Title: On spectral properties of high-dimensional spatial-sign covariance matrices in elliptical distributions with applications,
Abstract: Spatial-sign covariance matrix (SSCM) is an important substitute of sample
covariance matrix (SCM) in robust statistics. This paper investigates the SSCM
on its asymptotic spectral behaviors under high-dimensional elliptical
populations, where both the dimension $p$ of observations and the sample size
$n$ tend to infinity with their ratio $p/n\to c\in (0, \infty)$. The empirical
spectral distribution of this nonparametric scatter matrix is shown to converge
in distribution to a generalized Marčenko-Pastur law. Beyond this, a new
central limit theorem (CLT) for general linear spectral statistics of the SSCM
is also established. For polynomial spectral statistics, explicit formulae of
the limiting mean and covarance functions in the CLT are provided. The derived
results are then applied to an estimation procedure and a test procedure for
the spectrum of the shape component of population covariance matrices.