AStA Advances in Statistical Analysis

, Volume 99, Issue 1, pp 63–82 | Cite as

Efficiency of the pMST and RDELA location and scatter estimators

Original Paper

Abstract

The paper proposes two approaches to increase the efficiency of the pMST location and scatter estimator and of the RDELA location and scatter estimator. One approach is deduced from classical reweighting, commonly employed by established robust location and scatter estimators, and the other one is derived from Chebychev’s inequality. Simulation results suggest that both approaches are applicable to increase the efficiency of both estimators. Thereby the classical reweighting approach is outperformed by the approach based on Chebychev’s inequality. Using the latter, the performance of the pMST and RDELA estimator can be brought up to the level of the reweighted minimum covariance determinant and reweighted S-estimator.

Keywords

Efficiency Multivariate Location and scatter  Robust estimation 

References

  1. Becker, C., Liebscher, S., Kirschstein, T.: Multivariate outlier identification based on robust estimators of location and scatter. In: Robustness and Complex Data Structures—Festschrift in Honour of Ursula Gather, pp. 103–115 (2013)Google Scholar
  2. Bennett, M., Willemain, T.: Resistant estimation of multivariate location using minimum spanning trees. J. Stat. Comput. Simul. 69, 19–40 (2001)CrossRefMATHMathSciNetGoogle Scholar
  3. Croux, C., Haesbroeck, G.: Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. J. Multivar. Anal. 71, 161–190 (1999)CrossRefMATHMathSciNetGoogle Scholar
  4. Delaunay, B.: Sur la sphere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7, 793–800 (1934)Google Scholar
  5. Donoho, D.: Breakdown properties of multivariate location estimators. Ph.D. thesis, Department of Statistics, Harvard University (1982)Google Scholar
  6. Donoho, D., Huber, P.: The notion of breakdown point. A Festschrift for Erich Lehmann 157–184 (1983)Google Scholar
  7. Hubert, M., Debruyne, M.: Minimum covariance determinant. Wiley Interdiscip. Rev. Comput. Stat. 2(1), 36–43 (2010)CrossRefGoogle Scholar
  8. Joe, H.: Generating random correlation matrices based on partial correlations. J. Multivar. Anal. 97(10), 2177–2189 (2006)CrossRefMATHMathSciNetGoogle Scholar
  9. Jungnickel, D.: Graphs, networks and algorithms, 3rd edn. Springer (2008)Google Scholar
  10. Kaban, A.: Non-parametric detection of meaningless distances in high dimensional data. Stat. Comput. 22, 375–385 (2012)CrossRefMathSciNetGoogle Scholar
  11. Kirschstein, T., Liebscher, S., Becker, C.: Robust estimation of location and scatter by pruning the minimum spanning tree. J. Multivar. Anal. 120, 173–184 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. Liebscher, S., Kirschstein, T.: Restlos: robust estimation of location and scatter. R package version 0.1-2. (2013)Google Scholar
  13. Liebscher, S., Kirschstein, T., Becker, C.: Rdela—a delaunay-triangulation-based, location and covariance estimator with high breakdown point. Stat. Comput. 1–12 (2013). doi: 10.1007/s11222-012-9337-5
  14. Lopuhaä, H., Rousseeuw, P.: Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Stat. 19(1), 229–248 (1991)CrossRefMATHGoogle Scholar
  15. Penrose, M.: A strong law for the longest edge of the minimal spanning tree. Ann. Probab. 27(1), 246–260 (1999)CrossRefMATHMathSciNetGoogle Scholar
  16. Qiu, W., Joe, H.: Cluster Generation: random cluster generation (with specified degree of separation). R package version 1.3.1. (2013)Google Scholar
  17. R Core Team.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2013)Google Scholar
  18. Rousseeuw, P.: Multivariate estimation with high breakdown point. Math. Stat. Appl. 8, 283–297 (1985)CrossRefMathSciNetGoogle Scholar
  19. Rousseeuw, P., Croux, C., Todorov, V., Ruckstuhl, A., Salibian-Barrera, M., Verbeke, T., Koller, M., and Maechler, M.: Robustbase: basic robust statistics. R package version 0.9-8. (2013)Google Scholar
  20. Rousseeuw, P., Leroy, A.: Robust Regression and Outlier Detection. Wiley, New York (1987)CrossRefMATHGoogle Scholar
  21. Rousseeuw, P., van Driessen, K.: A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212–223 (1999)CrossRefGoogle Scholar
  22. Saw, J.G., Yang, M.C., Mo, T.C.: Chebyshev inequality with estimated mean and variance. Am. Stat. 38(2), 130–132 (1984)MathSciNetGoogle Scholar
  23. Todorov, V., Filzmoser, P.: An object-oriented framework for robust multivariate analysis. J. Stat. Softw. 32(3), 1–47 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Martin-Luther-UniversityHalle/SaaleGermany

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