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A generalization of Tyler's M-estimators to the case of incomplete data

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  • Frahm, Gabriel
  • Jaekel, Uwe

Abstract

Many different robust estimation approaches for the covariance or shape matrix of multivariate data have been established. Tyler's M-estimator has been recognized as the 'most robust' M-estimator for the shape matrix of elliptically symmetric distributed data. Tyler's M-estimators for location and shape are generalized by taking account of incomplete data. It is shown that the shape matrix estimator remains distribution-free under the class of generalized elliptical distributions. Its asymptotic distribution is also derived and a fast algorithm, which works well even for high-dimensional data, is presented. A simulation study with clean and contaminated data covers the complete-data as well as the incomplete-data case, where the missing data are assumed to be MCAR, MAR, and NMAR.

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  • Frahm, Gabriel & Jaekel, Uwe, 2010. "A generalization of Tyler's M-estimators to the case of incomplete data," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 374-393, February.
    • Handle: RePEc:eee:csdana:v:54:y:2010:i:2:p:374-393
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    References listed on IDEAS

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    1. Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
    2. Frahm, Gabriel & Jaekel, Uwe, 2007. "Tyler's M-estimator, random matrix theory, and generalized elliptical distributions with applications to finance," Discussion Papers in Econometrics and Statistics 2/07, University of Cologne, Institute of Econometrics and Statistics.
    3. Sebastian Kring & Svetlozar T. Rachev & Markus Höchstötter & Frank J. Fabozzi & Michele Leonardo Bianchi, 2009. "Multi-tail generalized elliptical distributions for asset returns," Econometrics Journal, Royal Economic Society, vol. 12(2), pages 272-291, July.
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    5. Lutz Dümbgen & David E. Tyler, 2005. "On the Breakdown Properties of Some Multivariate M‐Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 247-264, June.
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    7. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
    8. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    9. Serneels, Sven & Verdonck, Tim, 2008. "Principal component analysis for data containing outliers and missing elements," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1712-1727, January.
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    3. Kim, Seungkyu & Park, Seongoh & Lim, Johan & Lee, Sang Han, 2023. "Robust tests for scatter separability beyond Gaussianity," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).

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