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Suppose $X_1,...,X_n$ are iid from a continuous distribution with pdf $$f(x) = \lambda e^{-\lambda(x-\theta)},\:\: x>\theta,\: \theta \in \mathbb{R}$$

What is the distribution of $\frac{nX_{(1)}}{\sum_{j=1}^nX_j}$? Note $X_{(1)} = \min(X_1,..,X_n)$.

Note: I am trying to define the rejection region of an LRT and I am stuck at the point where I'd have to determine the distribution of the ratio of random variables above. My major challenge is a result of the fact that the numerator and the denominator are not independent.

Xi'an
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papi
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    You could use the fact that $X_{(1)}$ and $\sum_{j=1}^n (X_j-X_{(1)})$ are independent, with well-known distributions of their own (Typo in the pdf; should be $x>\theta$). – StubbornAtom Aug 13 '21 at 21:10
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    https://stats.stackexchange.com/questions/252692/distribution-given-sum/252784#252784 shows you how to find this. – whuber Aug 13 '21 at 22:35
  • Check `dirichlet` as a keyword – Xi'an Aug 14 '21 at 05:24

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