Problem 1 (yau-contest 2023)

The sequence $(X_1, X_2, \ldots, X_n, \ldots)$ is a Dirichlet process with base distribution $G_0$ and concentration parameter $\alpha_0>0$ if $G_0$ is a probability distribution on $R$ and satisfies:

- $X_1 \sim G_0$.
- Conditional on $X_1, X_2, \ldots, X_n$, the distribution of $X_{n+1}$ is $\alpha_0 G_0+\sum_{i=1}^n \delta_{X_i}$, appropriately normalized, where $\delta_x$ is the Dirac measure with probability $1$ on singleton $\{x\}$.

Assume that $G_0$ has finite first and second moments.

(a) Derive the distribution of $X_n$, $n\ge 1$.

(b) Let $Y_n=I(X_n>0)$. Prove or disprove $(Y_n)_{n\ge 1}$ forms a Dirichlet Process. If $(Y_n)_{n\ge 1}$ forms a Dirichlet Process, determine its concentration parameter and its base distribution.

Problem 2 (yau-contest 2023)

Suppose $X$ and $Y$ are non-negative random variables on a probability space $(\Omega,\mathcal{F},P)$. Let $H(x,y)$ be a function on $[0,\infty)^2$ such that $E(|H(X,Y)|)<\infty$. Define function $\phi(u)=u/(1+u)$ for $u\ge 0$. For integer $n=0,1,2,\ldots$, let
$$U_n=\sum_{j=1}^{2^n} \frac{1}{2^n} I\left( \frac{j-1}{2^n} \le \phi(X) < \frac{j}{2^n} \right),\qquad V_n=E(H(X,Y)\mid U_n).$$
Prove or disprove that there exists a random variable $Z$ such that as $n\to\infty$, $V_n$ converges to $Z$ almost surely. If there exists such a $Z$, show the expression of $Z$ (in the sense of almost surely).

Problem 3 (yau-contest 2023)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. random variables with a uniform distribution on $(0,1)$. Define the events $A_1,A_2,\ldots$ by
$$A_n=\{X_n=\max(X_1,\ldots,X_n)\}.$$
Define $R_n=\sum_{k=1}^n I(A_k)$. Let $(m_n)$ be a sequence of positive numbers such that $\lim_{n\to\infty} m_n=\infty$. Compute the following limit
$$\lim_{n\to\infty} P\big(|R_n-\log n|>m_n\sqrt{\log n}\big).$$

Problem 4 (yau-contest 2023)

Denote by $\mathcal{B}$ the Borel sigma field on the real line $R$, and let $(\Omega,\mathcal{F})$ be a measurable space. Define a mapping $Q(t,A)$ for $t\in R$ and $A\in\mathcal{F}$ such that $Q(t,\cdot)$ is a probability measure on $(\Omega,\mathcal{F})$ for each $t\in R$, and $Q(\cdot,A)$ is a Borel function for each $A\in\mathcal{F}$. Denote by $\Pi$ a probability measure on $(R,\mathcal{B})$, $P$ a probability measure on $(\Omega,\mathcal{F})$, and $T$ a random variable on $(\Omega,\mathcal{F},P)$. Assume that $\Pi$, $P$ and $T$ satisfy $\Pi=P\circ T^{-1}$, and for $A\in\mathcal{F}$,
$$P(A)=\int_R Q(t,A)\,\Pi(dt),$$
where $P\circ T^{-1}$ denotes the induced probability measure by $T$. Prove or disprove the following statement:
For any $A\in\mathcal{F}$, $P(A\mid T)=Q(T,A)$ almost surely.

Problem 5 (yau-contest 2023)

Four statisticians I, II, III and IV play a sequence of games. For each game, the winning probabilities of I, II, III and IV are $(1-\theta)/2$, $(1-\theta)/2$, $\theta/2$ and $\theta/2$, respectively, where $0<\theta<1$. There is only one winner in each game and no tie is allowed. Assume that outcomes of games are independent of each other. For a fixed integer $r\ge 2$, the stopping rule is to terminate as soon as one of the following conditions hold: (1) I and II together win $r$ games; (2) III and IV together win $r+1$ games. At the time of termination, let $X_1,X_2,X_3$ and $X_4$ denote the numbers of games won by I, II, III and IV, respectively.
(a) Prove or disprove the statistic $T=(X_1+X_2, X_3+X_4)$ is complete.
(b) Find a uniformly minimum variance unbiased estimator of $\theta$.

Problem 6 (yau-contest 2023)

A system of interest involves three random variables, $X$, $Y$, and $S$, where $S$ has a Poisson distribution with mean $2\lambda$, for a parameter $\lambda>0$, and where $X$ and $Y$ are conditionally independent Bernoulli variables, given $S=s$, with
$$P(X=1\mid S=s)=\frac{1}{2^{s+1}}\quad\text{and}\quad P(Y=1\mid S=s)=\frac{\theta}{2^s},$$
where $\theta\in(0,1)$ is a second parameter. The random variable $S$ is unobservable. We have $n$ i.i.d. copies $(X_i,Y_i)$ of $(X,Y)$.
(a) An intuitive estimator for $\theta$ is $\hat{\theta}_n=\bar{Y}_n/(2\bar{X}_n)$, where $\bar{X}_n=\frac{1}{n}\sum_{i=1}^n X_i$ and $\bar{Y}_n=\frac{1}{n}\sum_{i=1}^n Y_i$. Derive its asymptotic distribution.
(b) Consider the hypothesis testing problem
$$H_0:\ \theta=\frac{1}{2}\quad\text{versus}\quad H_1:\ \theta\ne \frac{1}{2}.$$
Construct an exact test statistic. As $\theta$ moves away from $1/2$, describe all sources of increasing power that you can think of.