In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.
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Statement
Suppose U1, ..., Un are i.i.d. standard normally distributed random variables, and an identity of the form
can be written, where each Qi is a sum of squares of linear combinations of the Us. Further suppose that
where ri is the rank of Qi. Cochran's theorem states that the Qi are independent, and each Qi has a chi-squared distribution with ri degrees of freedom. Here the rank of Qi should be interpreted as meaning the rank of the matrix B(i), with elements Bj,k(i), in the representation of Qi as a quadratic form:
Less formally, it is the number of linear combinations included in the sum of squares defining Qi, provided that these linear combinations are linearly independent.
Proof
We first show that the matrices B(i) can be simultaneously diagonalized and that their non-zero eigenvalues are all equal to +1. We then use the vector basis that diagonalize them to simplify their characteristic function and show their independence and distribution.
Each of the matrices B(i) has rank ri and thus ri non-zero eigenvalues. For each i, the sum
Therefore B(i) and C(i) can be simultaneously diagonalized. This can be shown by first diagonalizing B(i). In this basis, it is of the form:
Thus the lower
Thus there exists an orthogonal matrix S such that for all i,
Let
The characteristic function of Qi is:
This is the Fourier transform of the chi-squared distribution with ri degrees of freedom. Therefore this is the distribution of Qi.
Moreover, the characteristic function of the joint distribution of all the Qis is:
From this it follows that all the Qis are i.i.d.
Sample mean and sample variance
If X1, ..., Xn are independent normally distributed random variables with mean μ and standard deviation σ then
is standard normal for each i. It is possible to write
(here
and expand to give
The third term is zero because it is equal to a constant times
and the second term has just n identical terms added together. Thus
and hence
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 distributions with n − 1 and 1 degree of freedom respectively. This shows that the sample mean and sample variance are independent. This can also be shown by Basu's theorem, and in fact this property characterizes the normal distribution – for no other distribution are the sample mean and sample variance independent.
Distributions
The result for the distributions is written symbolically as
Both these random variables are proportional to the true but unknown variance σ2. Thus their ratio does not depend on σ2 and, because they are statistically independent. The distribution of their ratio is given by
where F1,n − 1 is the F-distribution with 1 and n − 1 degrees of freedom (see also Student's t-distribution). The final step here is effectively the definition of a random variable having the F-distribution.
Estimation of variance
To estimate the variance σ2, one estimator that is sometimes used is the maximum likelihood estimator of the variance of a normal distribution
Cochran's theorem shows that
and the properties of the chi-squared distribution show that
the expected value of
Alternative formulation
The following version is often seen when considering linear regression. Suppose that