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Citing Wikipedia,

From the relations between a beta and an F-distribution, Wilks' lambda can be related to the F-distribution when one of the parameters of the Wilks lambda distribution is either 1 or 2, e.g.,

$$\frac{1 - \Lambda(p, m, 1)}{\Lambda(p, m, 1)} \sim \frac{p}{m - p + 1} F_{p, m-p+1},$$ and $$\frac{1 - \sqrt{\Lambda(p, m, 2)}}{\sqrt{\Lambda(p, m, 2)}} \sim \frac{p}{m - p + 1} F_{2p, 2(m-p+1)}.$$

My textbook, Applied Multivariate Statistical Analysis (6 edition), also includes a similar conclusion, but it's not comprehensive either. What's more, it gives the exact distribution in terms of "No. of groups" instead of degree of freedom, which confuses me.

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I would like to know other cases of this conclusion. To paraphrase, what's Wilks' lambda's exact distribution when $p$ or $m$ is 1 or 2? I tried searching for John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press on Google, only to be blocked by paywalls.

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