(e) (5%) Why do you think statisticians are interested in the convergence of random variables?

(f) (10%) How does (a) to (e) give you any thought on the estimation for \(\theta\)? Briefly describe your thoughts, and justify your thoughts.

2. (50%) Let $X1, X2, \cdots, X_n$ be a random sample from a normal distribution $$f(x; \theta) = (2\pi\theta)^{-1/2} \exp \left( -\frac{1}{2\theta} (x - \mu)^2 \right), x \in R; \mu \in R, \theta > 0.$$ Let the distribution function of $f(x; \theta)$ be denoted as $\Phi(x; \mu, \theta)$. (a) (10%) Find the maximum likelihood estimators \(\hat{\mu},\hat{\theta}\) of \(\mu ,\theta\)

(b) (10%) Find the Rao-Cramer lower bound of \(\hat{\mu}\) and \(\hat{\theta}\), respectively.

(c) (10%) Find the asymptotic distribution of \(\hat{\mu},\hat{\theta}\)

(d) (10%) Find the asymptotic distribution of \(\Phi(x; \hat{\mu}, \hat{\theta})\). [Hint: Use Delta method and the derivatives with respect to \(\mu\) and \(\theta\) do not need to be expanded.]

(e) (10%) Find the Likelihood ratio test (LR test) for testing \( H0: \theta = \theta0 \) versus \( H0: \theta \neq \theta0 \), where \( \theta_0 \) is a pre-specified value and provide the corresponding critical value for the LR test.

112年 - 112 國立成功大學_碩士班招生考試_統計學系﹕數學#130253

111年 - 111 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#146986

110年 - 110 國立中央大學_碩士班招生考試_統計研究所/不分組(一般生、在職生):數理統計#105405

110年 - 110 國立中央大學_碩士班招生考試_財務金融學系/乙組(一般生):數理統計#105304

110年 - 110 國立政治大學_碩士班暨碩士在職專班招生考試_統計學系:數理統計學#103311

109年 - 109 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#146988