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., \( P[X_i \leq \xi_p] = p \).

(e) (5%) Suppose a random sample of size 49 is collected as follows:

0.06 0.06 0.09 0.15 0.17 0.17 0.18 0.21 0.23 0.29 0.29 0.31 0.34 0.34 0.37
0.41 0.41 0.42 0.44 0.45 0.46 0.47 0.47 0.49 0.49 0.49 0.50 0.53 0.56 0.57
0.57 0.58 0.58 0.59 0.59 0.60 0.63 0.64 0.66 0.68 0.68 0.71 0.74 0.78 0.81
0.85 0.85 0.91 0.97.

We obtain $\sum{i=1}^{49} xi = 23.84$ and $\sum{i=1}^{49} \log(xi) = -43.052$. Find $\hat{\xi}{0.6}$ and $\hat{\xi}{0.6}$.