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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.\ D.\ Qualifying Exam\\
  Thursday, August 24, 2006

\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 a $2^{5-2}$ fractional factorial design with generators
{\bf I=1234} and {\bf I=135}.  After analyzing the results from this
design the researcher decides to perform a second $2^{5-2}$ design
exactly the same as the first but with signs changed in column {\bf 3}
of the design matrix.

\begin{parts}

  \PP What is the defining relation of the first design? What is the
  resolution of the first design?  List the aliases of main effects in
  this design.  

  \PP What is the defining relation of the second design? What is the
  resolution of the second design?  List the aliases of two-term
  interactions of factors {\bf 1, 2,} and {\bf 3} in this design.

  \PP What is the defining relation of the combined design (designs 1
  and 2 together)?  What is the resolution of the combined design?
  (Hints: combine the defining relations of the two designs together
  and remove all the defining words that have both``$+$" and ``$-$"
  signs.)

  \PP Can you modify the first and second designs such that the
  combined design has a higher resolution?

\end{parts}


\QQ Consider the logistic regression model
\begin{eqnarray}\label{logit}
  \pi(x) & = & \frac{\exp(\alpha + \beta x)}{1+\exp(\alpha + \beta x)}, 
\hspace{0.2in} -\infty < x < \infty.
\end{eqnarray}

\begin{parts}

  \PP For given constants $\alpha$ and $\beta>0$, show that $\pi(x)$
  is a distribution function.  What is the density function of this
  distribution function?

  \PP Show that the logistic density function
  \begin{eqnarray*}
    f(x) = \frac{\exp(x)}{[1+\exp(x)]^2}, \hspace{0.2in} -\infty < x < \infty
  \end{eqnarray*}
  is symmetrical about zero. Based on this fact, derive the
  expectation of a random variable $X$ that has $\pi(x)$ ($\beta>0$)
  as its distribution function.

\end{parts}



\QQ Consider the following multiple linear regression model based on
centered data
\begin{eqnarray}\label{MR}
  Y_i-\bar{Y} & = & \beta_1 (X_{1i}-\bar{X}_1) +\cdots +
  \beta_p(X_{pi}-\bar{X}_p) +e_i, \hspace{0.1in} i=1, \ldots, n.
\end{eqnarray}
where $\{e_1, \ldots, e_n\}$ are iid normally distributed with mean
zero and variance $\sigma^2$.

\begin{parts}
  \PP Derive the least squares estimates of $(\beta_1, \ldots,
  \beta_p)$ in model (\ref{MR}) using matrix notation. What is the
  least squares estimate of the variance $\sigma^2$?

  \PP When the $p$ predictors are orthogonal, i.e.,
  \begin{eqnarray*}
    \sum_{i=1}^n(X_{ji}-\bar{X}_j)(X_{ki} -\bar{X}_k) = 0, &
    \mbox{for} &  k\neq j.
  \end{eqnarray*}
  show that the least squares estimates of $(\beta_1, \ldots,
  \beta_p)$ based on model (\ref{MR}) are the same as those based on
  the simple linear regression models:
  \[Y_i-\bar{Y}=\beta_j (X_{ji}-\bar{X}_j) +e_i, \hspace{0.1in} i=1,
  \ldots, n; \hspace{0.05in} j=1, \ldots, p.\]

  \PP Under the condition in part (b), show that the least squares
  estimates $\{\hat{\beta}_1, \ldots, \hat{\beta}_p, \hat{\sigma}^2\}$
  are independent of each other.
\end{parts}



\topic{Probability}


\QQ Let $U$ be a Uniform(0, 1) random variable. Define
\begin{displaymath}
Y_n = n 1_{(0, 1/n)}(U) - n1_{[1/n, 2/n)}(U).
\end{displaymath}
Use Vitali's theorem to show that $\{Y_n\}_n$ is not uniformly
integrable, but that $\int Y_n dP \rightarrow 0$.

\QQ Let $X_1, \ldots, X_n$ be i.i.d. random variables with common
cumulative distribution function $F(x) = 1 - e^{-x}$. Define $Z_n =
\max_{ 1 \leq i \leq n}X_i$. Show that
\begin{displaymath}
\mbox{limsup}\, \frac{Z_{n}}{\log n} \geq 1, \ \ \ \mbox{a.s.} 
\end{displaymath}
State the results you are using at every step.

\medskip 
\noindent {\it Hint:} You may want to use the known inequality 
$ 1 - x \leq e^{-x}$. 



\QQ Let $X(t)$ denote Brownian motion with drift $\mu$, and let $T_a$
denote the time until this process hits the level $a$.  Derive a
partial differential equation which $f(a,t)\equiv P(T_a \leq t)$ satisfies.



