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Breakdown properties of location estimates based on halfspace depth and projected outlyingness. (English) Zbl 0776.62031

The authors investigate multivariate generalizations of the median, trimmed mean and so-called \(W\)-estimates introduced by C. F. Mosteller and J. W. Tukey [Data analysis and regression. (1977)]. The estimates are based on a geometric construction related to “projection pursuit” by applying the concept of the depth of a point \(x\) in a \(d\)-dimensional data set. All the estimates are affine equivariant and have high breakdown point. This combination of equivariance and robustness is interesting because many of the classical location estimators lack one or both of these properties.
The generalization of the median has a breakdown point of at least \(1/(1+d)\) and it can be as high as 1/3 assuming a centrosymmetric distribution. Under very mild conditions the \(W\)-estimates have a breakdown point close to 1/2, even in high dimensions. In contrast, various estimators based on rejecting outliers and taking the mean of the remaining observations (such as iterative ellipsoidal trimming, convex hull peeling) never have a breakdown point exceeding \(1/(1+d)\).

MSC:

62F35 Robustness and adaptive procedures (parametric inference)
62H12 Estimation in multivariate analysis
62H99 Multivariate analysis
62G05 Nonparametric estimation
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