Fast highly efficient and robust one-step \(M\)-estimators of scale based on \(Q_n\). (English) Zbl 1506.62169

Summary: A parametric family of \(M\)-estimators of scale based on the Rousseeuw-Croux \(Q_n\)-estimator is proposed; estimator’s bias and efficiency are studied. A low-complexity one-step \(M\)-estimator is obtained allowing a considerably faster computation with greater than 80% efficiency and the highest possible 50% breakdown point. Analytical and Monte Carlo modeling results confirm the effectiveness of the proposed approach.


62-08 Computational methods for problems pertaining to statistics
62F35 Robustness and adaptive procedures (parametric inference)


robcor; Find
Full Text: DOI


[1] Bickel, P. J.; Lehmann, E. L., Descriptive statistics for nonparametric models. III. dispersion, Ann. Statist., 4, 1139-1158, (1976) · Zbl 0351.62031
[2] Croux, C.; Rousseeuw, P., Time-efficient algorithms for two highly robust estimators of scale, Comput. Statist., 1, 411-428, (1992)
[3] Hampel, F. R.; Ronchetti, E. M.; Rousseeuw, P. J.; Stahel, W. A., Robust statistics: the approach based on influence functions, (1986), John Wiley & Sons, Inc. New York · Zbl 0593.62027
[4] Hoare, C. A.R., Algorithm 65: find, Commun. ACM, 4, 321-322, (1961)
[5] Huber, P. J., Robust statistics, (1981), John Wiley & Sons, Inc. New York · Zbl 0536.62025
[6] Rainville, E., 1996. A comparison between several one-step \(M\)-estimators of location and dispersion in the presence of a nuisance parameter, Master’s Thesis. The University of British Columbia.
[7] Rousseeuw, P.; Croux, C., Alternatives to the Median absolute deviation, J. Amer. Statist. Assoc., 88, 1273-1283, (1993) · Zbl 0792.62025
[8] Rousseeuw, P.; Croux, C., The bias of \(k\)-step \(M\)-estimators, Statist. Probab. Lett., 20, 411-420, (1994) · Zbl 0801.62036
[9] Shevlyakov, G., Andrea, K., Choudur, L., Smirnov, P., Ulanov, A., Vassilieva, N., 2013a. Robust versions of the Tukey boxplot with their application to detection of outliers. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2013, Vancouver, BC, Canada, pp. 6506-6510.
[10] Shevlyakov, G.; Lyubomishchenko, N.; Smirnov, P., Some remarks on robust estimation of power spectra, (Proceedings of the 10th International Conference on Computer Data Analysis and Modeling, (2013), Publishing center of BSU Minsk, Belarus), 97-104
[11] Shevlyakov, G. L.; Smirnov, P. O., Robust estimation of the correlation coefficient: an attempt of survey, Aust. J. Stat., 40, 147-156, (2011)
[12] Smirnov, P.O., 2014. Robust correlations. R package version 0.1-6. http://CRAN.R-project.org/package=robcor.
[13] Smirnov, P., Shevlyakov, G., 2010. On approximation of the \(\operatorname{Q}_n\)-estimate of scale by fast \(M\)-estimates. In: Book of Abstracts: International Conference on Robust Statistics, ICORS 2010, Prague, Czech Republic, pp. 94-95.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.