In multivariate statistics it is used to do hypothesis tests for Hotelling's $T^2$ distribution, but no textbooks prove this. Is there any proof for it?
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1$\begingroup$ I thought it might not take me long to write a proof of this, but I'll have to return to this later. $\endgroup$– Michael HardyCommented Jun 24 at 23:43
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First this fact was proved by Hoteling himself -- see formula (25). See also e.g. formula (28.40) in The Advanced Theory of Statistics, Volume 2, 1946 by Kendall.
answered Jun 24 at 12:36
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$\begingroup$ Thank you, that's a great proof, but I noticed that the proof is very long and use probability density function for proof. Is that the only way to prove that? Is there any proof using more linear algebra or something like Wishart distribution? $\endgroup$ Commented Jun 25 at 7:25
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1$\begingroup$ @LunaSakura : Did you look at the proof of formula (28.40) in The Advanced Theory of Statistics, Volume 2, 1946 by Kendall? It is elementary and takes only one page. The Wishart distribution is (very) multidimensional and would be a great overkill. $\endgroup$ Commented Jun 25 at 16:14
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$\begingroup$ yes, but I didn't found the relationship between that formula and my question, also I don't understand the reason of formula (28.39). I would appreciate it if you could explain it more. $\endgroup$ Commented Jun 26 at 2:49