Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions
Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 1, p. 129-144

If the space 𝒬 of quadratic forms in n is splitted in a direct sum 𝒬 1 ...𝒬 k and if X and Y are independent random variables of n , assume that there exist a real number a such that E(X|X+Y)=a(X+Y) and real distinct numbers b 1 ,...,b k such that E(q(X)|X+Y)=b i q(X+Y) for any q in 𝒬 i . We prove that this happens only when k=2, when n can be structured in a Euclidean Jordan algebra and when X and Y have Wishart distributions corresponding to this structure.

Si l’espace 𝒬 des formes quadratiques sur n est décomposé en une somme directe 𝒬 1 ...𝒬 k et si X et Y sont des variables aléatoires indépendantes de n , supposons qu’il existe un nombre réel a tel que E(X|X+Y)=a(X+Y) ainsi que des nombres réels distincts b 1 ,...,b k tels que E(q(X)|X+Y)=b i q(X+Y) pour tout q de 𝒬 i . Nous montrons que cela n’arrive que pour k=2, que lorsque n peut être structuré en algèbre de Jordan euclidienne et que lorsque X et Y suivent des lois de Wishart correspondant à cette structure.

DOI : https://doi.org/10.24033/bsmf.2603
Classification:  60H10,  62H05
Keywords: symmetric cones, random matrices, characterization of Wishart laws
@article{BSMF_2011__139_1_129_0,
     author = {Letac, G\'erard and Weso\l owski, J.},
     title = {Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {139},
     number = {1},
     year = {2011},
     pages = {129-144},
     doi = {10.24033/bsmf.2603},
     zbl = {1213.62089},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2011__139_1_129_0}
}
Letac, G.; Wesołowski, J. Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions. Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 1, pp. 129-144. doi : 10.24033/bsmf.2603. http://www.numdam.org/item/BSMF_2011__139_1_129_0/

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