Let X1,..., Xn be a sample from N(, 2), and consider the UMP invariant level- test of

Let X1,..., Xn be a sample from N(ξ, σ2), and consider the UMP invariant level-α test of H : ξ/σ ≤ θ0 (Section 6.4). Let αn(F) be the actual significance level of this test when X1,..., Xn is a sample from a distribution F with E(Xi) = ξ, V ar(Xi) = σ2 < ∞. Then the relation αn(F) → α will not in general hold unless θ0 = 0. Hint: First find the limiting joint distribution of √n(X¯ − ξ) and

√n(S2 − σ2).

Section13.3