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ON TYLER'S M-FUNCTIONAL OF SCATTER

IN HIGH DIMENSION

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LUTZ DÜMBGEN

*Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294*

D-69120 Heidelberg, Germany
(Received June 4, 1996; revised October 16, 1997)

**Abstract.**
Let *y*_{1}, *y*_{2}, ... , *y*_{n} \in **R**^{q}
be independent, identically distributed random vectors with
nonsingular covariance matrix *Sigma*, and let *S* = *S*(*y*_{1},
... ,
*y*_{n}) be an estimator for *Sigma*. A quantity of
particular interest is the condition number of
*Sigma*^{-1}*S*. If the *y*_{i} are Gaussian and *S* is the
sample covariance matrix, the condition number of
*Sigma*^{-1}*S*, i.e. the ratio of its extreme eigenvalues,
equals 1 + *O*_{p}((*q*/*n*)^{1/2}) as *q* \to \infty and *q*/*n* \to 0.
The present paper shows that the same result can be achieved
with two estimators based on Tyler's (1987, *Ann.
Statist.*, **15**, 234-251) M-functional of scatter,
assuming only elliptical symmetry of \cal L(*y*_{i}) or less. The
main tool is a linear expansion for this M-functional which
holds uniformly in the dimension *q*. As a by-product we
obtain continuous Fréchet-differentiability with respect
to weak convergence.

*Key words and phrases*:
Differentiability,
dimensional asymptotics, elliptical symmetry,
M-functional, scatter matrix, symmetrization.

**Source**
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