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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.\ D.\ Qualifying Exam\\
  Friday, August 22, 2008

\end{center}

\vspace{.1in}

Please submit solutions to at most {\bf seven} problems.  You have
four hours.  No one is expected to answer all the problems correctly.
Partial credit will be given.  All problems are worth an equal amount
of credit.

\vspace{.2in}

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

\vspace{.2in}

\topic{Applied Statistics}


\QQ Consider the following unbalanced random one-way model:
\begin{eqnarray*}
Y_{ij} & = & \mu +\alpha_i +\epsilon_{ij}, \hspace{0.1in}
\hspace{0.2in} i=1,\ldots, k; \hspace{0.1in} j=1, \ldots, n_i,
\end{eqnarray*}
where  $\{\alpha_{i}, i=1, \ldots, k\}$ are iid  $N(0,\sigma_\alpha^2)$,
$\{\epsilon_{ij}, i=1, \ldots, k; j=1, \ldots, n_i\}$ are iid
 $N(0, \sigma_\epsilon^2)$, and the $\alpha_{i}$'s and $\epsilon_{ij}$'s are
 independent. Define
 \[ \mbox{SSA} = \sum_{i=1}^kn_i(\bar{y}_{i\cdot}-\bar{y}_{\cdot\cdot})^2 = \sum_{i=1}^k\frac{y_{i\cdot}^2}{n_i}
 - \frac{y_{\cdot\cdot}^2}{\sum_{i=1}^kn_i},\]
\[ \mbox{SSE} = \sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y}_{i\cdot})^2,\]
where
\[y_{i\cdot}=\sum_{j=1}^{n_i}y_{ij}, \hspace{0.1in} \bar{y}_{i\cdot}=\frac{y_{i\cdot}}{n_{i}}; \hspace{0.4in}
y_{\cdot\cdot} =\sum_{i=1}^k\sum_{j=1}^{n_i}y_{ij}, \hspace{0.1in}\bar{y}_{\cdot\cdot}=
\frac{y_{\cdot\cdot}}{\sum_{i=1}^kn_i}.\]


\begin{parts}
\PP
 Find the expectations and variances of $y_{i\cdot}$ and $y_{\cdot\cdot}$.
 What are the distributions of $y_{i\cdot}$ and $y_{\cdot\cdot}$?

\PP Find the expectation of SSA.

\PP What is the distribution of SSE? Find the expectation and variance of SSE.


\end{parts}

\QQ Consider the linear regression model:
\[Y = X\beta+\xi,\]
 where $Y =(y_{1},  \ldots, y_{n})'$,  $\xi=(\xi_{1}, \ldots, \xi_n)'$,
$\beta=(\beta_1, \ldots, \beta_p)'$,  and
  $X$  is an $n\times p$ full-rank matrix.
 $\{\xi_i, i=1, \ldots, n\}$  are assumed to be iid $N(0,\sigma^2)$ variables.
 The least squares estimate of $\sigma^2$ is given by
 \[\hat{\sigma}^2 = \frac{1}{n-p}\left[Y'(I-H)Y\right],\]
 where $H$ is the projection matrix $H=X(X'X)^{-1}X'$.


\begin{parts}

\PP
 Suppose that $A=(a_{ij})$ and $B=(b_{ij})$ are two $n\times n$ matrices. Show that trace($AB$) = trace($BA$)
 where trace($A$) =$\sum_{i=1}^na_{ii}$.

\PP
  Let $Z =(Z_{1},  \ldots, Z_{n})'$ be a random vector with  $\mbox{E}(Z)= \vecc{\mu}$ and
  $\mbox{Cov}(Z) = V$. Define $Q=Z'AZ$. Using the result in (a), show that
  \[ \mbox{E} (Q) = \mbox{trace}(AV) + \vecc{\mu}'A\vecc{\mu}.\]

\PP
Using the result in (b), find $\mbox{E}(\hat{\sigma}^2)$. Is $\hat{\sigma}^2$ an unbiased estimate of $\sigma^2$?

