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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.D.\ Qualifying Exam\\
  Monday--Tuesday, January 4--5, 2016

\end{center}

\vspace{.1in}

{\bf Put your solution to each problem on a separate sheet of paper.}

\vspace{.2in}


\QQ{5106} Find the maximum likelihood estimate of $\theta$ where
$\theta$ is a parameter in the multinomial distribution:
\begin{displaymath}
  (x_1, x_2, x_3, x_4) \sim M(n; 0.4(1+\theta), 0.3\theta, 0.6(1-2\theta), 0.5\theta)
\end{displaymath}

\begin{parts}

  \PP Choose a variable for the missing data and use the EM algorithm
  for iteratively estimating $\theta$.  Let $\theta^{(m)}$ be the
  current value of the unknown.  Derive the mathematical formula to
  update for $\theta^{(m+1)}$.

  \PP Let $L(\theta)$ denote the likelihood with the parameter
  $\theta$.  Prove that
  \begin{displaymath}
 L(\theta^{(m+1)}) \ge  L(\theta^{(m)}). 
  \end{displaymath}

\end{parts}


\QQ{5106} Let $x = (x_1, \cdots, x_n)$ be a given binary, Markovian
sequence.  In particular,
$$P(x_1 = 0) = 0.5,\; P(x_1 = 1) = 0.5,$$ and $$P(x_i = x_{i-1}) = p,\; P(x_i = 1 - x_{i-1}) = 1-p,$$ $i = 2, \cdots, n$.
Let $y= (y_1, \cdots, y_n)$ be a noisy observation of $x$.  That is,
$$y_i = x_i + e_i,$$
with $e_i \sim N(0, \sigma^2)$, $i = 1, \cdots, n$.

Assuming $p$ and $\sigma^2$ known, we can use the Maximum A Posteriori
method to reconstruct $x$ from $y$ in the following form:
$$\hat x =  \mbox{argmax}_{\{x_i\}} \sum_{i = 1}^n (-\frac{(y_i-x_i)^2}{2\sigma^2}) + \sum_{i=2}^n \log(1_{x_i = x_{i-1}}p + 1_{x_i \neq x_{i-1}}(1-p))$$

Write out a pseudocode for a Dynamic Programming procedure to compute
$\hat x$ in the computational order of $n$.

\pagebreak

\QQ{5166} Let $\{X_{1}, X_{2}, \ldots, X_{n}\}$  be  a sample generated from the model:
 \begin{eqnarray*}
 X_i & = & \epsilon_i + \theta \epsilon_{i-1},
 \end{eqnarray*}
 where the $\epsilon_i$'s are independent and identically distributed
 random variables with mean zero and variance $\sigma^2$.  Let
 $\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_{i}$.

\begin{parts}
\PP
 Find the variance of $\bar{X}$.

\PP
Let  $\dsp s^2  =\frac{\sum_{i=1}^{n}(X_{i}-\bar{X})^2}{n-1}.$
Find E($s^2$).

\PP
Compare Var$(\bar{X})$ with E($s^2/n$) for $\theta=0$,  $\theta>0$, and  $\theta<0$.
\end{parts}

\QQ{5166}

Consider a $2^{5-2}$ fractional factorial design with generators {\bf
  4=12} and {\bf 5=13}. After analyzing the results from the design
the experimenter decided to perform a second $2^{5-2}$ design exactly
the same as the first one but switching the signs of column {\bf 1}
from ``$+$" to``$-$" (a fold-over design).

\begin{parts}
\PP
 What are the advantages and disadvantages of a fractional factorial design compared with a full factorial design?


\PP What are the defining relation and design resolution of the first design? List the confounding patterns of all the two-factor interactions in the first design.


\PP What are the defining relation and design resolution of the combined  design? \newline List the confounding patterns of all the two-factor interactions
in the combined design.

