Suppose X1,..., Xn are i.i.d. N(, 1). Consider Hodges superefficient estimator of (unpublished, but cited in

Suppose X1,..., Xn are i.i.d. N(θ, 1). Consider Hodges’ superefficient estimator of θ (unpublished, but cited in Le Cam (1953)), defined as follows.

Let ˆ

θn be 0 if |X¯ n| ≤ n−1/4; otherwise, let ˆ

θn = X¯ n. For any fixed θ, determine the limiting distribution of n1/2(θˆ

n − θ). Next, determine the limiting distribution of n1/2(ˆ

θn − θn) under θn = hn−1/2. Is ˆ

θn regular?