A weak characterization of real Wishart matrices by quadratic forms

Gabriel Fraisse; Sylvie Viguier-Pla

doi:10.1016/j.crma.2016.03.011

Probability theory/Statistics

A weak characterization of real Wishart matrices by quadratic forms

Presented by: the Editorial Board

Gabriel Fraisse ¹; Sylvie Viguier-Pla ^1,²

¹ Université de Perpignan, IUT, Domaine d'Auriac, Carcassonne, France
² Équipe de statistique et probabilités, IMT, UMR 5219, université Paul-Sabatier, Toulouse, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 623-627

Abstract (Original language)
Résumé

Let M be a random symmetric real p-matrix of Wishart distribution with k degrees of freedom and scale parameter Σ. The distribution of M can usually be characterized by the distribution of $({}^{t}u_{1} M u_{1}, \dots, {}^{t}u_{p} M u_{p})$ , for any Σ-orthogonal basis $(u_{1}, \dots, u_{p})$ of $R^{p}$ . We propose to weaken this characterization, showing that, when $k < p$ , it is sufficient to know the distribution of $({}^{t}u_{1} M u_{1}, \dots, {}^{t}u_{k + 1} M u_{k + 1})$ .

Soit M une p-matrice aléatoire réelle symétrique de loi de Wishart à k degrés de liberté et de paramètre d'échelle Σ. On peut caractériser la loi de M par la loi de $({}^{t}u_{1} M u_{1}, \dots, {}^{t}u_{p} M u_{p})$ , pour toute base Σ-orthogonale $(u_{1}, \dots, u_{p})$ de $R^{p}$ . Nous proposons une caractérisation plus faible de la loi de M, montrant que, si $k < p$ , il suffit de connaître la loi de $({}^{t}u_{1} M u_{1}, \dots, {}^{t}u_{k + 1} M u_{k + 1})$ .

Received: 2011-05-12
Accepted: 2016-03-14
Published online: 2016-04-14

DOI: 10.1016/j.crma.2016.03.011

Author's affiliations:

Gabriel Fraisse ¹; Sylvie Viguier-Pla ^{1, 2}

¹ Université de Perpignan, IUT, Domaine d'Auriac, Carcassonne, France
² Équipe de statistique et probabilités, IMT, UMR 5219, université Paul-Sabatier, Toulouse, France

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     title = {A weak characterization of real {Wishart} matrices by quadratic forms},
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     language = {en},
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Gabriel Fraisse; Sylvie Viguier-Pla. A weak characterization of real Wishart matrices by quadratic forms. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 623-627. doi: 10.1016/j.crma.2016.03.011

References
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