Estimation of the inverse scatter matrix for a scale mixture of Wishart matrices under Efron-Morris type losses

Djamila Boukehil (1, 2) , Dominique Fourdrinier (3, 4) , Fatiha Mezoued (2) , William Strawderman (5)

1 LITIS - Laboratoire d'Informatique, du Traitement de l'Information et des Systèmes
2 ENSSEA - Ecole Nationale Supérieure de Statistique et d'Economie Appliquée [Tipaza]
3 LITIS - Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes
4 National Research Tomsk State University, Tomsk, Russia
5 Department of Statistics and Biostatistics Rutgers University

We consider estimation of the inverse scatter matrix Σ −1 for a scale mixture of Wishart matrices under various Efron-Morris type losses, tr[{Σ −1 − Σ −1 } 2 S k ] for k = 0, 1, 2..., where S is the sample covariance matrix. We improve on the standard estimators a S + , where S + denotes the Moore-Penrose inverse of S and a is a positive constant, through an unbiased estimator of the risk difference between the new estimators and a S +. Thus we demontrate that improvements over the standard estimators under a Wishart distribution can be extended under mixing. We give a unified treatement of the two cases where S is invertible (S + = S −1) and where S is singular.

Mots clés: AMS 2010 subject classifications: 62H12; 62F10; 62C99 Key words and phrases: Elliptically symmetric distributions; scale mixtures of normals; scale mixtures of Wisharts; Stein-Haff type identity; covariance matrix; scatter matrix; SURE estimators * Djamila Boukehil is PHD student; École Nationale

Domaines: Statistiques [stat]

Djamila Boukehil, Dominique Fourdrinier, Fatiha Mezoued, William Strawderman. Estimation of the inverse scatter matrix for a scale mixture of Wishart matrices under Efron-Morris type losses. 2020. ⟨hal-02503972⟩