題組內容

3. (50%) Let $X1, X2, \dots, X_n$ be a random sample of size $n$ from a distribution that has pdf

$$f(x) = \theta(1 - x)^{\theta-1}, \quad 0 < x < 1, \theta > 0.$$

(c) (10%) Is the MLE from (a) the UMVUE of $ 1 / \theta $? If yes, show it. If not, find the UMVUE of $ 1 / \theta $. Hint: Consider $ Y{i} = \ln (1 - X{i}) $.

題組內容

(c) (10%) Is the MLE from (a) the UMVUE of \( 1 / \theta \)? If yes, show it. If not, find the UMVUE of \( 1 / \theta \). Hint: Consider \( Y{i} = \ln (1 - X{i}) \).

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