Problem 6. Let X have density fθ with a < θ < b. Suppose we are interested in testing H0 : θ ≥ θ0versus H1 : θ < θ0, where θ0 belongs to the interval (a, b). Suppose the test φ(X) is NP(fθ2 , fθ1 ) foreverya<θ1 <θ2 <b.

(1) Show that the power function π of the test φ(X) is nonincreasing. Recall π is defined byπ(θ) = Eθ(φ(X)), a < θ < b.

(2) Find the size of the test φ(X).

(3) Suppose δ(X) is another test with satisfies Eθ0(δ(X)) ≤ Eθ0(φ(X)). Show that Eθ(δ(X)) ≤π(θ) holds for every θ < θ0.

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