Question: 2. UMP test for U(0, 8). Let X = (Xl .. , X;,) be a sample from the uniform distribution on (0, 8). (i) For

2. UMP test for U(0, 8). Let X = (Xl" .. , X;,) be a sample from the uniform distribution on (0, 8). (i) For testing H : 8 80 against K : 8 > 80 any test is UMP at level a for which Esoep(X) =

a, Esep(X) a for 8 80 , and ep(x) = 1 when max(xl' " '' x,,) > 80 , (ii) For testing H: 8 = 80 against K : 8 ,;: 80 a unique UMP test exists, and is given by ep( x) = 1 when max(xl" ' " x,,) > 80 or max(x ••. . . , XII) 80 ':;a . and ep (x) = ° otherwise.

[(i): For each () > ()o determine the ordering established by r(x) = PIJ(x)/PlJo(x) and use the fact that many points are equivalent under this ordering. (ii): Determine the UMP tests for testing () = ()o against () < ()o and combine this result with that of part (i).]

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