This work is concerned with the comparative analysis of a variety of robust estimates of location under the generalized Gaussian and the Student t- and the Tukey gross-error distributions in the univariate and multivariate cases. The chosen set of estimates comprises the sample mean, sample median, classical robust Maronna, Huber, and Hampel M-estimates, Meshalkin–Shurygin stable M-estimates with redescending score functions, and a low-complexity two-step estimate with the preliminary rejection of outliers by the Tukey boxplot rule followed by the use of the sample mean to the cleaned data — almost all of them are examined in the univariate and multivariate versions. The estimate performance is evaluated by efficiency, bias, and mean squared error. For univariate distributions with light and heavy tails, the best results are exhibited by the Huber, Hampel, and Meshalkin–Shurygin and two-step estimates of location. In the multivariate case, the Huber and two-step estimates perform best.
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References
T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley, New York (2003).
D. F. Andrews, P. J. Bickel, F.R. Hampel, P. J. Huber, W. H. Rogers, and J. W. Tukey, Robust Estimates of Location, Princeton Univ. Press, Princeton (1972).
H. Cramér, Mathematical Methods of Statistics, Princeton University Press, Princeton (1974).
F.R. Hampel, Contributions to the Theory of Robust Estimation, Ph.D. Thesis, University of California, Berkeley (1968).
F.R. Hampel, E. Ronchetti, P. J. Rousseeuw, and W.A. Stahel, Robust Statistics. The Approach Based on Influence Functions, Wiley, New York (1986).
P. J. Huber, “Robust estimation of a location parameter,” Ann. Math. Stat., 35, No. 1, 73–101 (1964).
P. J. Huber, Robust Statistics, Wiley, New York (1981).
P. J. Huber and E. Ronchetti, Robust Statistics, Wiley, New York (2009).
R.A. Maronna, “Robust M-estimators of multivariate location and scatter,” Ann. Stat., 4, No. 1, 51–67 (1976).
R. A. Maronna, R.D. Martin, and V.Y. Yohai, Robust Statistics. Theory and Methods, Wiley, New York (2006).
L.D. Meshalkin, “Some mathematical methods for the study of non-communicable diseases,” in: Proc. 6th Intern. Meeting of Uses of Epidemiol. in Planning Health Services, Vol. 1, Primosten, Yugoslavia (1971), pp. 250–256.
G. L. Shevlyakov, S. Morgenthaler, and A.M. Shurygin, “Redescending M-estimators,” J. Stat. Plan. Inference, 138, 2906–2916 (2008).
G. L. Shevlyakov and N. O. Vilchevski, Robustness in Data Analysis, De Gruyter, Boston (2011).
G. L. Shevlyakov, “Asymptotically stable tests with application to robust detection,” in: Recent Advances in Robust Statistics: Theory and Applications, C. Agostinelli, A. Basu, P. Filzmoser, and D. Mukherjee (eds.), Springer, India (2016), pp. 185–199.
A. M. Shurygin, “New approach to optimization of stable estimation,” in: Proc. 1 US/Japan Conf. on Frontiers of Statist. Modeling, H. Bozdogan (ed.), Kluwer Academic Publishers, The Netherlands (1994), pp. 315–340.
A.M. Shurygin, Applied Stochastics, Finansy i Statistika, Moscow (2000).
J. W. Tukey, Exploratory Data Analysis, Addison-Wesley, Reading (1977).
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Proceedings of the XXXIV International Seminar on Stability Problems for Stochastic Models, Debrecen, Hungary.
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Shevlyakov, G.L., Shagal, A. & Shin, V.I. A Comparative Study of Robust and Stable Estimates of Multivariate Location. J Math Sci 237, 831–845 (2019). https://doi.org/10.1007/s10958-019-04210-3
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DOI: https://doi.org/10.1007/s10958-019-04210-3