# 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. 1.

$Q_{i}$ is a quadratic form

2. 2.

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

3. 3.

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

4. 4.

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

Then any two of the following imply the third:

1. 1.

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

2. 2.

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

3. 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 CochransTheorem 2013-03-22 14:33:01 2013-03-22 14:33:01 CWoo (3771) CWoo (3771) 8 CWoo (3771) Theorem msc 62J10 msc 62H10 msc 62E10 Fisher’s theorem