\end{parts}

\pagebreak

\QQ{5167} In a one-way layout random-effects experiment, consider the
following linear model:
\begin{eqnarray*}
Y_{ti} & = & \mu +\tau_t + \epsilon_{ti}, \hspace{0.2in} t=1,\ldots, k;
\hspace{0.1in} i=1, \ldots, n_t,
\end{eqnarray*}
where  $\mu$ is the overall mean,     $\{\tau_{t}, t=1, \ldots, k\}$ are iid  $N(0,\sigma_\tau^2)$,
$\{\epsilon_{ti}, t=1, \ldots, k; i=1, \ldots, n_t\}$ are iid
 $N(0, \sigma_\epsilon^2)$, and the $\tau_{t}$'s and $\epsilon_{ti}$'s are
 independent.

\begin{parts}
  \PP Let $N=\sum_{t=1}^kn_t$,
  $\bar{Y}_{t\cdot}=\frac{1}{n_t}\sum_{i=1}^{n_t}Y_{ti}$, and
  $\bar{Y} = \frac{1}{N}\sum_{t=1}^k\sum_{i=1}^{n_t}Y_{ti}$.
  Calculate the expected values and variances of $\bar{Y}_{t\cdot}$
  and $\bar{Y} $.

\PP
Find Cov($\bar{Y}_{t\cdot}$, $\bar{Y}$).


\PP Let $S_T = \sum_{t=1}^kn_t(\bar{Y}_{t\cdot}-\bar{Y})^2$,
$S_R = \sum_{t=1}^k\sum_{i=1}^{n_t}(Y_{ti}-\bar{Y}_{t\cdot})^2$, \newline
and $S_D = \sum_{t=1}^k\sum_{i=1}^{n_t}(Y_{ti}-\bar{Y})^2$.
Show that $S_D = S_T + S_R$. \newline
Find the expected values of $S_T$ and $S_R$.
\end{parts}



\QQ{5167} Consider the linear regression model with $p$ predictors:
\[Y = X\vecc{\beta}+\vecc{e},\] where $Y =(y_{1}, \ldots, y_{n})'$,
$\vecc{e}=(e_{1}, \ldots, e_n)'$ with $e_1, e_2, \ldots, e_n$ iid $N(0,\sigma^2)$, $\vecc{\beta}=(\beta_0, \beta_1, \ldots, \beta_p)'$, and $X$ is
an $n\times (p+1)$ full-rank matrix.  Let $\hat{\vecc{\beta}}$ be the
least squares estimate of $\vecc{\beta}$.


\begin{parts}
\PP
Define  $\hat{Y}=X\hat{\vecc{\beta}}$ and $\hat{\vecc{e}} = Y-\hat{Y}$.
Find the distribution of $\hat{\vecc{e}}$.

\PP Let $\vecc{1} =(1, 1, \cdots, 1)'$, a column of ones with length
$n$ .  Define \[SS_{reg} = (\hat{Y} - \bar{Y}\vecc{1})'(\hat{Y} -
\bar{Y}\vecc{1}).\] Find the distribution of $SS_{reg}$ under $H_0:
\vecc{\beta}=0$.

\PP
 Show that the statistic
\[ F = \frac{SS_{reg}/p}{\hat{\vecc{e}}'\hat{\vecc{e}}/(n-p-1)}\]
has an $F$-distribution under $H_0: \vecc{\beta}=0$.

\end{parts}

\pagebreak

\mbox{}

\vspace{-1cm}

{\bf Put your solution to each problem on a separate sheet of paper.}

\vspace{.2in}


\QQ{5326} An urn contains $R$ red balls and $G$ green balls.  Ed draws
out $K$ balls one by one, randomly and with{\bf OUT} replacement.
Whenever Ed draws two red balls in a row he wins a dollar.  To be
precise, Ed receives a dollar after the $i^\text{th}$ draw if he draws
red balls on draws $i-1$ and $i$.  Note that under these rules, if Ed
draw 3 red balls in a row he receives 2 dollars; if he draws 4 red
balls in a row he receives 3 dollars, etc.  Let $X$ be the amount of
Ed's total winnings.

\begin{parts}

\PP Find $EX$.

\PP Find $\text{Var}(X)$.
  
