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Constrained \(M\)-estimation for multivariate location and scatter. (English) Zbl 0862.62048

Summary: Consider the problem of estimating the location vector and scatter matrix from a set of multivariate data. Two standard classes of robust estimates are \(M\)-estimates and \(S\)-estimates. The \(M\)-estimates can be tuned to give good local robustness properties, such as good efficiency and a good bound on the influence function at an underlying distribution such as the multivariate normal. However, \(M\)-estimates suffer from poor break-down properties in high dimensions. On the other hand, \(S\)-estimates can be tuned to have good breakdown properties, but when tuned in this way, they tend to suffer from poor local robustness properties.
In this paper a hybrid estimate called a constrained \(M\)-estimate is proposed which combines both good local and good global robustness properties.

MSC:

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