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)dΛ(µ) , where Λ is a probability distribution on the real line. Suppose that Λ is symmetric about 0; that is, Λ{E} = Λ{−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 Λ is not symmetric.