density for order stat, sufficient stat, MLE question
Pls take a look, this is an exam practice question but no answer is provided.
Let be a random sample from a distribution with density function
where is an unknown parameter.
(a) Let . Find the density function of X.
(b) find a simple sufficient statistic for .
(c) find the maximum likelihood estimator for .
(d) compute the mean square error of the maximum likelihood estimator .
(a) I take it X is the maximum Y in the sample ie so I find the density of it
(b) to find sufficient statistics for , I will use Factorisation theorem, ie factorising the joint density function of the sample
the second component depends on the sample only; the first component depends on and on the sample through and can be a sufficient statistic for .
?? does this makes sense ??
(c) here is where I am stuck and cannot move on to (d)
To find MLE for , I will find maximum of log-likelihood of the density of Y
subject to a constraint
this can be equal to zero only if n=0 and this is not what I am looking for here...
Can you help me to spot where I've gone wrong?