Convergence and fluctuations of Regularized Tyler estimators

Handle URI:
http://hdl.handle.net/10754/581781
Title:
Convergence and fluctuations of Regularized Tyler estimators
Authors:
Kammoun, Abla ( 0000-0002-0195-3159 ) ; Couillet, Romain; Pascal, Frederic; Alouini, Mohamed-Slim ( 0000-0003-4827-1793 )
Abstract:
This article studies the behavior of regularized Tyler estimators (RTEs) of scatter matrices. The key advantages of these estimators are twofold. First, they guarantee by construction a good conditioning of the estimate and second, being a derivative of robust Tyler estimators, they inherit their robustness properties, notably their resilience to the presence of outliers. Nevertheless, one major problem that poses the use of RTEs in practice is represented by the question of setting the regularization parameter p. While a high value of p is likely to push all the eigenvalues away from zero, it comes at the cost of a larger bias with respect to the population covariance matrix. A deep understanding of the statistics of RTEs is essential to come up with appropriate choices for the regularization parameter. This is not an easy task and might be out of reach, unless one considers asymptotic regimes wherein the number of observations n and/or their size N increase together. First asymptotic results have recently been obtained under the assumption that N and n are large and commensurable. Interestingly, no results concerning the regime of n going to infinity with N fixed exist, even though the investigation of this assumption has usually predated the analysis of the most difficult N and n large case. This motivates our work. In particular, we prove in the present paper that the RTEs converge to a deterministic matrix when n → ∞ with N fixed, which is expressed as a function of the theoretical covariance matrix. We also derive the fluctuations of the RTEs around this deterministic matrix and establish that these fluctuations converge in distribution to a multivariate Gaussian distribution with zero mean and a covariance depending on the population covariance and the parameter.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Convergence and fluctuations of Regularized Tyler estimators 2015:1 IEEE Transactions on Signal Processing
Publisher:
Institute of Electrical and Electronics Engineers (IEEE)
Journal:
IEEE Transactions on Signal Processing
Issue Date:
26-Oct-2015
DOI:
10.1109/TSP.2015.2494858
Type:
Article
ISSN:
1053-587X; 1941-0476
Additional Links:
http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7307228
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorKammoun, Ablaen
dc.contributor.authorCouillet, Romainen
dc.contributor.authorPascal, Fredericen
dc.contributor.authorAlouini, Mohamed-Slimen
dc.date.accessioned2015-11-05T06:26:26Zen
dc.date.available2015-11-05T06:26:26Zen
dc.date.issued2015-10-26en
dc.identifier.citationConvergence and fluctuations of Regularized Tyler estimators 2015:1 IEEE Transactions on Signal Processingen
dc.identifier.issn1053-587Xen
dc.identifier.issn1941-0476en
dc.identifier.doi10.1109/TSP.2015.2494858en
dc.identifier.urihttp://hdl.handle.net/10754/581781en
dc.description.abstractThis article studies the behavior of regularized Tyler estimators (RTEs) of scatter matrices. The key advantages of these estimators are twofold. First, they guarantee by construction a good conditioning of the estimate and second, being a derivative of robust Tyler estimators, they inherit their robustness properties, notably their resilience to the presence of outliers. Nevertheless, one major problem that poses the use of RTEs in practice is represented by the question of setting the regularization parameter p. While a high value of p is likely to push all the eigenvalues away from zero, it comes at the cost of a larger bias with respect to the population covariance matrix. A deep understanding of the statistics of RTEs is essential to come up with appropriate choices for the regularization parameter. This is not an easy task and might be out of reach, unless one considers asymptotic regimes wherein the number of observations n and/or their size N increase together. First asymptotic results have recently been obtained under the assumption that N and n are large and commensurable. Interestingly, no results concerning the regime of n going to infinity with N fixed exist, even though the investigation of this assumption has usually predated the analysis of the most difficult N and n large case. This motivates our work. In particular, we prove in the present paper that the RTEs converge to a deterministic matrix when n → ∞ with N fixed, which is expressed as a function of the theoretical covariance matrix. We also derive the fluctuations of the RTEs around this deterministic matrix and establish that these fluctuations converge in distribution to a multivariate Gaussian distribution with zero mean and a covariance depending on the population covariance and the parameter.en
dc.language.isoenen
dc.publisherInstitute of Electrical and Electronics Engineers (IEEE)en
dc.relation.urlhttp://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7307228en
dc.rights(c) 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.en
dc.titleConvergence and fluctuations of Regularized Tyler estimatorsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalIEEE Transactions on Signal Processingen
dc.eprint.versionPost-printen
dc.contributor.institutionLaboratoire des Signaux et Systemes (L2S, UMR CNRS 8506) CentraleSupelec-CNRS-Universite Paris-Sud, 91192 Gif-sur-Yvette, Franceen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorKammoun, Ablaen
kaust.authorAlouini, Mohamed-Slimen
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