題組內容

1. (40%) Let $X_1, X_2, \cdots, X_n$ be a random sample from the distribution with a probability density function
$f(x; \theta) = \theta(1 - x)^{\theta - 1}, 0 < x < 1; \theta > 0.$

 

(e) (8%) Consider \( H_0: \theta = 2 \) versus \( H_a: \theta > 2 \). Suppose a random sample of size 50 is collected as follows:
0.68 0.21 0.53 0.06 0.47 0.58 0.56 0.29 0.81 0.42 0.85 0.09 0.97 0.57 0.17 0.64 0.49 0.41 0.59 0.47 0.18 0.17 0.31 0.41 0.58 0.66 0.74 0.49 0.71 0.50 0.45 0.23 0.68 0.60 0.06 0.91 0.63 0.57 0.49 0.34 0.59 0.46 0.29 0.34 0.15 0.78 0.37 0.44 0.85 0.38

We find $50 - \sum_{i=1}^{50} x_i = 25.77$ and $\sum_{i=1}^{50} \log(1 - x_i) = -17.43$. Construct a Wald-type test to these hypotheses. Use this test to make inference about this sample given the significance level 0.05.