(d) (8%) Based on T_n(X), find the unique minimum variance unbiased estimator of θ.

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. 2. (10%) Let $X_1, X_2, \cdots, X_n$ be a random sample from the distribution with a probability density function $$f(x; \theta) = \theta_1 e^{-\theta_1(x - \theta_2)}, x > 0; \theta_1 > 0; \theta_1 \in R.$$ Find the maximum likelihood estimators of $\theta_1$ and $\theta_2$. Table 1: Useful quantiles of the standard normal distribution Table 2: Useful percentiles of the Chi-square distribution $P[X \le x]$ where $X$ is a chi-square random variable with degrees of freedom $r$ 3. (30%) Let $Y_i i = 1, 2, \dots, n$, be i.i.d. discrete uniform distribution on $\{1, 2, 3, 4\}$. And let $$X_{ki} = \begin{cases} 1, & \text{if } Y_i = k \\ 0, & \text{if } Y_i \neq k \end{cases} \quad \text{for } k = 1, 2, 3, 4.$$ (a) $(5\%)$ Suppose that $\tilde{X}_k = \sum_{i=1}^{n} X_{ki}$. Find the mean and variance of $\tilde{X}_k$.

(d) (8%) Based on \( T_{n}(X) \), find the unique minimum variance unbiased estimator of \( \theta \).

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