Cochran's theorem

In the statistics of the set of Cochran is used in the analysis of variance. The phrase goes back to the Scottish mathematician William Gemmell Cochran.

It is believed U1, ..., Un are stochastically independent standard normal random variables, and it is

Where ri is the rank of Qi. The set of Cochran says that the Qi are independent with a chi -square distribution with r degrees of freedom.

The set of Cochran is the inverse of the set of Fisher.

Example

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

Is a standard normal distribution for each i

Now you can write the following

In order to recognize this identity, one has to multiply on both sides and note that applies

And expanded to show

The third term is zero, since this is equal to a constant times

, and the second term is composed only of n identical terms, which have been joined together.

Combining the above results, and then dividing by σ2, then we obtain:

Now the rank of Q2 is just equal to 1 ( it is the square of only one linear combination of the standard normal random variables). The rank of Q1 is equal to n - 1, and therefore the conditions of the theorem of Cochran are met.

The set of Cochran then states that Q1 and Q2 are independent, with a chi -square distribution with n - 1 and 1 degree of freedom.

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

To estimate the variance σ2, a commonly used estimator is used

The set of Cochran shows that

Which shows that the mean value of of σ2 ( n - 1) / n.

Both distributions are proportional to the true but unknown variance σ2; Therefore, their ratio is independent of σ2, and because they are independent, one obtains

Where F1, n is the F- distribution with 1 and n degrees of freedom is (see also Student's t-distribution).

• Random variable
• Set ( mathematics)