Question: 12. Let X (X., X) be a sample from the uniform distribution U(0,0 + 1). (i) For testing H:00, against K: 0> 0 at level

12. Let X (X., X) be a sample from the uniform distribution U(0,0 + 1). (i) For testing H:00, against K: 0> 0 at level a there exists a UMP test which rejects when min(X,,..., X) > + C

(a) or max(XX) + 1 for suitable C(a). (ii) The family U(0, 0+ 1) does not have monotone likelihood ratio. [Ad- ditional results for this family are given in Birnbaum (1954) and Pratt (1958).] [(ii) By Theorem 2, monotone likelihood ratio implies that the family of UMP tests of H:00 against K: 0>0 generated as a varies from 0 to 1 is independent of 80]

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