Let X1,...,Xn be i.i.d. N(, 1). For testing = 0 against > 0, let n

Let X1,...,Xn be i.i.d. N(θ, 1). For testing θ = 0 against θ > 0, let φn be the UMP level α test. Let φ˜n be the test which rejects if X¯n ≥ bn/n1/2 or X¯n ≤ −an/n1/2, where bn = z1−α + n−1/4 and an is then determined to meet the level constraint. Are the tests asymptotically equivalent? Show that, for all

θ ≥ 0, 1 − Eθ(φn)

1 − Eθ(φ˜n)

→ 0 as n → ∞ .

How do you interpret this result? [Lehmann (1949)]