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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References:

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