50 Hodges'Example of Super-Efficiency Suppose thatXn=(X1,...,Xn)where the Xi's are i.i.d. Normal(uo,1).We can show that I(uo)=1 for all uo.Consider the following estimator of uo, n|xn≥n-1/4 )|xml<n-1/4 Hodges showed that: VHECN. N(0,1)ifo≠0 The latter variance makes the asymptotic distribution of the MLE inadmissible.50 Hodges’ Example of Super-Efficiency Suppose that Xn = (X1,...,Xn) where the Xi’s are i.i.d. Normal(µ0, 1). We can show that I(µ0) = 1 for all µ0. Consider the following estimator of µ0, µ ˆ(Xn) = ⎧⎨⎩ X¯ n |X¯ n| ≥ n−1/4 0 |X¯ n| < n−1/4 Hodges showed that: √n(ˆµ(Xn) − µ0) →D ⎧⎨⎩ N(0, 1) if µ0 = 0 0 if µ0 = 0 The latter variance makes the asymptotic distribution of the MLE inadmissible