# On spectral properties of high-dimensional spatial-sign covariance matrices in elliptical distributions with applications

## TL;DR

This paper studies the spectral behavior of the spatial-sign covariance matrix in high-dimensional elliptical distributions, deriving a generalized Marčenko-Pastur law and a CLT for spectral statistics, with applications to covariance matrix spectrum estimation and testing.

## Contribution

It introduces a new asymptotic spectral analysis of SSCM in high dimensions, including a CLT for linear spectral statistics and explicit formulas for polynomial cases, extending robust covariance estimation methods.

## Key findings

- Empirical spectral distribution converges to a generalized Marčenko-Pastur law.
- Established a CLT for linear spectral statistics of SSCM.
- Provided explicit formulas for mean and covariance in polynomial spectral statistics.

## 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 Mar\v{c}enko-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.