Suppose that X1, . . . , Xn form a random sample from a distribution that involves a parameter whose value is unknown, and

Suppose that X1, . . . , Xn form a random sample from a distribution that involves a parameter θ whose value is unknown, and the joint p.d.f. or the joint p.f. fn(x|θ) has a monotone likelihood ratio in the statistic T = r(X). Let θ0 be a specified value of θ, and suppose that the following hypotheses are to be tested:
H0: θ ≥ θ0,
H1: θ <θ0.
Let c be a constant such that Pr(T ≤ c|θ = θ0) = α0. Show that the test procedure which rejects H0 if T ≤ c is a UMP test at the level of significance α0.