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On estimation of the diagonal elements of a sparse precision matrix / Samuel Balmand in Electronic Journal of Statistics, vol 10 n° 1 (January 2016)    

Public [article]  inElectronic Journal of Statistics > vol 10 n° 1 (January 2016) . - pp 1551 - 1579 Titre : On estimation of the diagonal elements of a sparse precision matrix Type de document : Article/Communication Auteurs : Samuel Balmand , Auteur ; Arnak Dalalyan, Auteur Année de publication : 2016 Article en page(s) : pp 1551 - 1579 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Statistiques [Termes IGN] calcul matriciel [Termes IGN] estimateur [Termes IGN] matrice creuse [Termes IGN] matrice de covariance [Termes IGN] matrice diagonale [Termes IGN] méthode du maximum de vraisemblance (estimation) [Termes IGN] régression linéaire [Termes IGN] résidu Résumé : (Auteur) In this paper, we present several estimators of the diagonal elements of the inverse of the covariance matrix, called precision matrix, of a sample of independent and identically distributed random vectors. The main focus is on the case of high dimensional vectors having a sparse precision matrix. It is now well understood that when the underlying distribution is Gaussian, the columns of the precision matrix can be estimated independently form one another by solving linear regression problems under sparsity constraints. This approach leads to a computationally efficient strategy for estimating the precision matrix that starts by estimating the regression vectors, then estimates the diagonal entries of the precision matrix and, in a final step, combines these estimators for getting estimators of the off-diagonal entries. While the step of estimating the regression vector has been intensively studied over the past decade, the problem of deriving statistically accurate estimators of the diagonal entries has received much less attention. The goal of the present paper is to fill this gap by presenting four estimators —that seem the most natural ones— of the diagonal entries of the precision matrix and then performing a comprehensive empirical evaluation of these estimators. The estimators under consideration are the residual variance, the relaxed maximum likelihood, the symmetry-enforced maximum likelihood and the penalized maximum likelihood. We show, both theoretically and empirically, that when the aforementioned regression vectors are estimated without error, the symmetry-enforced maximum likelihood estimator has the smallest estimation error. However, in a more realistic setting when the regression vector is estimated by a sparsity-favoring computationally efficient method, the qualities of the estimators become relatively comparable with a slight advantage for the residual variance estimator. Numéro de notice : A2016--107 Affiliation des auteurs : LASTIG MATIS+Ext (2012-2019) Thématique : MATHEMATIQUE Nature : Article nature-HAL : ArtAvecCL-RevueIntern DOI : 10.1214/16-EJS1148 Date de publication en ligne : 31/05/2016 En ligne : http://dx.doi.org/10.1214/16-EJS1148 Format de la ressource électronique : URL Article Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=84707 [article]

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