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I am reading 'Applied multivariate statistical analysis' by Richard Johnson and I do not understand that first to explain this test, talks about the t-student distribution.

enter image description here

And then out of nowhere he plugs the f-distribution: enter image description here

I even read this post: Relationship between F and Student's t distributions

But, i am not able to follow the explanation. Why has he plugging the f-distribution out of nowhere? and if the result is a mix of t and f distributions, why isn't this a complete new distribution with a different name? (I would like to be able to understand this graphically)

gunes
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Chicago1988
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1 Answers1

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It's because the second part is for multivariate analysis in $p$ dimensions, unlike the first screenshot which deals with all scalars. It proceeds with an analogy from univariate case. In multivariate case, you won't be able to compare a statistics of the form $T=n(\bar{X}-\mu_0)S^{-1}$ against a threshold because it is $p$ dimensional. However, by analogy, the univariate case is generalized to multivariate case, i.e. $$t^2=\frac{(\bar X-\mu_0)^2}{S/\sqrt{n}}=n(\bar X-\mu_0)\frac{1}{S}(\bar X-\mu_0)\rightarrow T^2=n\mathbf{(\bar X -\mu_0)^TS^{-1}(\bar X - \mu_0)}$$ And, the statistics obtained, i.e. $T^2$ is a scalar which we can compare against a threshold. Its distribution is Hotelling's T-squared.

gunes
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