\end{parts}

\QQ In a generalized linear model (GLM), the response variable $Y$ is
assumed to have a density function with the exponential dispersion
form:
\begin{eqnarray*}
f(y; \theta, \phi) & = & \exp\{[y\theta-b(\theta)]/a(\phi)+c(y,\phi)\}.
\end{eqnarray*}
Consider the gamma density
\[f(y; \lambda, \beta) = \left\{\begin{array}{ll}
   \frac{\beta^\lambda}{\Gamma(\lambda)}y^{\lambda-1}e^{-\beta y}, & y>0, \\
   0,                                                             & y\leq 0,\\
   \end{array}\right.\]
   where $\beta>0$ and $\lambda>0$.

\begin{parts}

\PP
 Write the gamma density in the exponential dispersion form. Identify $\theta$, $b(\theta)$, $a(\phi)$, and $c(y,\phi)$
 in terms of $\beta$ and $\lambda$.

 \PP Let $l(\theta, y) =[y\theta-b(\theta)]/a(\phi)+c(y,\phi)$.
 Derive the mean and variance of $Y$ (in the general case) from the
 relations $E\left( \frac{\partial l}{\partial \theta}\right) = 0$ and
 $E\left( \frac{\partial^2 l}{\partial \theta^2}\right) + E\left(
   \frac{\partial l}{\partial \theta}\right)^2 = 0$. What are the mean
 and variance when $Y$ has the gamma density?

\end{parts}

\QQ \mbox{}

\begin{parts}

\PP For the following autoregressive (AR) process
in which $\{a_t\}$ is a white noise process, check the stationarity
condition.  Then calculate the autocorrelation $\rho_{k}$ for
$k=0,1,2,3$ (hint: one root of the following polynomial is $-2$):

$$X_{t}+2.1X_{t-1}+1.6X_{t-2}+0.4X_{t-3}=a_{t}.$$

\PP For the following moving-average (MA) process
in which $\{a_t\}$ is a white noise process, check the invertibility
condition.  Then calculate the autocorrelation $\rho_{k}$ for
$k=0,1,2,3$:

$$X_{t}=1+a_{t}+1.5 a_{t-1}+0.6a_{t-2}.$$
  
\end{parts}

\topic{Probability}

\QQ \mbox{}

\begin{parts}

\PP  Let $\mathcal A$ be a collection of sets on a space $\Omega$.
Show that $\mathcal A$ is a $\sigma$-field if and only if $\mathcal A$ is a $\pi$/$\lambda$ system. (Carefully state the definitions of a $\sigma$-field and $\lambda$-systems.)\\

\PP Let $P$ and $Q$ be two probability measures on $(\mathbb R, \mathcal B)$ with \[ P(-\infty,x]=Q(-\infty,x]\ \text{ for all } x\in\mathbb R.\] Show that $P(B)=Q(B)$ for all $B\in \mathcal B$, the collection of all Borel sets on $\mathbb R$. (Carefully state what result on $\pi$/$\lambda$ systems you use.) \\

\end{parts}


\QQ


\begin{parts}
\PP
Let $X_1, X_2, \ldots$ be i.i.d.\ with $\mathbb E X_i=0$ and $\mathbb E|X_i|<\infty$,  $ i=1,2,\cdots$. 
Show that \begin{enumerate}
\item the sequences $X_i$ and $Y_i := X_i 1\{ |X_i|\le i\}$, $i=1,2,\ldots$  are tail equivalent
and
\item $\displaystyle \left|  \frac1n \sum_{i=1}^n \mathbb E Y_i \right| \to 0\;\;$ as $n\to\infty$.\\
\end{enumerate}
\PP Let $S_n$ be a Binomial random variable with parameters $n$ and $p_n$ and Var$(S_n)\to\infty$ as $n\to\infty$.
Show that \[ \frac{S_n- \mathbb E S_n}{\sqrt{\text{Var}(S_n)}}\]
converges in distribution to a standard normal random variable as $n\to\infty$ by writing $S_n$ as a sum of independent random variables and verifying Lyapounov's condition.
\\
\end{parts}



\QQ Let $X_1,X_2,\dots$ be i.i.d.\ with common distribution
$N(1,1)$. Let $V_n = \frac{1}{n} \sum_1^n X_i$ and $M_n =$ median
of $\{X_1,\dots,X_n\}$ be the mean and median based on the first
$n$ random variables.
\begin{parts}
  \PP Write down the limiting distributions of $\sqrt{n}(V_n -
  1)$ and $\sqrt{n}(M_n - 1)$.

