On Tyler’s \(M\)-functional of scatter in high dimension. (English) Zbl 0912.62061

Dümbgen, Lutz

Ann. Inst. Stat. Math. 50, No. 3, 471-491 (1998).

Summary: Let \({\mathbf y}_1,{\mathbf y}_2,\dots, {\mathbf y}_n\in \mathbb{R}^q\) be independent, identically distributed random vectors with nonsingular covariance matrix \(\Sigma\), and let \(S= S({\mathbf y}_1,\dots, {\mathbf y}_n)\) be an estimator for \(\Sigma\). A quantity of particular interest is the condition number of \(\Sigma^{-1}S\). If the \({\mathbf y}_i\) 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 D. E. Tyler’s [Ann. Stat. 15, 234-251 (1987; Zbl 0628.62053)] \(M\)-functional of scatter, assuming only elliptical symmetry of \({\mathcal L}({\mathbf y}_i)\) 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.

MSC:

62H12

Estimation in multivariate analysis

Keywords:

M-functional; differentiability; dimensional asymptotics; elliptical symmetry; scatter matrix; symmetrization

Citations:

Zbl 0628.62053