Cochran's theorem

In statistics , Cochran's theorem is used in analysis of variance . The sentence goes back to the Scottish mathematician William Gemmell Cochran .

It is assumed that random variables are stochastically independent, standard normally distributed , and it is true ${\displaystyle U_{1},\dots U_{n},}$

${\displaystyle \sum _{i=1}^{n}U_{i}^{2}=Q_{1}+\cdots +Q_{k},}$

where each represents the sum of the squares of linear combinations of the s. It is also assumed that ${\displaystyle Q_{i}}$ ${\displaystyle U}$

${\displaystyle r_{1}+\cdots +r_{k}=n,}$

where is the rank of . Cochran's theorem states that they are independent with a chi-square distribution with degrees of freedom . ${\displaystyle r_{i}}$ ${\displaystyle Q_{i}}$ ${\displaystyle Q_{i}}$ ${\displaystyle r_{i}}$

Cochran's theorem is the reverse of Fisher's theorem .

example

If there are independent normally distributed random variables with expected value and standard deviation , then ${\displaystyle X_{1},\dots X_{n},}$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$

${\displaystyle U_{i}=(X_{i}-\mu )/\sigma \;}$

is standard normal distributed for each . ${\displaystyle i}$

Now you can write the following

${\displaystyle \sum _{i=1}^{n}U_{i}^{2}=\sum \limits _{i=1}^{n}\left({\frac {X_{i}-{\overline {X}}}{\sigma }}\right)^{2}+n\left({\frac {{\overline {X}}-\mu }{\sigma }}\right)^{2}}$

In order to recognize this identity, one must multiply by on both sides and note that applies ${\displaystyle \sigma }$

${\displaystyle \sum _{i=1}^{n}(X_{i}-\mu )^{2}=\sum _{i=1}^{n}(X_{i}-{\overline {X}}+{\overline {X}}-\mu )^{2}}$

and expanded to show

${\displaystyle \sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}+\sum _{i=1}^{n}({\overline {X}}-\mu )^{2}+2\sum _{i=1}^{n}(X_{i}-{\overline {X}})({\overline {X}}-\mu ).}$

The third term is zero because the factor

${\displaystyle \sum _{i=1}^{n}({\overline {X}}-X_{i})=0}$

is, and the second term consists only of identical terms that have been joined together. ${\displaystyle n}$

If you combine the above results and then divide through , you get: ${\displaystyle \sigma ^{2}}$

${\displaystyle \sum _{i=1}^{n}\left({\frac {X_{i}-\mu }{\sigma }}\right)^{2}=\sum _{i=1}^{n}\left({\frac {X_{i}-{\overline {X}}}{\sigma }}\right)^{2}+n\left({\frac {{\overline {X}}-\mu }{\sigma }}\right)^{2}=Q_{1}+Q_{2}.}$

Now the rank of just equals 1 (it is the square of just one linear combination of the standard normally distributed random variables). The rank of is equal , and therefore the conditions of Cochran's theorem are satisfied. ${\displaystyle Q_{2}}$ ${\displaystyle Q_{1}}$ ${\displaystyle n-1}$

Cochran's theorem then states that and are independent, with a chi-square distribution with and degrees of freedom. ${\displaystyle Q_{1}}$ ${\displaystyle Q_{2}}$ ${\displaystyle n-1}$ ${\displaystyle 1}$

This shows that the mean and the variance are independent; Furthermore applies

${\displaystyle ({\overline {X}}-\mu )^{2}\sim {\frac {\sigma ^{2}}{n}}\chi _{1}^{2}.}$

A commonly used estimator is used to estimate the unknown population variance ${\displaystyle \sigma ^{2}}$

${\displaystyle {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}\right)^{2}.}$

Cochran's theorem shows that

${\displaystyle {\hat {\sigma }}^{2}\sim {\frac {\sigma ^{2}}{n}}\chi _{n-1}^{2},}$

which shows that the expectation of is equal to . ${\displaystyle {\hat {\sigma }}^{2}}$ ${\displaystyle \sigma ^{2}{\frac {n-1}{n}}}$

Both distributions are proportional to the true but unknown variance. Therefore, their ratio is independent of , and because they are independent, one obtains ${\displaystyle \sigma ^{2}}$ ${\displaystyle \sigma ^{2}}$

${\displaystyle {\frac {\left({\overline {X}}-\mu \right)^{2}}{{\frac {1}{n}}\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}\right)^{2}}}\sim F_{1,n}}$,

where the F-distribution with and represents degrees of freedom (see also Student's t-distribution ). ${\displaystyle F_{1,n}}$ ${\displaystyle 1}$ ${\displaystyle n}$

literature

• Cochran, WG: The distribution of quadratic forms in a normal system, with applications to the analysis of covariance . Mathematical Proceedings of the Cambridge Philosophical Society 30 (2): 178-191, 1934.
• Bapat, RB: Linear Algebra and Linear Models . Second edition (1990). Jumper. ISBN 978-0-387-98871-9