  \PP Let
$$ T_n =\left\{ \begin{array}{ll}
V_n  & {\rm if\ \ } |V_n| \le n^{-\frac{1}{4}}\\
M_n  & {\rm if\ \ } |V_n| >  n^{-\frac{1}{4}}\\
\end{array}
\right.
$$
in other words,
\[T_n = V_n I(|V_n| \le n^{-\frac{1}{4}}) + M_n I( |V_n| >  n^{-\frac{1}{4}}).\]
Obtain the limiting distribution of $\sqrt{n} (T_n-1) $. {\it
  Hint: First compute the limit of $E(I(|V_n| \le
  n^{-\frac{1}{4}}))= P(|V_n| \le
  n^{-\frac{1}{4}})$.}

\end{parts}


\QQ A clumsy climber is on a ladder with 6 rungs.  At any time, he is
either on the ground (state 0) or on one of the 6 rungs (states 1 to
6).  Let $i$ denote his current state, and suppose that his state at
the next time is determined by the following rule: If $1 \leq i \leq
5$, he either steps up one rung, stays where he is, steps down one
rung, or falls to the ground, with probabilities .4, .3, .2, .1,
respectively.  If $i=0$, he steps up one rung (with probability 1),
and if $i=6$, he steps down one rung (with probability 1).  Note that,
if $i=1$, he ends up on the ground (either by ``stepping'' or by
``falling'') with probability .3.  Let $X_n$ denote the climber's
state at time $n$.

\begin{parts}
  

\PP Write down the transition probability matrix for this Markov
chain.

\vspace{.1cm}

In the remaining parts, write out matrix expressions for your answers.
Any matrices appearing in your answers should be written out
explicitly.

\PP Suppose the climber is currently on rung $i$ where $i \geq 1$.
What is his expected number of visits to rung $j$ (where $j \geq 1$)
before visiting the ground (state 0)?  


\PP Let $\tau$ be the first time (after time zero) that the climber
reaches either the ground (state 0) or the top of the ladder (state
6).  That is, $\tau = \inf \{ n \geq 1 : X_n \in \{0,6\}\,\}$.
Assuming $1 \leq i,j \leq 5$ and $m>0$, evaluate the following:
\begin{displaymath}
E_i \sum_{k=0}^{\vecc{m}} I(X_k = j, \tau>k)
\;\;\;\;\;\;\;\;\textrm{(Note that the sum is only up to $m$.)}
\end{displaymath}


\PP If the climber starts from the ground (state 0), what is the
probability he will reach the top of the ladder (state 6) before ever
returning to the ground?

\end{parts}

\pagebreak

\topic{Theoretical Statistics}


\QQ Suppose we observe $X_1, \ldots, X_n$ iid from the density
$f(x \G \theta) = \frac{1}{2} e^{-|x-\theta|}$ where
$\theta$ is unknown.

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

\PP Is the statistic you found above complete?

\PP What would be your answers to (a) and (b) if instead
$f(x \G \theta) = \frac{1}{2\beta}
e^{-|x-\alpha|/\beta}$ where $\theta=(\alpha,\beta)$ is unknown?

\end{parts}

\QQ Suppose that we have two independent random samples:
$X_1,\ldots,X_n$ are exponential($\theta$) and $Y_1,\ldots,Y_m$ are
exponential($\mu$).

\begin{parts}

  \PP Find the likelihood ratio test (LRT) of $H_0 : \theta=\mu$
  versus $H_1: \theta \neq \mu$.

  \PP Show that the test in part (a) can be based on the statistic
  \begin{displaymath}
    T = \frac{\sum X_i}{\sum X_i + \sum Y_i}\,.
  \end{displaymath}

   \PP Find the distribution of $T$ when $H_0$ is true.

