Problem 1 (yau-contest 2014)
Section: Probability and Statistics

Suppose that $X_n$ converges to $X$ in distribution and $Y_n$ converges to a constant $c$ in distribution. Show that

(a) $Y_n$ converges to $c$ in probability;

(b) $X_nY_n$ converges to $cX$ in distribution.

Problem 2 (yau-contest 2014)
Section: Probability and Statistics

Let $X$ and $Y$ be two random variables with $|Y|>0$, a.s.. Let $Z=X/Y$.

(a) Assume the distribution function of $(X,Y)$ has the density $p(x,y)$. What is the density function of $Z$?

(b) Assume $X$ and $Y$ are independent and $X$ is $N(0,1)$ distributed, $Y$ has the uniform distribution on $(0,1)$. Give the density function of $Z$.

Problem 3 (yau-contest 2014)
Section: Probability and Statistics

Let $(\Omega,\mathcal F,P)$ be a probability space.

(a) Let $\mathcal G$ be a sub $\sigma$-algebra of $\mathcal F$, and $\Gamma \in \mathcal F$. Prove that the following properties are equivalent:

(i) $\Gamma$ is independent of $\mathcal G$ under $P$,

(ii) for every probability $Q$ on $(\Omega,\mathcal F)$, equivalent to $P$, with $\frac{dQ}{dP}$ being $\mathcal G$ measurable, we have $Q(\Gamma)=P(\Gamma)$.

(b) Let $X,Y,Z$ be random variables and $Y$ is integrable. Show that if $(X,Y)$ and $Z$ are independent, then $E[Y|X,Z]=E[Y|X]$.

Problem 4 (yau-contest 2014)
Section: Probability and Statistics

Let $X_1,\ldots,X_n$ be i.i.d. $N(0,\sigma^2)$, and let $M$ be the mean of $|X_1|,\ldots,|X_n|$.

1. Find $c \in R$ so that $\hat\sigma = cM$ is a consistent estimator of $\sigma$.

2. Determine the limiting distribution for $\sqrt{n}(\hat\sigma - \sigma)$.

3. Identify an approximate $(1-\alpha)\%$ confidence interval for $\sigma$.

4. Is $\hat\sigma = cM$ asymptotically efficient? Please justify your answer.

Problem 5 (yau-contest 2014)
Section: Probability and Statistics

The shifted exponential distribution has the density function

$$f(y;\phi,\theta)=\frac{1}{\theta}\exp\{-(y-\phi)/\theta\},\qquad y>\phi,\ \theta>0.$$

Let $Y_1,\ldots,Y_n$ be a random sample from this distribution. Find the maximum likelihood estimator (MLE) of $\phi$ and $\theta$ and the limiting distribution of the MLE.

You may use the following Rényi representation of the order statistics: Let $E_1,\ldots,E_n$, be a random sample from the standard exponential distribution (i.e., the above distribution with $\phi=0,\theta=1$). Let $E_{(r)}$ denote the $r$-th order statistics. According to the Rényi representation,

$$E_{(r)} \overset{D}{=} \sum_{j=1}^r \frac{E_j}{n+1-j},\qquad r=1,\ldots,n.$$

Here, the symbol $\overset{D}{=}$ denotes equal in distribution.