I'm stuck trying to write a proof for finding the sampling distribution (pdf) of a geometric mean.
Suppose $x_1 x_2 ... x_n$ are IID $U(0,1)$, find the pdf of: $$\left(\prod_{i=1}^n x_i \right)^\frac{1}{n}$$
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Lee David Chung Lin
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Aubrey
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1Hint: What is the distribution of $y_i = -\log x_i$? – heropup Jan 20 '18 at 23:55
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I can use the MGF method to show that $-log(x_i)$ ~ exp(1) – Aubrey Jan 21 '18 at 00:11
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1So what is the distribution of $\sum -\log(x_i)$ or $\frac1n\sum -\log(x_i)$ ? – Henry Jan 21 '18 at 01:26
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This post here gives the distribution of $\prod_{i=1}^n X_i=Z$(say). Now all you have to do is to perform a single variable transformation to find the distribution of $Z^{1/n}$. – StubbornAtom Jan 21 '18 at 09:59
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For the result see here https://math.stackexchange.com/a/3392284/198592 – Dr. Wolfgang Hintze Oct 19 '19 at 14:01