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
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

@article{CRMATH_2016__354_6_623_0,
     author = {Gabriel Fraisse and Sylvie Viguier-Pla},
     title = {A weak characterization of real {Wishart} matrices by quadratic forms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {623--627},
     publisher = {Elsevier},
     volume = {354},
     number = {6},
     year = {2016},
     doi = {10.1016/j.crma.2016.03.011},
     language = {en},
}

TY  - JOUR
AU  - Gabriel Fraisse
AU  - Sylvie Viguier-Pla
TI  - A weak characterization of real Wishart matrices by quadratic forms
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 623
EP  - 627
VL  - 354
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2016.03.011
LA  - en
ID  - CRMATH_2016__354_6_623_0
ER  -

%0 Journal Article
%A Gabriel Fraisse
%A Sylvie Viguier-Pla
%T A weak characterization of real Wishart matrices by quadratic forms
%J Comptes Rendus. Mathématique
%D 2016
%P 623-627
%V 354
%N 6
%I Elsevier
%R 10.1016/j.crma.2016.03.011
%G en
%F CRMATH_2016__354_6_623_0

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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.011/

References
Cited by

[1] T. Bodnar; S. Mazur; K. Podgórski Singular inverse Wishart distribution and its application to portfolio theory, J. Multivar. Anal., Volume 143 (2016), pp. 314-326

[2] M.L. Eaton Multivariate Statistics, Wiley, New York, 1983

[3] P. Graczyk; G. Letac; H. Massam The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution, J. Theor. Probab., Volume 18 (2005) no. 1, pp. 1-42

[4] L. Nachbin The Haar Integral, D. Van Nostrand Company, Inc., Princeton, NJ, Toronto, London, 1965

[5] G. Saporta Probabilités, Analyse des données et statistique, Technip, Paris, 2006

[6] Y. Xiao; Y.-C. Ku; P. Bloomfield; S.K. Ghosh On the degrees of freedom in MCMC-based Wishart models for time series data, Stat. Probab. Lett., Volume 98 (2015), pp. 59-64

Cited by Sources:

Comments - Policy