Argue that the proposed estimators λˆ and λ˜ below are both consistent and asymptotically normal. Then, give their asymptotic variances V(λˆ) and V(λ˜), and decide if one of them is always bigger than the other.

Question

Let X1,…,Xn∼i.i.d. Poiss(λ), for some λ>0. Let λ^=X¯¯¯¯n and λ~=−ln(Y¯¯¯¯n), where Yi=1{Xi=0},i=1,…,n.

V(λˆ)=

V(λ˜)