Robustness properties of S-estimators of multivariate location and shape in high dimension

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104 Scopus citations


For the problem of robust estimation of multivariate location and shape, defining S-estimators using scale transformations of a fixed ρ function regardless of the dimension, as is usually done, leads to a perverse outcome: estimators in high dimension can have a breakdown point approaching 50%, but still fail to reject as outliers points that are large distances from the main mass of points. This leads to a form of nonrobustness that has important practical consequences. In this paper, estimators are defined that improve on known S-estimators in having all of the following properties: (1) maximal breakdown for the given sample size and dimension; (2) ability completely to reject as outliers points that are far from the main mass of points; (3) convergence to good solutions with a modest amount of computation from a nonrobust starting point for large (though not near 50%) contamination. However, to attain maximal breakdown, these estimates, like other known maximal breakdown estimators, require large amounts of computational effort. This greater ability of the new estimators to reject outliers comes at a modest cost in efficiency and gross error sensitivity and at a greater, but finite, cost in local shift sensitivity.

Original languageEnglish (US)
Pages (from-to)1327-1345
Number of pages19
JournalAnnals of Statistics
Issue number3
StatePublished - Jun 1996


  • Breakdown point
  • Efficiency
  • Gross error sensitivity
  • Local shift sensitivity
  • Outlier rejection

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability


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