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

TL;DR

This paper demonstrates that certain conditional expectation properties of quadratic forms in Euclidean spaces imply the underlying space has a Jordan algebra structure, leading to Wishart distributions.

Contribution

It establishes a novel link between quadratic regression properties and Euclidean Jordan algebras, characterizing Wishart distributions through these algebraic structures.

Findings

Conditional quadratic regression implies Jordan algebra structure.

Wishart distributions naturally arise in this algebraic context.

The space splits into exactly two components under these conditions.

Abstract

If the space $\mathcal{Q}$ of quadratic forms in $\mathbb{R}^n$ is splitted in a direct sum $\mathcal{Q}_1\oplus...\oplus \mathcal{Q}_k$ 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.