Assume X1,..., Xn are i.i.d. N(0, 2). Let 2 n be the maximum likelihood estimator of

Assume X1,..., Xn are i.i.d. N(0, σ2). Let σˆ 2 n be the maximum likelihood estimator of σ2 given by σˆ 2 n = n i=1 X2 i /n.

(i) Find the limiting distribution of √n(σˆ n − σ).

(ii) For a constant

c, let Tn,c = c n

i=1 |Xi|/n. For what constant c is Tn,c a consistent estimator of σ?

(iii) Determine the limiting distribution of √n(Tn,c − σ) with c chosen as your consistent estimator.

(iv) Determine the limiting distribution of √n log(σˆ n/Tn,c) (again with c chosen from (ii) above).