Question: Suppose that X1, . . . , Xn form a random sample from the 2 distribution with unknown degrees of freedom ( = 1,

Suppose that X1, . . . , Xn form a random sample from the χ2 with unknown degrees of freedom θ (θ = 1, 2, . . .), and it is desired to test the following hypotheses at a given level of significance α0 (0 < α0 < 1):
H0: θ ≤ 8,
H1: θ ≥ 9.
Show that there exists a UMP test, and the test specifies rejecting H0 if log Xi ≥ k for some appropriate constant k.

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