\end{parts}

\QQ{5326} Consider the Gambler's Ruin problem: a gambler is betting on
the tosses of a {\bf fair} coin, winning \$1 when it comes up heads
and losing \$1 when it comes up tails.  The gambler has an initial
fortune of $z$ dollars and a goal of $g$ dollars, i.e., starting with
$z$ dollars, he continues playing until he reaches $g$ dollars or goes
broke (loses all his money).

\begin{parts}
  
  \PP Let $\psi(z)$ denote the probability of reaching the goal $g$ as
  a function of the initial fortune $z$.  Use the Law of Total
  Probability to find an equation that $\psi(z)$ must satisfy.  Then
  find $\psi(z)$ by obtaining a solution of this equation which
  satisfies $\psi(0)=0$ and $\psi(g)=1$.

  \PP {\bf Given that the gambler reached the goal}, what is the
  probability that his first six tosses had a {\bf total} of four
  heads and two tails?  (Use your answer to part (a).  To avoid
  complications, assume $6 \leq z \leq g-6$ so that the gambler makes
  at least six tosses.)

\end{parts}


\QQ{5327} Let $X_i$ be independently distributed as
$\mbox{Poisson}(\beta N_i)$ for $i=1, 2, \ldots, n$ where $\beta$ is
an unknown positive number and $N_1, N_2, \ldots, N_n$ are known
positive constants.

\begin{parts}
  
\PP Find the maximum likelihood estimate (MLE) of $\beta$. Call it $T_1$.

\PP Show that $T_1$ is unbiased for $\beta$. 

\PP Observe that $\dsp T_2 = \frac{1}{n} \sum_{i=1}^n \frac{X_i}{N_i}$ is
also an unbiased estimator of $\beta$.  Which one of $T_1$ and $T_2$
has smaller variance?  Prove your claim.

\end{parts}


\QQ{5327}
Let $X_1, X_2,\ldots ,X_n$ be i.i.d  from a distribution with pdf given by 
\begin{eqnarray*}
f(x \mid \theta) =
\begin{cases}
 \theta^{-c} c x^{c-1} e^{-(x/\theta)^c}, \,  \text{if}\,   x > 0\\
 0, \, \text{otherwise}, 
 \end{cases}
\end{eqnarray*}
where $c > 0$ is a known constant and $\theta > 0$ is the unknown parameter. 

\begin{parts}
  
\PP  Find a complete sufficient statistic for $\theta$.    

\PP Find the minimum variance unbiased estimator (MVUE) for $\theta$.   (Hint:  Find an unbiased statistic by taking a suitable power of the complete sufficient statistic.)

\PP  Find the  most powerful (MP) test of size $\alpha$ for testing
\begin{eqnarray*}
H_0: \theta = \theta_0, \quad \text{vs} \quad H_1:  \theta = \theta_1,
\end{eqnarray*}
where $0 < \theta_0 < \theta_1$ are known constants. 

\end{parts}


\QQ{6346} Suppose $(\Omega, \mathcal{F}_0, \mu)$ is a probability
space, $\mathcal{F}$ is a $\sigma$-field with
$\mathcal{F} \subset \mathcal{F}_0$, $X:\Omega \rightarrow \mathbb{R}$
is measurable with respect to $\mathcal{F}_0$, and
$\mathbb{E}|X|<\infty$.

\begin{parts}
  
\PP Define conditional expectation, $\field{E}(X|\mathcal{F})$.

\PP Let $Y=I_A$ and $X=I_B$ for some $A, B\in \mathcal{F}_0$ with $0<\mu(A)<1$.  Let $\mathcal{G}=\sigma(Y)$.  Find $\field{E}(X|\mathcal{G})$.

\PP What happens if $A$ and $B$ are the same set?

\end{parts}

\QQ{6346} Let $B(t)$ be standard Brownian motion.

\begin{parts}
  
  \PP Show that the joint density of
  $B(t_1)=X_1, B(t_2)=X_2, \ldots, B(t_n)=X_n$ where
  $0<t_1 < \cdots <t_n$ is a product of univariate normal densities.

\PP Find $E[B(t)|B(s)]$, $0\leq s \leq t$.

\PP Define $X(t)=B(1/t), t>0$, and $X(0)=0$.  Describe the process
$X(t)$ in terms of its mean and covariance.  Is $X(t)$ Brownian
motion?  Is it a Gaussian process?

\end{parts}

\end{document}

