(Received June 4, 1996; revised October 16, 1997)
Abstract. Let y1, y2, ... , yn \in Rq be independent, identically distributed random vectors with nonsingular covariance matrix Sigma, and let S = S(y1, ... , yn) be an estimator for Sigma. A quantity of particular interest is the condition number of Sigma-1S. If the yi are Gaussian and S is the sample covariance matrix, the condition number of Sigma-1S, i.e. the ratio of its extreme eigenvalues, equals 1 + Op((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(yi) 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.