\end{parts}


%General Linear Models
\QQ

For the exponential family, the density function may be written:
\begin{equation}
\label{ExponentialFamily}
 f(y,\theta)=\exp\{a(y)b(\theta)+c(\theta)+d(y)\}
 \end{equation}



\begin{parts}

\PP  Find $\,\displaystyle \frac{\partial}{\partial \theta}f(y,\theta)\,.$

\PP Use this result to find $E(a(Y))$ where $Y$ has density
  $f(y,\theta)$.

\PP  Find the score function in terms of the functions
$a(y),b(\theta),c(\theta),d(y)$ and  their derivatives.

\PP  Show that the expected value of the score function is $0$.

\end{parts}

\pagebreak

% Vic: Multivariate Analysis question:
\QQ
The data matrix $\bf{x}$ for a random sample of size $n = 4$ from
a bivariate normal distribution is given by $\bf{x} = \left(
\begin{smallmatrix}
2  & 1 \\ 2  & 3 \\ 1 &  2 \\ 3 &  2
\end{smallmatrix}
\right).$

\begin{parts}
\PP Describe and sketch a 95\% confidence region for
the mean vector $\mu$ of this distribution. (Use the attached table.)

\PP Test $\mu = ( 3.5 \
\ 2.5 )^T$ at level $\alpha = 0.05.$

\end{parts}


\QQ For $i=1,\ldots,n$ exchangeable subjects/observations, the CPO
(conditional predictive ordinate) of the observation/subject $i$ is
given by the cross-validated probability:
\begin{displaymath}
  CPO_i = E_{\theta} 
\left[ f_i(y_{i,obs} \G \theta) \G \tilde{y}_{-i,obs} \right]
  = \int_{\Theta} f_i(y_{i,obs} \G \theta)
p(\theta \G \tilde{y}_{-i,obs})\,d\theta
\end{displaymath}
where $\tilde{y}_{-i,obs}$ is the observed data vector minus the
$i$-th observation, $f_i(y_{i,obs} \G \theta)$ is the sampling
density of observation $i$ evaluated at the observed value
$y_{i,obs}$, and $p(\theta \G \tilde{y}_{obs}) \propto \pi(\theta)
\prod_{j=1}^n f_j(y_{j,obs} \G \theta)$ is the posterior density
given the observed data $\tilde{y}_{obs}$.  An important measure of influence of the $i$-th observation on the posterior of $\theta$ is the K-L distance
\begin{displaymath}
  KL(p,p_{-i}) = \int_{\Theta} 
  \left[ \log \left\{\frac{p(\theta \G \tilde{y}_{obs})}
    {p(\theta \G \tilde{y}_{-i,obs})}\right\}\right]
 p(\theta \G \tilde{y}_{obs})
    \,d\theta\,.
\end{displaymath}

\begin{parts}
  
\PP Show that 
\begin{displaymath}
  (CPO_i)^{-1} = E_{\theta} \left[
    \{f_i(y_{i,obs} \G \theta)\}^{-1} \G \tilde{y}_{obs} \right]
  =\int_{\Theta}\{f_i(y_{i,obs} \G \theta)\}^{-1} 
  p(\theta \G \tilde{y}_{obs})\,d\theta\,.
\end{displaymath}

\PP How can you approximate $CPO_{i}$ if you have $N$ (large number)
of samples from the full posterior $p(\theta \G \tilde{y}_{obs})$?

\PP Show that 
\begin{displaymath}
  KL(p,p_{-i}) = \log \left[\int_{\Theta}\{f_i(y_{i,obs} \G \theta)\}^{-1} 
    p(\theta \G \tilde{y}_{obs})\,d\theta \right]
  + \int_{\Theta}\left[\log\{f_i(y_{i,obs} \G \theta)\} \right]
  p(\theta \G \tilde{y}_{obs})\,d\theta
\end{displaymath}

\PP What is the relationship between $CPO_i$ and $KL(p,p_{-i})$?

