1. ## max likelihood

suppose that Y1,...Yn are random samples from a uniform distribtution with
f(y l θ ) = 1/ (2θ +1) for 0<y<2θ +1.

find the max likelihood for θ.

my working:

L = product of 1/ (2θ +1) n times
= [1/ (2θ +1) ]^n

taking ln and then differentiation, i get (-2n) / (2θ +1)

but if i set that to 0, i do not get an answer.

may i know what went wrong?

2. Point to remember: When the range of the variable depends upon the unknown parameter of the distribution, special care should be taken. Here, $0; ie $y$ depends on $\theta$. So, just a differentiation won't do the job.

Hint: Do you realize the fact that the pdf will be maximum when $\theta$ is minimum? Also, do you know what are "order statistics"?

3. solve for theta where $y_{(n)}=2\theta +1$
where that y_n is the max of the y's known as the largest order stat.

4. ya i did realise that the pdf would be max when θ is min. i do not know what order stats are but based on wiki, i believe that if Yn is the largest among then it is greater than all other Yi of size n.

does that mean that P( Yn < fixed value ) = P(all Y < fixed value ) =[ Y/ (2θ +1) ] ^n

and then we differentiate [ Y/ (2θ +1) ] ^n to get the pdf? but i thought the pdf was already given in the question as 1/ (2θ +1)?

sorry, im new to this topic and am confused. thanks for helping

5. Then what you get after solving the equation $y_{(n)}=2\theta+1$ becomes your MLE. Can you get that?

6. no. i think im lost..what do you mean by solving the equation Y = (2θ +1)..

7. I mean express $\theta$ in terms of $y_{(n)}$.