Cochran's theorem

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. Let U1, ..., UN be i.i.d. standard normally distributed random variables, and . Let be symmetric matrices. Define ri to be the rank of . Define , so that the Qi are quadratic forms. Further assume . Cochran's theorem states that the following are equivalent: the Qi are independent each Qi has a chi-squared distribution with ri degrees of freedom. Often it's stated as , where is idempotent, and is replaced by . But after an orthogonal transform, , and so we reduce to the above theorem. Claim: Let be a standard Gaussian in , then for any symmetric matrices , if and have the same distribution, then have the same eigenvalues (up to multiplicity). Claim: . Lemma: If , all symmetric, and have eigenvalues 0, 1, then they are simultaneously diagonalizable. Now we prove the original theorem. We prove that the three cases are equivalent by proving that each case implies the next one in a cycle (). If X1, ..., Xn are independent normally distributed random variables with mean μ and standard deviation σ then is standard normal for each i. Note that the total Q is equal to sum of squared Us as shown here: which stems from the original assumption that . So instead we will calculate this quantity and later separate it into Qi's. 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 with the matrix of ones which has rank 1. In turn given that . This expression can be also obtained by expanding in matrix notation. It can be shown that the rank of is as the addition of all its rows is equal to zero. Thus the conditions for Cochran's theorem are met.