Suppose P{I = 1} = p = 1 P{I = 2}. Given I = i, X N(, 2 i ), where 2 1

Suppose P{I = 1} = p = 1 − P{I = 2}. Given I = i, X ∼

N(θ, σ2 i ), where σ2 1 < σ2 2 are known. If p = 1/2, show that, based on the data (X, I), there does not exist a UMP test of θ = 0 vs θ > 0. However, if p is also unknown, show a UMPU test exists. [See Examples 10.20-21 in Romano and Siegel (1986).]