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# 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 with $r_{i}$ degrees 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*.

Defines:

Fisher's theorem

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

62J10*no label found*62H10

*no label found*62E10

*no label found*

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