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.