ON TYLER’S M-FUNCTIONAL OF SCATTER IN HIGH DIMENSION

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} ∈ R^{q} be independent, identically distributed random vectors with nonsingular covariance matrix Σ, and let S = S(y_{1}, …, y_{n}) be an estimator for Σ. A quantity of particular interest is the condition number of Σ^{−1}S. If the y_{i} are Gaussian and S is the sample covariance matrix, the condition number of Σ^{−1}S, i.e. the ratio of its extreme eigenvalues, equals 1 + O_{p}((q/n)^{1/2}) as q → ∞ 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 Mfunctional 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.