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On the breakdown properties of some multivariate \(M\)-functions. (English) Zbl 1089.62056

Affine equivariant robust \(M\)-estimates of multivariate location and scatter matrices are considered. The multivariate \(M\)-estimate corresponds to the maximum likelihood estimate derived for the location-scatter class of elliptical \(t\)-distributions. It is shown that the breakdown point of this estimator equals \(1/(q+\nu)\) where \(q\) is the dimension of the data space and \(\nu\) is the number of degrees of freedom of the \(t\)-distribution. The Taylor \(M\)-estimate for scatter corresponds to \(\nu=0\). It’s breakdown point is \(1/q\), but this estimate of scatter presumes the “center” of the distribution. The authors consider a symmetrized version of this estimate and demonstrate that it’s breakdown point is \(1-\sqrt{1-1/q}\).

MSC:

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