題組內容

  1. (40%) Let \( X_1, X_2, \dots, X_n \) be a random sample from the distribution with a probability density function

\[ f(x; \theta) = \theta x^{\theta-1}, 0 < x < 1; \theta > 0. \]
Let the distribution function be denoted as \( F(x; \theta) \). For \( 0 < p < 1 \), define the \( p \)th quantile of this distribution to be
\( \xi_p = F^{-1}(p) \), i.e., \( P[X_i \leq \xi_p] = p \).

(d) (10%) Let \( 0 < X{(1)} \leq X{(2)} \leq \dots \leq X{(n)} \) denote the order statistics \( X{1}, X{2}, \ldots, X{n} \). Find \( E[F(X{(k)})] \), where \( k \doteq p(n + 1) \). Based on the result, find another estimator \( \hat{\xi}{p} \) of \( \xi_{p} \)?