(d) (5%) Prove that \(\theta_{n}\) is a consistent estimator of \(\theta\).

(e) (5%) Find an unbiased estimator \(\tilde{\theta}_n\) of \(\theta\).

(f) (5%) Find the MLE of the median of the distribution for this sample.

Table 1: Area under the standard normal curve II. A. Let $X_1, X_2, \ldots, X_n$ be a random sample from a $U(0, \theta)$ population. (a) (7%) Consider $H_0: \theta = \theta_0$ versus $H_1: \theta \neq \theta_0$ and define $X_{(n)} = \max_{1 \leq i \leq n} X_i$. Show that $$T^*(X): \text{Reject } H_0 \text{ if } X_{(n)} > \theta_0 \text{ or } X_{(n)} \leq \theta_0 \alpha^{1/n}$$ is a uniformly most powerful (UMP) test of size $\alpha$.

(b) (7%) Find the uniformly most accurate (UMA) \(1 - \alpha\) confidence interval. Prove your claim.

(c) (7%) Is your finding in Problem b the shortest length confidence interval (C.I.)? Prove your claim.

B. Suppose that the random variables $Y_1, Y_2, \ldots, Y_n$ satisfy $$Y_i = \beta x_i + \varepsilon_i, \quad i = 1, 2, \ldots, n,$$ where $x_1, x_2, \ldots, x_n$ are fixed constants, and $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ are iid $N(0, \sigma^2)$ with $\sigma^2$ unknown. (a) (7%) Find a two-dimensional sufficient statistic for \((\beta, \sigma^2)\).

(b) (7%) Find the MLE of \(\beta\), and show that it is an unbiased estimator of \(\beta\).

(c) (7%) Find the UMVUE of \(\beta\).

C. (8%) Feller Weak Law of Large Numbers Suppose that $X_1, X_2, \ldots, X_n$ are iid random variables and define $S_n = \sum_{j=1}^n X_j$. If $$\lim_{x \to \infty} x P[|X_1| > x] = 0,$$ then $$\frac{S_n}{n} - E[X_1 1_{[|X_1| \leq n]}] \xrightarrow{p} 0 \text{ as } n \to \infty,$$ where $1_{[\cdot]}$ is an indicator function. Let $F$ be a symmetric distribution and suppose $$1 - F(x) = \frac{e}{2x \ln x}, \quad x \geq e.$$ Suppose that $\{X_n, n \geq 1\}$ is an iid sequence of random variables with comment distribution $F$. Show that the sample mean converges in probability to 0.

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