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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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