Problem 1 (qiuzhen-qualify 2025-spring)

Let \{U_n\} be i.i.d. with uniform distribution on [0,1]. For $n \ge 2$, define

$$X_n := \min\{U_1, \ldots, U_n\}, \quad Y_n := \max\{U_1, \ldots, U_n\}.$$

(1) Derive the joint distribution of $(X_n, Y_n)$.

(2) Calculate $E[X_n \mid \sigma(Y_n)]$.

Problem 2 (qiuzhen-qualify 2025-spring)

Suppose $(X,Y,Z)$ is a three-dimensional mean-zero Gaussian vector with covariance matrix

$$\begin{pmatrix}
1 & \rho & \tau \\
\rho & 1 & 0 \\
\tau & 0 & 1
\end{pmatrix}.$$

(1) Show that $\rho^2 + \tau^2 \le 1$.

(2) Show that there exist real numbers $a,b,c$ and a standard Gaussian random variable $W$ independent of $(Y,Z)$ such that $X = aY + bZ + cW$ and derive $a,b,c$.

Problem 3 (qiuzhen-qualify 2025-spring)

Suppose $\{\xi_j\}$ are i.i.d. Bernoulli random variables with $\Pr[\xi_1 = 1] = \Pr[\xi_1 = -1] = 1/2$. Set $S_n = \sum_{j=1}^n \xi_j$ and define $\tau = \min\{n : S_n = -a \text{ or } S_n = b\}$ for $a, b \in Z_{>0}$.

(1) Calculate $E[\mathrm{e}^{\lambda S_\tau}]$ for $\lambda \in \mathbb{R}$.

(2) Calculate $E[s^\tau]$ for $s \in (0,1)$.

Problem 4 (qiuzhen-qualify 2025-spring)

Suppose $\{\xi_j\}$ are i.i.d. Bernoulli random variables with $\Pr[\xi_1 = 1] = \Pr[\xi_1 = -1] = 1/2$. Set

$$X_n = \sum_{j=1}^n 2^{j-1} \xi_j.$$

(1) Define $T = \min\{j \ge 1 : \xi_j = +1\}$. Derive the law of $T$ and calculate $E[T]$.

(2) Show that $\{X_n\}$ is a martingale and calculate $E[X_T]$.

Problem 5 (qiuzhen-qualify 2025-spring)

Let $\{B_t\}_{t\ge 0}$ be a one-dimensional Brownian motion starting from the origin (i.e., $B_0 = 0$). For $a \in (0,\infty)$, define

$$T_a := \inf\{t \ge 0 : |B_t| = a\}.$$

(1) Prove that $T_a < \infty$ a.s.

(2) Prove that $T_a$ and $\mathbf{1}_{\{B_{T_a} = a\}}$ are independent.

Problem 6 (qiuzhen-qualify 2025-spring)

Suppose $X$ and $Y$ are i.i.d. with $E[X] = 0$ and $\operatorname{var}(X) = 1$. Suppose further that

$$E\big[(X - Y)^2 \mid \sigma(X + Y)\big] = 2.$$

Show that $X$ and $Y$ are independent standard Gaussian random variables.

Problem 7 (qiuzhen-qualify 2025-spring)

Let $X_1, \ldots, X_n$ ($n \ge 2$) be i.i.d. uniform $U(0, \theta)$ where $0 < \theta < \infty$.

(a) Under squared error loss $l(\theta, \delta(X)) = (\delta(X) - \theta)^2$, derive the generalized Bayes rule for the improper prior $\pi(\theta) = 1$, $0 < \theta < +\infty$.

(b) Is the Bayes rule derived in (a) consistent for $\theta$?

Problem 8 (qiuzhen-qualify 2025-spring)

Let $X_1, \ldots, X_n$ be i.i.d. $\mathrm{Gamma}(2025, \theta)$ with $\theta > 0$ unknown. Recall that the probability density function of a $\mathrm{Gamma}(a,\theta)$ is

$$f(x \mid a, \theta) = \frac{1}{\Gamma(a)\, \theta^{a}} x^{a-1} \mathrm{e}^{-x/\theta}, \quad x > 0,$$

where $\Gamma(a) = (a-1)!$ for a positive integer $a$.

(a) Consider the hypothesis testing problem $H_0 : \theta = 1$ vs. $H_1 : \theta \ne 1$. Derive the UMP level $\alpha = 0.05$ test if it exists. If it doesn't exist, please argue why it does not exist.

(b) Derive the MLE for $1/\theta$; what is the asymptotic distribution of this MLE for $1/\theta$?

Problem 9 (qiuzhen-qualify 2025-spring)

Let $(X_1, Y_1), \ldots, (X_n, Y_n)$ be i.i.d. random bivariate vectors. Let $\rho$ be the correlation coefficient between $X_i$ and $Y_i$. Let

$$\hat{\rho} = \frac{1}{(n-1)\sqrt{S_X^2 S_Y^2}} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y}),$$

where $\bar{X} = \sum_i X_i / n$, $\bar{Y} = \sum_i Y_i / n$, $S_X^2 = \sum_i (X_i - \bar{X})^2 / (n-1)$, $S_Y^2 = \sum_i (Y_i - \bar{Y})^2 / (n-1)$.

(a) Assume that $E|X_i|^4 < \infty$ and $E|Y_i|^4 < \infty$, prove that $\sqrt{n}(\hat{\rho} - \rho) \to \mathcal{N}(0, c)$ for some value $c$. Please identify $c$ in terms of moments of $(X_i, Y_i)$, however, there is no need to simplify your calculation on $c$.

(b) Further, assume that $(X_i, Y_i)$ follows a bivariate normal distribution that

$$\begin{pmatrix} X_i \\ Y_i \end{pmatrix} \sim N\!\left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \right).$$

The probability density function of the above bivariate normal distribution is given by

$$f(x,y \mid \rho) = \frac{1}{2\pi \sqrt{1 - \rho^2}} \exp\left\{ -\frac{1}{2(1 - \rho^2)}[x^2 - 2\rho xy + y^2] \right\}.$$

Find a minimal sufficient statistic for $\rho$. Is this minimal sufficient statistic complete or not? Please explain.

Problem 10 (qiuzhen-qualify 2025-spring)

Let $X_1, \ldots, X_n$ be i.i.d. from $N(0, \sigma^2)$ with $\sigma > 0$ unknown. Let $W = \frac{1}{n} \sum_{i=1}^n |X_i|$. Recall $\chi^2_k$ has probability density function

$$\frac{1}{2^{k/2}\, \Gamma(k/2)} x^{k/2-1} \mathrm{e}^{-x/2}, \quad x \ge 0.$$

(a) Find an unbiased estimator of $\sigma$ based on $W$ and identify its limiting distribution.

Problem 11 (qiuzhen-qualify 2025-spring)

Let $X_1,\ldots,X_n$ be iid from exponential distribution with density $f(x;\lambda,\alpha)=\lambda e^{-\lambda(x-\alpha)} I_{\{x\ge \alpha\}}$, where $\lambda>0$ and $-\infty<a<\infty$ are unknown parameters.

(a) Find the MLE of $(\lambda,\alpha)$, say $(\hat{\lambda},\hat{\alpha})$.

(b) What distribution does $\frac{\sqrt{n}(\hat{\alpha}-\alpha)}{n(\hat{\alpha}-\alpha)}$ converge to? You will receive partial credits if you can identify the distribution(s) of the numerator and/or denominator.