## A distribution-free M-estimator of multivariate scatter.(English)Zbl 0628.62053

An m-dimensional distribution has known center $$\underset \tilde{} t$$; in this paper a parameter V of multivariate scatter is defined as proportional to the covariance matrix. With a sample $$\underset \tilde{} x_ 1,...,\underset \tilde{} x_ n$$ it is proposed to estimate V by the solution to $m ave\{(\underset \tilde{} x_ i-\underset \tilde{} t)(\underset \tilde{} x_ i-\underset \tilde{} t)'/(\underset \tilde{} x_ i-\underset \tilde{} t)'V_ n^{-1}(\underset \tilde{} x_ i- \underset \tilde{} t)\}=V_ n.$ This is studied as an affine invariant M- estimator of scatter.
A constructive proof of the existence of a solution is given for finite samples satisfying certain conditions. For continuous populations the estimator is shown to be strongly consistent and asymptotically normal, with its asymptotic distribution being distribution-free with respect to the class of continuous elliptically distributed populations. The last part of the paper considers the case where $$\underset \tilde{} t$$ is unknown and hence has to be estimated together with V.
Reviewer: R.Mentz

### MSC:

 62H12 Estimation in multivariate analysis 62E20 Asymptotic distribution theory in statistics 62F35 Robustness and adaptive procedures (parametric inference) 62G05 Nonparametric estimation
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