\end{parts}

\pagebreak

\topic{Biostatistics}


% Logistic Regression

\QQ Assume that one has a random sample of $n$ observations
$(\vecc{x}_i,y_i)$ for $i=1,\ldots,n$ where
\begin{displaymath}
\vecc{x}_i=\left ( \begin{matrix}
              1 \\
              x_{i1} \\
              x_{i2} \\
              \vdots \\
              x_{ip} \\
            \end{matrix} \right )
\;\;\;\mbox{and}\;\;\;
y_i=\begin{cases}
0 \text{  if the observation is not an event}\\
1 \text{  if the observation is an event}\\
\end{cases}
\end{displaymath}
Suppose you fit a logistic model to the data and derive
$\hat{\vecc{\beta}}$, the maximum likelihood estimator of
$\vecc{\beta}= (\beta_0, \beta_1, \beta_2,\ldots,\beta_p)^{\prime}$
(the true vector of parameters for the logistic model).  Let
$$\hat{p_i}=\frac{1}{1+\exp\{-\hat{\vecc{\beta}}'\vecc{x}_i\}}$$
denote the estimated probability that the $i^{th}$ individual develops disease.
Show that in this case, the following must be true:
$$\sum\limits_{i=1}^n(y_i-\hat{p_i})=0$$
That is, the expected number of cases must equal the observed number of cases.

% Clinical Trials
\QQ
Assume that an investigator wants to do a clinical trial testing
the difference between two normal means. She is willing to assume:

\begin{itemize}
\item $\sigma^2$ is the variance, assumed equal for the two groups.
\item $\Delta=\mu_1-\mu_2$ is the minimum difference that she thinks is clinically
important.
\item She is willing to accept a type I error of $\alpha$.
\end{itemize}


Because of cost considerations, the total sample size $n=n_1+n_2$ is
fixed. (Here $n_1$ and $n_2$ are the number to be randomized to the
first and second group, respectively.)\\

What are the values of $n_1$ and $n_2$ that maximize the power of the
study?

\pagebreak

% Statistics for Epidemiology
\QQ

\begin{parts}

\PP
When considering a sample with $k$ age groups the data are:

\begin{tabular}{cccc}
\hline
Age Group & \# at Risk & \# Events & Rate$=\frac{\text{\# Events}}{\text{\# at Risk}}$\\
\hline
1 & $n_1$ & $d_1$ & $r_1$\\
2 & $n_2$ & $d_2$ & $r_2$\\
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$\\
k & $n_k$ & $d_k$ & $r_k$\\
\hline
Total & $n$ & $d$ & $r$\\
\end{tabular}


Show that the crude rate, $r$, can be decomposed as:


$$r= \sum \limits_{i=1}^{k} p_i r_i$$

Where $p_i=n_i/n$ is the proportion of people in the $i^{th}$
age-group.

\PP
For two sample data, the data may be viewed as:

\begin{tabular}{cccc}
\hline
Age Group & \# at Risk & \# Events & Rate$=\frac{\text{\# Events}}{\text{\# at Risk}}$\\
\hline
\multicolumn{4}{c}{Sample 1}\\
1 & $n_{11}$ & $d_{11}$ & $r_{11}$\\
2 & $n_{21}$ & $d_{21}$ & $r_{21}$\\
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$\\
k & $n_{k1}$ & $d_{k1}$ & $r_{k1}$\\
\hline
Total & $n_1$ & $d_1$ & $r_1$\\
\hline
\multicolumn{4}{c}{Sample 2}\\
1 & $n_{12}$ & $d_{12}$ & $r_{12}$\\
2 & $n_{22}$ & $d_{22}$ & $r_{22}$\\
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$\\
k & $n_{k2}$ & $d_{k2}$ & $r_{k2}$\\
\hline
Total & $n_2$ & $d_2$ & $r_2$\\

\end{tabular}

Decompose the difference, $r_1-r_2$ into two sums, the 
first involving the difference in age specific rates, $r_{i1}-r_{i2}$, and the
second involving differences in the proportion of individuals in the different
age groups, $p_{i1}-p_{i2}$

\PP  Comment on what this means and suggest a way how we might deal with this.

