Let be a random sample from the density function
where both and are positive unknown parameters.
i. Find a minimum sufficient statistic for .
ii. Find the maximum likelihood estimator for .
iii. Find the maximum likelihood estimator for .
My attempt at the answer.
i. Joint density for a sample :
Now I need to look at the ratio of two densities,
which will be contant as a function of if and only if , and
or and .
Therefore, is a minimal sufficient statistic for .
ii and iii. Finding MLE for
Use the likelihood function derived in (i)
MLE for .
I note that the likelihood function , if is an independent variable and all other variables held constant, , is increasing and does not attain local maximum. Using the condition , I claim that a attains maximum when it equals the minimum order statistic of the sample.
Therefore, can I say that MLE for is ?
MLE for .
I will maximise the log-likelihood function as follows.
Second order derivative is negative: therefore this is maximum.