Cochran’s theorem

Let X be multivariate normally distributed as $\boldsymbol{N_{p}(0,I)}$ such that

\textbf{X}^{\operatorname{T}}\textbf{X}=\sum_{i=1}^{k}Q_{i},

where each

1.

$Q_{i}$ is a quadratic form
2.

$Q_{i}=\textbf{X}^{\operatorname{T}}\textbf{B}_{i}\textbf{X}$ , where $\textbf{B}_{i}$ is a $p$ by $p$ square matrix
3.

$\textbf{B}_{i}$ is positive semidefinite
4.

$\operatorname{rank}(\textbf{B}_{i})=r_{i}$

Then any two of the following imply the third:

1.

$\sum_{i=1}^{k}r_{i}=p$
2.

each $Q_{i}$ has a chi square distribution (http://planetmath.org/ChiSquaredRandomVariable) with $r_{i}$ of freedom, $\chi^{2}(r_{i})$
3.

$Q_{i}$ ’s are mutually independent

As an example, suppose ${X_{1}}^{2}\sim\chi^{2}(m_{1})$ and ${X_{2}}^{2}\sim\chi^{2}(m_{2})$ . Furthermore, assume ${X_{1}}^{2}\geq{X_{2}}^{2}$ and $m_{1}>m_{2}$ , then

{X_{1}}^{2}-{X_{2}}^{2}\sim\chi^{2}(m_{1}-m_{2}).

This corollary is known as Fisher’s theorem.

Title	Cochran’s theorem
Canonical name	CochransTheorem
Date of creation	2013-03-22 14:33:01
Last modified on	2013-03-22 14:33:01
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 62J10
Classification	msc 62H10
Classification	msc 62E10
Defines	Fisher’s theorem