1. ## Maximum Likelihood Estimator

Let Y1<Y2<...<Yn be the order statistics of a random sample from a distribution with pdf $\displaystyle f(x; \theta) = 1$, $\displaystyle \theta - 0.5 < x < \theta + 0.5$. Show that every statistic u(X1,X2,...,Xn) such that $\displaystyle Y_n - 0.5<u(X_1,X_2,...,X_n)<Y_1 + 0.5$ is a mle of theta. In particular $\displaystyle (4Y_1 + 2Y_n + 1)/6, (Y_1 + Y_n)/2, (2Y_1 + 4Y_n - 1)/6$ are three such statistics.

I don't even know how to start this problem. Can someone please point me in the right direction?

2. The likelihood function is

$\displaystyle I(\theta-.5<X_{(1)}<X_{(n)}<\theta+.5)$

(which shows that the MLEs are the min and the max).

This fuction is either 0 or 1.
It's not calculus, you just need to make this equal to one.

JUST take your three $\displaystyle \hat \theta$s and make sure that these holds...

$\displaystyle \hat\theta-.5<Y_1$ and $\displaystyle Y_n<\hat\theta+.5$

SOLVE for $\displaystyle \hat\theta$ and you get that U!!!

I plugged in $\displaystyle \hat\theta={Y_n+Y_1\over 2}$ and these are equivalent to

$\displaystyle Y_n-Y_1<1$ which is true!

3. I have a question about
$\displaystyle I(\theta-.5<X_{(1)}<X_{(n)}<\theta+.5)$

Is that the same thing as $\displaystyle L( \theta) \theta-.5<X_{(1)}<X_{(n)}<\theta+.5)=$ (for a likelihood function) because I thought $\displaystyle I( \theta)$ was the fisher information function.

and what does the 'u' in u(X1,X2,...,Xn) suppose to represent?

4. Originally Posted by statmajor
$\displaystyle I(\theta-.5<X_{(1)}<X_{(n)}<\theta+.5)$

Is that the same thing as $\displaystyle L( \theta) \theta-.5<X_{(1)}<X_{(n)}<\theta+.5)=$ (for a likelihood function) because I thought $\displaystyle I( \theta)$ was the fisher information function.

and what does the 'u' in u(X1,X2,...,Xn) suppose to represent?
NO, I use I as an indicator function
I(A)=1 if the event A occurs
I(A)=0 if A does not

U(X1,...,Xn) is just a statistic, it's a function of the data
(That's the definition of a stat)

BY the way, this was a fun problem,
I would love to use it in a class some day,
but I doubt my students would understand it.

5. Sorry to bother you once again, but it seems that I got stuck once more.

You said to plug in $\displaystyle \hat\theta={Y_n+Y_1\over 2}$, and I plugged it into $\displaystyle \hat\theta-.5<Y_1$ which gave me $\displaystyle {Y_n+Y_1\over 2} -.5<Y_1$

Im not not sure how you got Y_n-Y_1<1[/tex](Probably doing something stupid).

6. Originally Posted by statmajor
Sorry to bother you once again, but it seems that I got stuck once more.

You said to plug in $\displaystyle \hat\theta={Y_n+Y_1\over 2}$, and I plugged it into $\displaystyle \hat\theta-.5<Y_1$ which gave me $\displaystyle {Y_n+Y_1\over 2} -.5<Y_1$

Im not not sure how you got Y_n-Y_1<1[/tex](Probably doing something stupid).
multiply that by two.....

$\displaystyle Y_n+Y_1 -1<2Y_1$

do I need to continue?

7. Originally Posted by matheagle
multiply that by two.....

$\displaystyle Y_n+Y_1 -1<2Y_1$

do I need to continue?