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In my homework I have $X_1,...,X_n$ which are all uniformly distributed on $(0,\theta)$

I have concluded that the $MLE=\hat{\theta}=max(X_1,...,X_n)$ because:

$L_x(\theta)= \frac{1}{\theta^n}$ so the likelihood is a decreasing function. However we also know that $\theta > max(X_1,...,X_n)$, hence $\hat{\theta}=max(X_1,...,X_n)$

But I need to find an exact 95% confidence interval for $\theta$. I think the book wants me to find a pivot

I have tried to set $P(X \in (0,X_n))=(\frac{X_n}{\theta})^n$ for some $\theta>X_n$

Therefore $P(X \in (X_n,\theta))=1-(\frac{X_n}{\theta})^n=0.95$ which gives me

$\theta=\frac{X_n}{0.05^{1/n}}$

However I am pretty sure this is not the way to go and it is not a standard method. There should be some more standard method but using $L,\ell_X, \ell_X'=0$ just gives me MLE=0 which is of no use

Any help/hint would be appreciated

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  • $\begingroup$ Typo, I have changed it now $\endgroup$ – Daniel Apr 20 '20 at 18:53
  • $\begingroup$ Can you find the distribution of $\hat\theta$? $\endgroup$ – StubbornAtom Apr 20 '20 at 19:05
  • $\begingroup$ No, it doesn't have any standard distribution. A simulation shows it looks exponential but that doesn't help much. I tried calculating the upper bound of $\theta$ as $X_n/0.05^{1/n}=\theta_{upper}$ $\endgroup$ – Daniel Apr 21 '20 at 6:39
  • $\begingroup$ Distribution is standard. A pivot for $\theta$ is $\hat\theta/\theta$ (it has a standard distribution). Here is one particular confidence interval (in fact the shortest length interval) based on this pivot. $\endgroup$ – StubbornAtom Apr 21 '20 at 18:28
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Your estimator is definitely wrong. Please check your work. As for the confidence interval it will probably be a one sided confidence interval. If you will use the basic definition of confidence interval you will be good to go. Hint: $P(max(X_1, X_2, X_3,.....,X_n)< c) = \prod_{i=1}^{n}{P(X_i<c)}$

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  • $\begingroup$ Sorry, made a typo, the interval is $(0,\theta)$ $\endgroup$ – Daniel Apr 20 '20 at 18:53
  • $\begingroup$ My answer still stands. Could you show your further work? $\endgroup$ – Nisarg Jain Apr 21 '20 at 17:28
  • $\begingroup$ I have updated my question with the work so far $\endgroup$ – Daniel Apr 21 '20 at 18:19
  • $\begingroup$ check example 10. this will solve most of your doubts. $\endgroup$ – Nisarg Jain Apr 21 '20 at 20:21
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What you did was correct but except for some typo, it is not $X_n$ but max($X_i$), and do not forget the lower bound max($X_i$). (you already stated that $\theta\geq \hat{\theta}$)

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