0
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.[citation needed]
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.
ExamplesSample mean and sample varianceIf X1, ..., Xn are independent normally distributed random variables with mean μ and standard deviation σ thenis standard normal for each i. It is possible to write
(here, summation is from 1 to n, that is over the observations). 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 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
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