(a) (8%) Find the maximum likelihood estimator T_n(X) of θ, where X = (X_1, X_2, , X_n)'.

(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.

題組內容

(a) (8%) Find the maximum likelihood estimator \( T_{n}(X) \) of \( \theta \), where \( X = (X_{1}, X_{2}, \dots, X_{n})' \).

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