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Some existing representations for the cumulative distribution function of positive definite quadratic forms in normal random variables lead to inefficient computational algorithms. These inefficiencies are overcome with the derivation of some alternative recurrence relations. Some additional results concerning the distribution function and some extensions are also provided which make it possible to compute exact significance levels for certain test statistics proposed for the analysis of variance of unbalanced data.
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Robert H. Brown (1986) studied this question.
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