\QQ Let $p_\theta(x)$ be a density on the real line that is Hellinger
differentiable at $\theta_0$, i.e., for $\theta\to\theta_0$,
\[ \sqrt{p_\theta(x)}= \sqrt{p_{\theta_0}(x)} + \frac12
(\theta-\theta_0) \tau_0(x) \sqrt{p_{\theta_0}(x)} +
r_{\theta,\theta_0}(x),\] with $\int
r_{\theta,\theta_0}^2(x)dx=o(|\theta-\theta_0|^2)$ and $\int
\tau_0^2(x)p_{\theta_0}(x)dx<+\infty$.  Show the following:

\begin{parts}
  \PP The generalized score function has mean zero: $$\int \tau_0(x)
  p_{\theta_0}(x)dx=0\,.$$

  \PP The squared Hellinger distance between $P_\theta$ and
  $P_{\theta_0}$ satisfies \[H^2(P_\theta,P_{\theta_0})=\frac14
  (\theta-\theta_0)^2 \int \tau_0^2(x) p_{\theta_0}(x)dx +
  o(|\theta-\theta_0|^2)\;\;\;\; \mbox{for $\theta\to\theta_0$.}\]

\end{parts}


\topic{Theoretical Statistics}


\QQ Consider the model $Y=X \beta+e$, where $e\sim N(0, \sigma^2I)$,
and the reduced model $Y=X_0 \gamma+e$, $C(X_0) \subset C(X)$.  Let
$M$ and $M_0$ be the perpendicular projection operators onto $C(X)$
and $C(X_0)$, respectively.

\begin{parts}
  \PP Show that $M-M_0$ is the perpendicular projection operator onto
  $C(M-M_0)$.


  \PP Show that the reduced model is true if and only if

  $\displaystyle \frac{Y^T(M-M_0)Y/r(M-M_0)}{Y^T(I-M)Y/r(I-M)}$ is
  distributed as $F(r(M-M_0),r(I-M),0)$.

\end{parts}



\QQ Suppose $X_1, X_2, \ldots, X_n$ are iid $N(\theta,\theta)$, that
is, the mean and variance are both equal to $\theta>0$.

\begin{parts}
  
\PP What is the method of moments (MOM) estimator of $\theta$?

\PP Is the MOM estimator asymptotically efficient?  Find the limiting
ratio of variances between the MOM estimator and the MLE as $n
\rightarrow \infty$.

\end{parts}


\pagebreak


\QQ Define $X \sim \mbox{DP}(\lambda)$ to mean that $X$ is an
integer-valued random variable which has the ``Double Poisson''
distribution with pmf (mass function) given by
\begin{displaymath}
  P(X=x) = 
  \begin{cases}
    \displaystyle \frac{\lambda^{|x|}e^{-\lambda}}{2(|x|!)}
    & \text{for $x \neq 0$}\\[12pt]
    e^{-\lambda} & \text{for $x=0$.}
  \end{cases}
\end{displaymath}
Answer the following.  Justify your answers.

\begin{parts}

  \PP Suppose $X_1, X_2, \ldots, X_n$ are iid $\text{DP}(\lambda)$.
  Find a complete sufficient statistic for $\lambda$.

  \PP Suppose $X_1, X_2, \ldots, X_n$ are independent with $X_i \sim
  \text{DP}(i\theta)$ (they are {\bf not} identically distributed).
  Find the best unbiased estimator of $\theta$.

\end{parts}

\topic{Biostatistics}



\QQ The proportional hazards model specifies that:
\[\lambda(t|x)=\lambda_0(t)\exp\{\vecc{X}'\vecc{\beta}\}\]

\begin{parts}

\PP Consider the following observations:

\begin{tabular}{ccc}
  time & $x$ & fail\\
  1 & 3 & 1\\
  3 & 2 & 0 \\
  5 & 4 & 1 \\
  9 & 6 & 0 \\
\end{tabular}

Write out the partial likelihood associated with the data.

\PP The above contained no ties.  Suppose the data were the
  following:

\begin{tabular}{ccc}
  time & $x$ & fail\\
  3 & 3 & 1 \\
  3 & 2 & 1 \\
  5 & 4 & 0 \\
  9 & 6 & 1 \\
\end{tabular}

Sketch at least two ways we could take the tie into account in forming
the partial likelihood contribution at time 3.

