### Abstract

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 language | English (US) |
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Pages (from-to) | 1327-1345 |

Number of pages | 19 |

Journal | Annals of Statistics |

Volume | 24 |

Issue number | 3 |

State | Published - Jun 1996 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability