cochran's theorem

In statistics, Cochran's theorem is used in the analysis of variance. Suppose U₁, ..., U_n are independent standard normally distributed random variables, and an identity of the form

\sum_{i=1}^n U_i^2=Q_1+\cdots + Q_k can be written where each Q_i is a sum of squares of linear combinations of the Us. Then if

r_i+\cdots +r_k=n where r_i is the rank of Q_i, Cochran's theorem states that the Q_i are independent, and Q_i has a chi-square distribution with r_i degrees of freedom. Cochran's theorem is the converse of Fisher's theorem.

Example

If X₁, ..., X_n are independent normally distributed random variables with mean μ and standard deviation σ then

U_i=(X_i-\mu)/\sigma

is standard normal for each i. It is possible to write

\sum U_i^2=\sum\left(\frac{X_i-\overline{X}}{\sigma}\right)^2 + n\left(\frac{\overline{X}-\mu}{\sigma}\right)^2 (here, summation is from 1 to n, that is over the observations). To see this identity, multiply throughout by

\sigma

and note that

\sum(X_i-\mu)^2= \sum(X_i-\overline{X}+\overline{X}-\mu)^2 and expand to give

\sum(X_i-\overline{X})^2+\sum(\overline{X}-\mu)^2+ 2\sum(X_i-\overline{X})(\overline{X}-\mu). The third term is zero because it is equal to a constant times

\sum(\overline{X}-X_i),

and the second term is just n identical terms added together. Combining the above results (and dividing by σ²), we have:

\sum\left(\frac{X_i-\mu}{\sigma}\right)^2= \sum\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 Q₂ is just 1 (it is the square of just one linear combination of the standard normal variables). The rank of Q₁ can be shown to be n − 1, and thus the conditions for Cochran's theorem are met. Cochran's theorem then states that Q₁ and Q₂ are independent, with Chi-squared distribution with n − 1 and 1 degree of freedom respectively. This shows that the sample mean and sample variance are independent; also

(\overline{X}-\mu)^2\sim \frac{\sigma^2}{n}\chi^2_1. To estimate the variance σ², one estimator that is often used is

\hat{\sigma^2}= \frac{1}{n}\sum\left( X_i-\overline{X}\right)^2 . Cochran's theorem shows that

\hat{\sigma^2}\sim \frac{\sigma^2}{n}\chi^2_{n-1} which shows that the expected value of

\hat{\sigma}^2

is σ²n/(n − 1). Both these distributions are proportional to the true but unknown variance σ²; thus their ratio is independent of σ² and because they are independent we have

\frac{\left(\overline{X}-\mu\right)^2} {\frac{1}{n}\sum\left(X_i-\overline{X}\right)^2}\sim F_{1,n} where F_1,n is the F-distribution with 1 and n degrees of freedom (see also Student's t-distribution).

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