\end{parts}

\pagebreak 

\QQ Consider the following two by two table representing the results
of a cohort study:

  \begin{center}
    \begin{tabular}{lc|c|c|c}

      &\multicolumn{1}{c}{\  }& \multicolumn{2}{ c }{Exposure }  &\\
      &\multicolumn{1}{c}{\  }& \multicolumn{1}{c}{$+$} & \multicolumn{1}{c}{$-$} & Total\\
      \cline{3-4}
      Disease    & $+$ & a & b & a+b\\
      \cline{3-4}
      & $-$ &   c & d & c+d \\
      \cline{3-4}
      Total &\multicolumn{1}{c}{\  }& \multicolumn{1}{c}{a+c} & \multicolumn{1}{c}{b+d} & \multicolumn{1}{c}{n} \\
    \end{tabular}
  \end{center}



\begin{parts}

\PP Define each of the following, both in terms of the table
  provided and in terms of probabilities:
  \begin{enumerate}
  \item Incidence
  \item Prevalence
  \item Odds Ratio
  \item Relative Risk
  \end{enumerate}

\PP

  Use the delta method to derive an approximate standard deviation for
  the odds ratio.

\end{parts}

\QQ Derive sample size formulae for the following scenarios.  In
  each, you may assume an $\alpha=.05$ (one-sided) and a $\beta=0.10$.
  You may also assume $\sigma$ is known and for the two-sample case,
  is the same for both populations.

  \begin{parts}

  \PP You wish to test the hypothesis that the mean in a population
    is some specified value, i.e. $\mu=\mu_0$ versus the alternative
    that the mean is some specified but alternative value
    $\mu=\mu_A>\mu_0$.  The test is to be based on a single sample of
    size $n$.

  \PP You wish to test the hypothesis that the means in two
    populations are the same, i.e. $\mu_1-\mu_2=0$ versus that they are
    different by some specified amount, i.e. $\mu_1-\mu_2=\Delta>0$.
    The test will be based on two samples of size $n$.

    \PP Suppose for some reason, it is much cheaper to select
    observations in one of the populations.  How would you change the
    derivation of the above two sample problem to take into account
    different sample sizes (i.e., $n_1 \ne n_2$)?

  \end{parts}



\pagebreak


\topic{Computational Statistics}


\QQ Let $y \in \real$ and $x \in \real^n$ be the response and the
predictor random variables in a linear regression problem, and one is
interested in solving for $b^* \in \real^n$ such that: ${b}^* =
\argmin_{b \in \real^n} E[|y - x^Tb|^2]$. However, the number of
predictors $n$ is very large and we will instead use another predictor
vector $z \in \real^d$ ($d << n$) and solve the alternate problem:
${\tilde b}^* = \argmin_{{\tilde b} \in \real^d} E[|y - z^T{\tilde
  b}|^2]$.

The criterion for choosing $z$ is:
$$
z = U^Tx\ \ \mbox{such that}\ \ E[\|x - Uz\|^2]\ \ \mbox{is minimized}
\ ,
$$
where $U \in \real^{n \times d}$, $U^TU = I_d$ ($d \times d$ identity
matrix), and $\|\cdot \|$ is the two-norm of a vector.

\begin{parts}
  \PP Derive an optimal $U$ and hence an optimal $z$ for which the
  criterion is achieved.

  \PP Suggest an algorithm for performing this linear dimension
  reduction step before performing linear regression.

  \PP Comment on the variation of the regression error $E[|y - z^T
  {\tilde b}^*|^2]$ versus $d$.

\end{parts}


\QQ Let $x_1, x_2, \dots, x_t$ denote the evolving state of a system
in $\real$, and $y_1, y_2, \dots, y_t$ denote our observations of this
system, also in $\real$. We make the following assumptions: (i)
$\{x_t\}$ is a Markov process and the one-step transitional density
$f(x_t|x_{t-1})$ is easy to sample from. (ii) Given $x_t$, the
observation $y_t$ is independent of all the previous states and
previous observations. The likelihood function $f(y_t|x_t)$ is given
and easy to evaluate for a given pair $(x_t, y_t)$.

\begin{parts}

\PP State the nonlinear filtering equations for this system. In other
words, write the relationship between the posterior density at time
$t+1$, i.e. $f(x_{t+1}|y_1,y_2,\dots,y_{t+1})$ and the posterior
density at time $t$, i.e. $f(x_t|y_1,y_2,\dots,y_t)$.

\PP We are interested in estimating the posterior mean \[\int x_{t+1}
f(x_{t+1}|y_1,y_2,\dots,y_{t+1}) dx_{t+1}\,.\] Suggest a sequential
Monte Carlo algorithm for this purpose, assuming that you have $n$
perfect samples, $x_t^1, x_t^2, \dots, x_t^n$, from the previous
posterior ($f(x_t|y_1,y_2,\dots,y_t)$).

\end{parts}

\end{document}

