題組內容

1. (50%) Let $X_1, X_2, \cdots, X_n$ be a random sample from an uniform distribution $$f(x; \theta) = \frac{1}{\theta_2 - \theta_1}, x \in (\theta_1, \theta_2); \theta_1 < \theta_2.$$ Let the distribution function of $f(x; \theta_1, \theta_2)$ be denoted as $F(x; \theta_1, \theta_2)$. The detailed derivation has to be provided for all the following problem. No point is given when only answers are given.

(d) \( (10\%) \) Show that the sample mid-point \( (\bar{\theta}_2 + \bar{\theta}_1) / 2 \) is more efficient than the sample mean.