110年 - 110 國立臺北大學_碩士班一般入學考試試題_統計學系：數理統計#146993

(a) \( (10\%) \) Find the maximum likelihood estimators \( \bar{\theta}_1, \bar{\theta}_2 \) of \( \theta_1, \theta_2 \).

(b) \( (10\%) \) Find the joint distribution of \( \bar{\theta}_1,\bar{\theta}_2 \)

(c) (10%) Show that the sample mean \( \frac{1}{n} \sum_{i=1}^{n} X_i \) and the sample mid-point \( (\bar{\theta}_2 + \bar{\theta}_1)/2 \) are unbiased estimator of the midpoint \( (\theta_1 + \theta_2)/2 \).

(d) \( (10\%) \) Show that the sample mid-point \( (\bar{\theta}_2 + \bar{\theta}_1) / 2 \) is more efficient than the sample mean.

(e) (10%) Find the maximum likelihood estimate \( \hat{M} \) of the median of \( F(x; \theta_1, \theta_2) \) and the corresponding distribution of \( \hat{M} \).

i. Power function;

ii. Unbiased test;

iii.Monotone likelihood ratio (MLR);

iv. Likelihood ratio test (LRT).

i. Let $X_1, X_2, \ldots, X_n$ be iid Beta($\theta, 1$). Find a Most Powerful test of $$H_0: \theta = 1 \text{ v.s. } H_1: \theta = 2.$$

ii. Let $X_1, X_2, \ldots, X_n$ be iid Poisson($\lambda$). Find a Uniformly Most Powerful test of $$H_0: \lambda \le \lambda_0 \text{ v.s. } H_1: \lambda > \lambda_0.$$

iii. Let $X_{ij} \sim f(x_{ij}|\theta_i) = \frac{1}{\theta_i} e^{\frac{x_{ij}}{\theta_i}}, i = 1, 2, j = 1, 2, \ldots, n$ be 2 independent random samples from two exponential populations. Find a likelihood ratio test of $$H_0: \theta_1 = \theta_2 \text{ v.s. } H_1: \theta_1 \neq \theta_2$$