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

Tyler, David E.

Ann. Stat. 15, 234-251 (1987).

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

Keywords:

Huber-type M-estimator; minimizing maximum asymptotic variance; multivariate scatter; covariance matrix; affine invariant M-estimator of scatter; finite samples; continuous populations; strongly consistent; asymptotically normal; distribution-free; continuous elliptically distributed populations