Estimation, UMVUE, Cramer-Rao lower bound

• Suppose thatX1,...,Xnis a random sample of size n from a Normal(,2)distribution,

where << is unknown and >0 is known.

I'm interested in estimation of ()=exp(t) for a fixed t=0 , not equal to zero.

a) Find UMVUE of ().

b) Find variance of the estimator in part a).

c) Find the Cramer-Rao lower bound (CRLB) for the variance of an unbiased estimator of ().

d) Show that the variance in part b) is larger than the bound in part c), but their ratio converges to 1 as n .