\end{parts}

\pagebreak

\QQ Let the survival time $T>0$ be an integer-valued (discrete) random
variable with finite mean residual lifetime $\mu_k = E[T-k \G T>k]$
and discrete hazard rate $h_k = P[T=k \G T \geq k]$ for
$k=0,1,2,\ldots$

\begin{parts}
  
\PP Show that $\mu_0 - (1-h_1)\mu_1 = 1$.

Hint: One approach is to write down the terms in the sums for $\mu_0$
and $\mu_1$ in two rows, multiply each term of $\mu_1$ by $(1-h_1)$,
and subtract the rows.

\PP Can you generalize the result of (a) to $\mu_k -
(1-h_{k+1})\mu_{k+1}$?

\PP Use the results of (a) and (b) to show that the Geometric
distribution is the only discrete distribution with constant mean
residual lifetime.

\PP Show that the exponential distribution is the only continuous
distribution for which the mean residual lifetime $r(t)$ is constant
for all $t>0$.

\end{parts}


\topic{Computational Statistics}

\QQ Let $x$ and $y$ be two continuous random variables such that the
marginal probability  density of $x$ is $N(a_0, \sigma_0^2)$ and the conditional probability
density of 
$y$ given $x$ is  $N(bx, \sigma_1^2)$. 
\begin{parts}

\PP We know that the posterior density 
$P(x|y)$ is also Gaussian. Derive an expression for its mean.

\PP We can use this result to form the ``update step'' in the Kalman filter as follows.
Let the predictive density of a process $x_t$, given all the previous observations 
$(y_1, y_2, \dots, y_{t-1})$, be Gaussian with mean $\mu_{t|t-1}$ and variance 
$\sigma^2_{t|t-1}$. What is the mean of the posterior density of $x_t$ given the updated
data $(y_1, y_2, \dots, y_{t-1}, y_t)$? We can assume that the conditional 
density of $y_{t}$, given $x_t$, is Gaussian with 
mean $b x_t$ and variance $\nu^2$.

Assume all quantities to be scalar in this problem.

\end{parts}

\pagebreak

\QQ Consider a multinomial distribution
parameterized by $\theta$ according to:
$$ (x_1, x_2, x_3, x_4) \sim {\cal
M}(n;0.5 + 0.25\theta, 0.25(1 - \theta), 0.25(1 - \theta),
0.25\theta) \ .
$$
Our goal is to develop an EM algorithm for estimating $\theta$ 
from the given values of $x_1, \dots, x_4$. 

\begin{parts}

  \PP Define two new random variables $y_1$ and $y_2$ such that $x_1 =
  y_1 + y_2$, where $(y_1, y_2) \sim {\cal M}(x_1; 0.5/T,
  0.25\theta/T)$ with $T=0.5+0.25\theta$. Set the complete data to be
  $y = (y_1, y_2, x_2, x_3, x_4)$. The vector $y$ has multinomial
  distribution according to:
$$
y \sim {\cal M}(n;0.5, 0.25\theta, 0.25(1 - \theta),
0.25(1-\theta), 0.25 \theta) \ .
$$
What is the log-likelihood of the complete data?

\PP  Here we will derive an EM algorithm to estimate $\theta$.
\begin{enumerate}
\item {\bf E-Step}:  First, what is the expected value of the log-likelihood
of the complete data  with respect to the density
function $f(y_1, y_2|\theta_m, x_1, x_2, x_3, x_4)$?
Call that function: $Q(\theta|\theta_m,y)$. (Drop terms that do not depend on $\theta$)

\item {\bf M-Step}: Next, solve for 
$$
\theta_{m+1} = \argmax_{\theta} Q(\theta|\theta_m, y) .
$$
\end{enumerate}

(Hint 1:  For a multinomial random variable 
$x \equiv  (x_1, x_2, \dots,x_k) \sim {\cal
M}(n;p_1, p_2, \dots, p_k))$, the likelihood function is given by:
$$
f(x|p_1, p_2,\dots,p_k) \propto p_1^{x_1}p_2^{x_2}\dots  p_k^{x_k} \ .
$$



(Hint 2: You can use the fact that $E[y_2|\theta_m,x_1, x_2, x_3, x_4] = {\theta_m
\over 2 + \theta_m}x_1$ .)

\end{parts}




\end{document}

