I have a problem that I can't solution. Let $\mathbf{X}=\{X_1,X_2,\ldots,X_n\}\sim\mathrm{Uniform}(0,\theta)$ and we have $H_0:\theta=\theta_0$ and $H_1:\theta>\theta_0$. We reject the $H_0$ when $X_{(n)}>c$. Find $\mathbf{p\textbf{-}value}$.
I know that $\mathbf{p\textbf{-}value}=\mathbb{Pr}_{\theta_0}(T(\mathbf{X})>T(\mathbf{x}))$
where $\mathbf{x}$ is the observed value of $\mathbf{X}$, but next I don't know what is observed value and how to find $T(\mathbf{x})$.
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Richard Hardy
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What is $X_{(n)}$? I'm assuming this is some sort of test statistic that is a function of $\mathbf X$. – mhdadk Mar 31 '23 at 19:56
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$X_{(n)}=\max\mathbf{X}$ – Samvel Safaryan Mar 31 '23 at 19:59
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1You might be interested into this previous question about the tank problem. – Sextus Empiricus Mar 31 '23 at 20:05
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2If you know what $T(\mathbf{X})$ is, then $T(\mathbf{x})$ is naturally determined (and is then treated as a non-random value). Can you find $T(\mathbf{X})$ for this problem? – Zhanxiong Mar 31 '23 at 20:46
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2Uniform distributions are fodder for many textbook questions, so searching for related posts turns up promising hits. – whuber Mar 31 '23 at 20:53