On Tyler's M-functional of scatter in high dimension

Let y(1),y(2),...,y(n) is an element of 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 + Op((q/n)(1/2)) as q --> infinity and q/n --> 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 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 Frechet-differentiability with respect to weak convergence.