Thread: Showing an estimate is an MLE

Let X be Poisson. Consider a random sample of size n from the X distribution, and let Y be the sample sum.

a) Show that Y/n is a maximum likelihood estimator of
b) Show that the estimator in part (a) is unbiased and consistent.
c) Also, show that Y/n is an efficient estimator of

I'm just confused on what I have to show for each part of the question. I know they each have their requirements for what you have to show for each, I just don't remember and I don't have my book with me to check.

Originally Posted by Janu42

Let X be Poisson. Consider a random sample of size n from the X distribution, and let Y be the sample sum.

a) Show that Y/n is a maximum likelihood estimator of
b) Show that the estimator in part (a) is unbiased and consistent.
c) Also, show that Y/n is an efficient estimator of

I'm just confused on what I have to show for each part of the question. I know they each have their requirements for what you have to show for each, I just don't remember and I don't have my book with me to check.

a) Your good on this one?

b1) Unbiased means E(lambHat) = lamb. (Definitely true)

b2) Consistent means lambHat -> lamb in probability. (True)

c) Compare to fisher (asymptotic) variance. Use Cramer-Rao bound. (They will be equal in this case)