You must show that . You need to note that this is the pdf of the Gamma distribution. And note in general,

Thus, .

Therefore, is unbiased.

Since we know that is unbiased, all we need to do is show that the MLE is . Then we can claim that its a minimum variance estimator.b) Show that theta hat =(1/r)*Ybar is a minimum variance estimator for theta.

Since , we see that

In this case, it would be best to maximize .

Therefore,

Now, to maximize, set

Therefore,

Since the MLE is equivalent to the unbiased estimator, we see that is the minimum variance estimator.

We can use some of the work we did for b) here.c) Find a sufficient statistic for theta.

Thanks in advance for any help.

We said that

Now applying the factorization theorem, we see that our choice for . Therefore, our . Thus, we can claim that is sufficient for .

Does this make sense?

If you need some additional clarifications, feel free to post back.