Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac; J. Wesołowski

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 1, page 129-144
  • ISSN: 0037-9484

Abstract

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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.

How to cite

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Letac, G., and Wesołowski, J.. "Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions." Bulletin de la Société Mathématique de France 139.1 (2011): 129-144. <http://eudml.org/doc/272502>.

@article{Letac2011,
abstract = {If the space $\mathcal \{Q\}$ of quadratic forms in $\mathbb \{R\}^n$ is splitted in a direct sum $\mathcal \{Q\}_1\oplus \ldots \oplus \mathcal \{Q\}_k$ and if $X$ and $Y$ are independent random variables of $\mathbb \{R\}^n$, 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 $E(q(X)|X+Y)=b_iq(X+Y)$ for any $q$ in $\mathcal \{Q\}_i.$ We prove that this happens only when $k=2$, when $\mathbb \{R\}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.},
author = {Letac, G., Wesołowski, J.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {symmetric cones; random matrices; characterization of Wishart laws},
language = {eng},
number = {1},
pages = {129-144},
publisher = {Société mathématique de France},
title = {Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions},
url = {http://eudml.org/doc/272502},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Letac, G.
AU - Wesołowski, J.
TI - Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 1
SP - 129
EP - 144
AB - If the space $\mathcal {Q}$ of quadratic forms in $\mathbb {R}^n$ is splitted in a direct sum $\mathcal {Q}_1\oplus \ldots \oplus \mathcal {Q}_k$ and if $X$ and $Y$ are independent random variables of $\mathbb {R}^n$, 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 $E(q(X)|X+Y)=b_iq(X+Y)$ for any $q$ in $\mathcal {Q}_i.$ We prove that this happens only when $k=2$, when $\mathbb {R}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.
LA - eng
KW - symmetric cones; random matrices; characterization of Wishart laws
UR - http://eudml.org/doc/272502
ER -

References

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