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Cochran's theorem


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A theorem, given by Cochran in 1934, concerning sums of chi-squared variables. Let Y represent an n×1 vector of independent standard normal random variables and let A1, A2,…, Ak be non-zero symmetric matrices such that , where Y′ is the transpose of Y. Write Qj=YAjY. Cochran's theorem, published in 1934, states that, if any one of the following three conditions is true, then so are the other two:(i) The ranks of A1, A2,…, Ak sum to n.(ii) Each of Q1, Q2,…, Qk has a chi-squared distribution.(iii) Each of Q1, Q2,…, Qk is independent of all the others.

(i) The ranks of A1, A2,…, Ak sum to n.

(ii) Each of Q1, Q2,…, Qk has a chi-squared distribution.

(iii) Each of Q1, Q2,…, Qk is independent of all the others.

Subjects: Probability and Statistics.


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