6.15 Let X1,...,Xn be iid with E(Xi) = , var(Xi) = 1, and E(Xi ) 4...

6.15 Let X1,...,Xn be iid with E(Xi) = θ, var(Xi) = 1, and E(Xi − θ)

4 = µ4, and consider the unbiased estimators δ1n = (1/n)X2 i − 1 and δ2n = X¯ 2 n − 1/n of θ 2.

(a) Determine the ARE e2,1 of δ2n with respect to δ1n.

(b) Show that e2,1 ≥ 1 if the Xi are symmetric about θ.

(c) Find a distribution for the Xi for which e2,1 < 1.

6.16 The property of asymptotic relative efficiency was defined (Definition 6.6) for estimators that converged to normality at rate √n. This definition, and Theorem 6.7, can be generalized to include other distributions and rates of convergence.