Suppose X1,..., Xn are i.i.d. N(, 2) with known. For testing = 0 versus

Suppose X1,..., Xn are i.i.d. N(ξ, σ2) with σ known. For testing

ξ = 0 versus ξ = 0, the average power of a test φ = φ(X1,..., Xn) is given by

∞

−∞

Eξ (φ)d(ξ) , where  is a probability distribution on the real line. Suppose that  is symmetric about 0; that is, {E} = {−E} for all Borel sets E. Show that, among α level tests, the one maximizing average power rejects for large values of |



i Xi|. Show that this test need not maximize average power if  is not symmetric.