Suppose X1,..., Xk are independent, with Xi N(i, 1). Consider testing the null hypothesis 1 ==

Suppose X1,..., Xk are independent, with Xi ∼ N(θi, 1). Consider testing the null hypothesis θ1 =···= θk = 0 against max |θi| ≥ δ, for some δ > 0.

Find a maximin level α test as explicitly as possible. Compare this test with the maximin test if the alternative parameter space were i θ 2 i ≥ δ2. Argue they are quite similar for small δ. Specifically, consider the power of each test against (δ, 0,..., 0)

and show that it is equal to α + Cαδ2 + o(δ2) as δ → 0, and the constant Cα is the same for both tests.

Section 8.6