(c-ii) \((3\%)\) Calculate the posterior mean and variance.

(c-iii) \((3\%)\) Denote \(\hat{p}B\) to be the posterior mean found in (c-ii). Find \(\alpha, \beta\) such that \(R(p, \hat{p}B)\) is constant.

(d) Consider three estimators $$\hat{p}_U = \text{UMVUE found in (b)},$$ $\hat{p}_B$ = the posterior mean with constant risk found in (c-iii), and $$\hat{p}T = \begin{cases} \hat{p}U \text{ with probability } \frac{n}{n+1}, \\ 1/2 \text{ with probability } \frac{1}{n+1}. \end{cases}$$ (d-i) \((3\%)\) Find \(\sup{p\in (0,1)}R(p,\hat{p}U)\) \(\sup{p\in (0,1)}R(p,\hat{p}B)\) , and \(\sup{p\in (0,1)}R(p,\hat{p}T)\) for a fixed \(n\)

(d-ii) \((3\%)\) Find the limiting distribution of \(\sqrt{n} (\hat{p}U - p)\) and \(\sqrt{n} (\hat{p}B - p)\), respectively.

(d-iii) \((3\%)\) Show that \(\hat{p}U\) and \(\hat{p}B\) are the consistent estimators of \(p\).

(d-iv) \((3\%)\) Show that \(\hat{p}_U\) is asymptotically efficient of \(p\).

(d-v) \((3\%)\) Find the asymptotic relative efficiency (ARE) of \(\hat{p}U\) with respect to \(\hat{p}B\).

4. (12%) Let X₁, X₂, ..., Xₙ be a random sample from N(μ, σ²), where both parameters are unknown. (a) \((4\%)\) Find a pivotal quantity of \(\mu\)

(b) \((4\%)\) Use the pivotal quantity of (a) to find a \(100(1 - \alpha)\%\) confidence interval (C.I.) for \(\mu\)

(c) \((4\%)\) Is your finding in (b) the shortest length C.I.? Prove your claim.

5. (8%) Let X be a sample of size 1 from a population P. (a) (4%) Find a uniformly most powerful (UMP) test of size α ∈ (0, 1/4) for H₀: P = P₀ versus H₁: P = P₁, where P₀ is N(0, 1) and P₁ is the double exponential distribution with the p.d.f. 1/4 e^(-|x|/2). [Hint: ∫₁^∞ (1/√2π) e^(-x²/2) dx ≈ 0.1587]

115年 - 115 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#147002

115年 - 113 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#146998

114年 - 114 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#147007

112年 - 112 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#147018

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 國立臺北大學_碩士班一般入學考試試題_統計學系：數理統計#146993

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