Correct answer: (C) A UMP region is {Σln(1−e^(−x_i))>a}.
Explanation
For every β>1, the likelihood ratio against β=1 is βⁿexp[(β−1)Σln(1−e^(−x_i))]. It increases with the same statistic for every alternative, so a common upper-tail critical region maximizes power uniformly. Choose its cutoff to give null rejection probability α. **C**.
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