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