C., Rousseeuw PJ., Hössjer O. (1994). Generalized S-estimators. J Amer Stat Assoc 89:1271–1281 · Zbl 0812.62073
[6] Davies PL. (1998). On locally uniformly linearizable high breakdown location and scale functionals. Ann Stat 26:1103–1125 · Zbl 0929.62059
[7] Devroye L., Lugosi G. (2001). Combinatorial methods in density estimation. Springer, Berlin Heidelberg New York · Zbl 0964.62025
[8] Fraiman R., Yohai VJ., Zamar RH. (2001). Optimal robust M-estimates of location. Ann Stat 29:194–223 · Zbl 1029.62019
[9] Hampel F. (1971). A general qualitative definition of robustness. Ann Math Stat 42:1887–1896 · Zbl 0229.62041
[10] Hössjer O., Croux C., Rousseeuw PJ. (1994). Asymptotics of generalized S-estimators. J Multivar Anal 51:148–177 · Zbl 0815.62012
[11] Huber PJ. (1964). Robust estimation of a location parameter. Ann Math Stat 35:73–101 · Zbl 0136.39805
[12] Huber PJ. (1981). Robust statistics. Wiley, New York · Zbl 0536.62025
[13] Lee AJ. (1990). U-Statistics, Theory and Practice. Marcel Dekker, New York · Zbl 0771.62001
[14] Rousseeuw PJ., Yohai VJ. (1984). Robust regression by means of S-estimators. In: Franke J., Hardle W., Martin D (eds). Robust and nonlinear time series. Lecture notes in statistics, vol 26. Springer, Berlin Heidelberg New York, p 256–272
[15] Salibian-Barrera M., Zamar RH. (2004). Uniform asymptotics for robust location estimates when the scale is unknown. Ann Stat 32:1434–1447 · Zbl 1047.62022
[16] Tukey JW. (1960). A survey of sampling from contaminated distributions. In: Olkin I (eds). Contributions to probability and statistics. Standford University Press, Stanford California
[17] Zielinski R. (1998). Uniform strong consistency of sample quantiles. Stat Prob Lett 37:115–119 · Zbl 0891.62035
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.