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Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. (English) Zbl 0733.62058

The finite-sample replacement breakdown point of a location estimator \(\underset{\tilde{}} t_ n\) at a collection \(\underset{\tilde{}} X\) is defined as the smallest fraction of outliers that can take the estimate over all bounds: \[ \epsilon^*(\underset{\tilde{}} t_ n,\underset{\tilde{}} X)=\min_{1\leq m\leq n}\{m/n :\;\sup \| \underset{\tilde{}} t_ n(\underset{\tilde{}} X)- \underset{\tilde{}} t_ n(\underset{\tilde{}} Y_ m)\| =\infty, \] where \(\underset{\tilde{}} Y_ m\) is \(\underset{\tilde{}} X\) with m replacements. In section 2 it is shown that \([(n+1)/2]/n\) (where [u] denotes the nearest integer less than or equal to u) is an upper bound for the breakdown point of a translation equivariant location estimator, and that the same bound holds for the \(L_ 1\)-estimator. In section 3, the role of the indicated bound is considered for affine equivariant estimators of location and covariance, in particular for minimum volume ellipsoid and S-estimators. In section 4, the breakdown point is related to a measure of performance based on large deviation probabilities, and in section 5 it is shown that one-step reweighting preserves the breakdown point.

MSC:

62H12 Estimation in multivariate analysis
62F35 Robustness and adaptive procedures (parametric inference)
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