%0 Generic
%A Dümbgen, Lutz
%C Heidelberg
%D 1997
%F heidok:20935
%K Differentiability; dimensional asymptotics; elliptical symmetry; M-functional; scatter matrix; symmetrization
%R 10.11588/heidok.00020935
%T The Asymptotic Behavior of Tyler's M-Estimator of Scatter in High Dimension
%U https://archiv.ub.uni-heidelberg.de/volltextserver/20935/
%X It is shown that Tyler's (1987) M-functional of scatter, whichis a robust surrogate for the covariance matrix of a distribution on R^p ,is Fr'echet-differentiable with respect to the weak topology. This propertyis derived in an asymptotic framework, where the dimension p may tend toinfinity. If applied to the empirical distribution of n i.i.d. randomvectors with elliptically symmetric distribution, the resulting estimatorhas the same asymptotic behavior as the sample covariance matrix in anormal model, provided that p tends to infinity and p/n tends to zero.
%Z This is an extended version of the paper "On Tyler's M-functional of scatter in  high dimension" which has been tentatively accepted for publication in the Annals of the  Institute of Statistical Mathematics (50 (1998), pp. 471-491).   The present version contains some additional results and more detailed proofs.