The spatial sign covariance operator: Asymptotic results and applications

TL;DR

This paper investigates the asymptotic properties of the spatial sign covariance operator in functional data analysis, providing tools for robust outlier detection and comparing scatter operators across populations.

Contribution

It derives the asymptotic distribution of the spatial sign covariance operator when location is unknown and develops tests for comparing scatter operators.

Findings

Asymptotic distribution of the spatial sign covariance operator is established.

New tests for differences between scatter operators are proposed.

Monte Carlo simulations demonstrate test effectiveness for small samples.

Abstract

Due to the increasing recording capability, functional data analysis has become an important research topic. For functional data the study of outlier detection and/or the development of robust statistical procedures has started recently. One robust alternative to the sample covariance operator is the sample spatial sign covariance operator. In this paper, we study the asymptotic behaviour of the sample spatial sign covariance operator when location is unknown. Among other possible applications of the obtained results, we derive the asymptotic distribution of the principal directions obtained from the sample spatial sign covariance operator and we develop test to detect differences between the scatter operators of two populations. In particular, the test performance is illustrated through a Monte Carlo study for small sample sizes.