1. ## MLE problem

find the MLE of the iid random variables of uniform distribution $\displaystyle U(2\theta,3\theta)$ where $\displaystyle \theta>0$

2. Obtain the max of

$\displaystyle {1\over \theta^n}I(2\theta<X_{(1)}<X_{(n)}<3\theta)$

where those are order stats and the I is an indicator function.
I had thought there would be two competing solutions, but the answer is quite simple.

3. Originally Posted by matheagle
Obtain the max of

$\displaystyle {1\over \theta^n}I(2\theta<X_{(1)}<X_{(n)}<3\theta)$

where those are order stats and the I is an indicator function.
I had thought there would be two competing solutions, but the naswer is quite simple.
i think the answer is$\displaystyle (\frac {X_{(1)}}{2},\frac{X_{(n)}}{3})$
but i do not know whether it is true?

4. Originally Posted by chialin4
i think the answer is$\displaystyle (\frac {X_{(n)}}{2},\frac{X_{(1)}}{3})$
but i do not know whether it is true?
neither, and the answer is not an interval.
you want theta as small as possible and that is $\displaystyle \frac {X_{(n)}}{3}$

5. Originally Posted by matheagle
neither, and the answer is not an interval.
you want theta as small as possible and that is $\displaystyle \frac {X_{(n)}}{3}$
but we do not know whether $\displaystyle \frac {X_{(n)}}{3}$ is greater than$\displaystyle \frac {X_{(1)}}{2}$ or not~
so which one should we chose?

6. Originally Posted by chialin4
but we do not know whether $\displaystyle \frac {X_{(n)}}{3}$ is greater than$\displaystyle \frac {X_{(1)}}{2}$ or not~
so which one should we chose?

$\displaystyle \frac {X_{(n)}}{3}<\theta<\frac {X_{(1)}}{2}$

7. Originally Posted by matheagle
$\displaystyle \frac {X_{(n)}}{3}<\theta<\frac {X_{(1)}}{2}$
so MLE is$\displaystyle \frac {X_{(n)}}{3}$?