Cochran's theorem
In statistics, Cochran's theorem, devised by William G. Cochran,^{1} is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.^{2}
Contents
Statement
Suppose U_{1}, ..., U_{n} are independent standard normally distributed random variables, and an identity of the form
can be written, where each Q_{i} is a sum of squares of linear combinations of the Us. Further suppose that
where r_{i} is the rank of Q_{i}. Cochran's theorem states that the Q_{i} are independent, and each Q_{i} has a chisquared distribution with r_{i} degrees of freedom.^{1} Here the rank of Q_{i} should be interpreted as meaning the rank of the matrix B^{(i)}, with elements B_{j,k}^{(i)}, in the representation of Q_{i} as a quadratic form:
Less formally, it is the number of linear combinations included in the sum of squares defining Q_{i}, provided that these linear combinations are linearly independent.
Examples
Sample mean and sample variance
If X_{1}, ..., X_{n} 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 is the sample mean). To see this identity, multiply throughout by and note that
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 Q_{2} is just 1 (it is the square of just one linear combination of the standard normal variables). The rank of Q_{1} can be shown to be n − 1, and thus the conditions for Cochran's theorem are met.
Cochran's theorem then states that Q_{1} and Q_{2} are independent, with chisquared 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.^{3}
Distributions
The result for the distributions is written symbolically as
Both these random variables are proportional to the true but unknown variance σ^{2}. Thus their ratio is does not depend on σ^{2} and, because they are statistically independent, the distribution of their ratio is given by
where F_{1,n − 1} is the Fdistribution with 1 and n − 1 degrees of freedom (see also Student's tdistribution). The final step here is effectively the definition of a random variable having the Fdistribution.
Estimation of variance
To estimate the variance σ^{2}, 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 chisquared distribution show that the expected value of is σ^{2}(n − 1)/n.
Alternative formulation
The following version is often seen when considering linear regression.^{citation needed} Suppose that is a standard multivariate normal random vector (here denotes the nbyn identity matrix), and if are all nbyn symmetric matrices with . Then, on defining , any one of the following conditions implies the other two:
 (thus the are positive semidefinite)
 is independent of for
See also
 Cramér's theorem, on decomposing normal distribution
 Infinite divisibility (probability)
This article needs additional citations for verification. (July 2011) 
References
 ^ ^{a} ^{b} Cochran, W. G. (April 1934). "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. doi:10.1017/S0305004100016595.
 ^ Bapat, R. B. (2000). Linear Algebra and Linear Models (Second ed.). Springer. ISBN 9780387988719.
 ^ Geary, R.C. (1936). "The Distribution of the "Student's" Ratio for the NonNormal Samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184. doi:10.2307/2983669. JFM 63.1090.03. JSTOR 2983669.

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