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

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

Zbl 0628.62053
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