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I am trying to go through the derivation of the Grubbs test from (1), section 5, and I have gone through it and have some very general understanding.

Intending to go through in more detail, I am a little stumped by what is going on in the very first equation:

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I don't quite understand the notation, and cannot find 'simultaneous density function' or gaussians with a factorial in the denominator of the scaling term. Further, what are the steps dx1, dx2 ... I don't understand the notation outside of the context of a differentiation.

Many Thanks

(1) https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-21/issue-1/Sample-Criteria-for-Testing-Outlying-Observations/10.1214/aoms/1177729885.full

Joseph
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1 Answers1

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If we integrate the density function, we get the probability.

By independence, if we do not impose the ordering constraint, the joint density function is

$$\prod_{i=1}^nf(x_i|\mu=0, \sigma^2)=\prod_{i=1}^n \frac1{\sqrt{2\pi}\sigma}\exp\left( -\frac{(x_i-0)^2}{2\sigma^2}\right)= \frac1{(\sqrt{2\pi}\sigma)^n}\exp\left(-\frac1{2\sigma^2}\sum_{i=1}^n x_i^2\right)$$

Now, since we impose the ordering constraint, we are considering the permutations as the same, hence we multiply by $n!$.

Siong Thye Goh
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  • Thanks for this Siong that makes sense. Could you shed some light on the notation in the original post, is the $dx_1..dx_n$ supposed to indicate that we are talking about infinitesimal steps in the region around $x_i$ as $P(X=x) = 0$? (and this is also indicated by the $d$ in $dF()$ on the left hand side)? – Joseph Jan 07 '22 at 23:56
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    $dF$ is usually in Lebesgue-Stieltjes integral, I understand it as when we integrate both sides to obtain the corresponding probability. – Siong Thye Goh Jan 08 '22 at 06:37