Skip to main content
SpringerLink
Log in

Efficiency of the pMST and RDELA location and scatter estimators

  • Original Paper
  • Published:
AStA Advances in Statistical Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • 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)

  • Bennett, M., Willemain, T.: Resistant estimation of multivariate location using minimum spanning trees. J. Stat. Comput. Simul. 69, 19–40 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Croux, C., Haesbroeck, G.: Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. J. Multivar. Anal. 71, 161–190 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Delaunay, B.: Sur la sphere vide. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk 7, 793–800 (1934)

    Google Scholar 

  • Donoho, D.: Breakdown properties of multivariate location estimators. Ph.D. thesis, Department of Statistics, Harvard University (1982)

  • Donoho, D., Huber, P.: The notion of breakdown point. A Festschrift for Erich Lehmann 157–184 (1983)

  • Hubert, M., Debruyne, M.: Minimum covariance determinant. Wiley Interdiscip. Rev. Comput. Stat. 2(1), 36–43 (2010)

    Article  Google Scholar 

  • Joe, H.: Generating random correlation matrices based on partial correlations. J. Multivar. Anal. 97(10), 2177–2189 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Jungnickel, D.: Graphs, networks and algorithms, 3rd edn. Springer (2008)

  • Kaban, A.: Non-parametric detection of meaningless distances in high dimensional data. Stat. Comput. 22, 375–385 (2012)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Liebscher, S., Kirschstein, T.: Restlos: robust estimation of location and scatter. R package version 0.1-2. (2013)

  • 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

  • Lopuhaä, H., Rousseeuw, P.: Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Stat. 19(1), 229–248 (1991)

    Article  MATH  Google Scholar 

  • Penrose, M.: A strong law for the longest edge of the minimal spanning tree. Ann. Probab. 27(1), 246–260 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Qiu, W., Joe, H.: Cluster Generation: random cluster generation (with specified degree of separation). R package version 1.3.1. (2013)

  • R Core Team.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2013)

  • Rousseeuw, P.: Multivariate estimation with high breakdown point. Math. Stat. Appl. 8, 283–297 (1985)

    Article  MathSciNet  Google Scholar 

  • 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)

  • Rousseeuw, P., Leroy, A.: Robust Regression and Outlier Detection. Wiley, New York (1987)

    Book  MATH  Google Scholar 

  • Rousseeuw, P., van Driessen, K.: A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212–223 (1999)

    Article  Google Scholar 

  • Saw, J.G., Yang, M.C., Mo, T.C.: Chebyshev inequality with estimated mean and variance. Am. Stat. 38(2), 130–132 (1984)

    MathSciNet  Google Scholar 

  • Todorov, V., Filzmoser, P.: An object-oriented framework for robust multivariate analysis. J. Stat. Softw. 32(3), 1–47 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Martin-Luther-University, Gr. Steinstrasse 73, 06099 , Halle/Saale, Germany

    Steffen Liebscher & Thomas Kirschstein

Authors
  1. Steffen Liebscher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Kirschstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steffen Liebscher.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liebscher, S., Kirschstein, T. Efficiency of the pMST and RDELA location and scatter estimators. AStA Adv Stat Anal 99, 63–82 (2015). https://doi.org/10.1007/s10182-014-0231-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10182-014-0231-7

Keywords

Search

Navigation