Question: Consider a random sample of size n from a Poisson distribution, Xi ~ POI (mu).

(a) Find the CRLB for the variances of unbiased estimators of mu.

(b) Find the CRLB for the variances of unbiased estimators of theta = e^-mu.

(c) Find a UMVUE of mu.

(d) Find the MLE theta hat of theta.

(e) Is theta hat an unbiased estimator of theta?

(f) Is theta hat asymptotically unbiased?

(g) Show that theta tilde = [(n-1)/n]^(sum of Xi) is an unbiased estimator of theta.

Hint: Note that Y = Sum of Xi ~ POI (n mu), and that E (theta tilde) and Var (theta tilde) are related to the MGF of Y.