Robust estimation and random matrices

Robust estimation and random matrices

Romain Couillet

Centrale Supélec 91192 Gif sur Yvette, France

Corresponding Author Email: 
romain.couillet@centralesupelec.fr
Page: 
273-320
|
DOI: 
https://doi.org/10.3166/TS.33.273-320
Received: 
19 January 2015
| |
Accepted: 
12 October 2015
| | Citation
Abstract: 

This article provides a technical survey of the recent advances between the fields of robust estimation of scatter and of large dimensional random matrix theory. An exposition of the theoretical results will be made which we shall apply to various contexts in the area of sta- tistics and signal processing at large. The theoretical results essentially show that, while robust estimators of scatter are implicitly defined and thus difficult objects to manipulate, in the large dimensional random matrix regime where both the population size and the number of samples are simultaneously large, these implicit robust estimators tend to behave similar to much sim- pler random matrix models, amenable to analysis. This induces that many statistical properties of these estimators could be unearthed which we shall discuss. In terms of applications, these robust estimators of scatter are long-standing structural elements to handle both outliers and heavy-tailed behavior in the observed data. These impulsiveness harnessing effects will be pre- cisely documented and shall be instrumental to develop improved robust statistics methods for detection and estimation in antenna arrays, portfolio optimization, etc.

Keywords: 

random matrix theory, robust estimation, array processing, statistics.

Extended Abstract
1. Introduction
2. Comportement asymptotique des estimateurs robustes de type Maronna
3. Propriétés de rejet de données erronées
4. Comportement asymptotique des estimateurs régularisés
5. Résultats de second ordre
6. Conclusions
Remerciements
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