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Bias-robust estimators of multivariate scatter based on projections

Author

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  • Maronna, Ricardo A.
  • Stahel, Werner A.
  • Yohai, Victor J.

Abstract

Equivariant estimation of the multivariate scatter of a random vector X can be derived from a criterion of (lack of) spherical symmetry g(X). The scatter matrix is V = (ATA)-1, where A is the transformation matrix which makes AX as spherical as possible, that is, which minimizes g(AX). The new class of projection estimators is based on making the spread of univariate projections as constant as possible by choosing g(X) = supu = 1 s(uTX) -1, where s is any robust scale functional. The breakdown point of such an estimator is at least that of s, independently of the dimension p of X. In order to study the bias, we calculate condition numbers based on asymptotics and on simulations of finite samples for a spherically symmetric X, contaminated by a point mass, with the median absolute deviation as the scale measure. The simulations are done for an algorithm which is designed to approximate the projection estimator. The bias is much lower than the one of Rousseeuw's MVE-estimator, and compares favorably in most cases with two M-estimators.

Suggested Citation

  • Maronna, Ricardo A. & Stahel, Werner A. & Yohai, Victor J., 1992. "Bias-robust estimators of multivariate scatter based on projections," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 141-161, July.
    • Handle: RePEc:eee:jmvana:v:42:y:1992:i:1:p:141-161
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    Citations

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    Cited by:

    1. Maronna, Ricardo A. & Yohai, Víctor J., 1994. "Robust estimation in simultaneous equations models," DES - Working Papers. Statistics and Econometrics. WS 3956, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Zhang, Jian, 2002. "Some Extensions of Tukey's Depth Function," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 134-165, July.
    3. Gather, Ursula & Davies, P. Laurie, 2004. "Robust Statistics," Papers 2004,20, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    4. Ma, Yanyuan & Genton, Marc G., 2001. "Highly Robust Estimation of Dispersion Matrices," Journal of Multivariate Analysis, Elsevier, vol. 78(1), pages 11-36, July.
    5. Croux, Christophe & Haesbroeck, Gentiane, 1997. "An easy way to increase the finite-sample efficiency of the resampled minimum volume ellipsoid estimator," Computational Statistics & Data Analysis, Elsevier, vol. 25(2), pages 125-141, July.
    6. Zhou, Weihua & Dang, Xin, 2010. "Projection based scatter depth functions and associated scatter estimators," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 138-153, January.
    7. David E. Tyler & Frank Critchley & Lutz Dümbgen & Hannu Oja, 2009. "Invariant co‐ordinate selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 549-592, June.
    8. Fekri, M. & Ruiz-Gazen, A., 2004. "Robust weighted orthogonal regression in the errors-in-variables model," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 89-108, January.
    9. Tyler, David E., 2010. "A note on multivariate location and scatter statistics for sparse data sets," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1409-1413, September.
    10. Hernández, Sonia & Yohai, Víctor J., 1999. "Locally and globally robust estimators in regression," DES - Working Papers. Statistics and Econometrics. WS 6348, Universidad Carlos III de Madrid. Departamento de Estadística.
    11. Juan, Jesús & Prieto, Francisco J., 1994. "A subsampling method for the computation of multivariate estimators with high breakdown point," DES - Working Papers. Statistics and Econometrics. WS 3952, Universidad Carlos III de Madrid. Departamento de Estadística.
    12. Schmitt, Eric & Öllerer, Viktoria & Vakili, Kaveh, 2014. "The finite sample breakdown point of PCS," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 214-220.

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