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Consider again the conditions of Exercise 19. Suppose now that we have a two-dimensional vector θ = (θ1, θ2), where θ1 and θ2 are real-valued parameters. Suppose also that A is a particular circle in the θ1θ2-plane, and that the hypotheses to be tested are as follows:
Show that if the test procedure δ is unbiased and of size α, and if its power function π(θ|δ) is a continuous function of θ, then it must be true that π(θ|δ) = α at each point θ on the boundary of the circle A. |
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