Cochran's theorem
In statistics, Cochran's theorem is used in the analysis of variance.
Suppose U1, ..., Un are independent standard normally distributed random variables, and an identity of the form
- <math>
\sum_{i=1}^n U_i^2=Q_1+\cdots + Q_k
<math>
can be written where each Qi is a sum of squares of linear combinations of the Us. Then if
- <math>
r_i+\cdots +r_k=n
<math>
where ri is the rank of Qi, Cochran's theorem states that the Qi are independent, and Qi has a chi-square distribution with ri degrees of freedom.
Cochran's theorem is the converse of F-distribution with 1 and n degrees of freedom (see also Student's t-distribution).
