
\documentclass[12pt]{article}
\setlength{\textheight}{9.0truein}

\setlength{\oddsidemargin}{0.1truein}
\setlength{\evensidemargin}{0.1truein}
\setlength{\textwidth}{6.3truein}
\setlength{\topmargin}{-0.5truein}

\newcommand{\comment}[1]{}
\newcommand{\vsph}{\vspace{.40cm}}
\newcommand{\vsp}{\vspace{.80cm}}
\newcommand{\vecc}[1]{\mbox{\boldmath $#1$}}

\newcounter{qnum}
\newcounter{pnum}[qnum]


\newcommand{\topic}[1]{\vsph \hrule \vsph
\mbox{} \hfill   { \it #1} \hfill \mbox{}
}

\newcommand{\QQ}{\vsph \hrule \vsph \stepcounter{qnum} 
                {\noindent \bf Problem \theqnum.} \hspace{.2cm} }


\newcommand{\PP}{\stepcounter{pnum} \item[{\bf (\alph{pnum})}]\, }

\newenvironment{parts}{
   \begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
   \setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
   }{\end{list}}

\def\god{{\buildrel {\cal D} \over \longrightarrow}}
\def\goas{{\buildrel {\rm a.s.} \over \longrightarrow}}
%\def\RR{{\hbox{I\kern-.22em\hbox{R}}}}

\def\r{\rightarrow}
\def\p{\stackrel{p}{\r}}
\def\d{\stackrel{d}{\r}}
\newcommand{\var}{\mbox{Var}}

\usepackage{amsfonts}

\newcommand{\RR}{\mathbb {R}}\setlength{\textheight}{9.0truein}


\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam\\
Monday, August 18, 2003

\end{center}

\vsp 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.

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

\vsph



\topic{Applied Statistics}

\QQ Suppose that $\{X_i, i=1, \ldots, n\}$ are independent random
variables with $\mbox{E}(X_i)=\mu_i$ and
$\mbox{Var}(X_i)=\sigma_i^2$. In ANOVA and regression,
transformations are often used to stabilize the variances of
$\{X_i\}$.

\begin{parts}
\PP
If $\sigma_i=g(\mu_i)$ and  the transformation $Y_i=f(X_i)$
is used to stabilize the variances, find an approximate  relationship
between $f(\cdot)$ and $g(\cdot)$ using the Taylor expansion.

\PP
If $\sigma_i = c \mu_i^\alpha$, which transformation $f(\cdot)$
will approximately stabilize the variance $\mbox{Var}(X_i)=\sigma_i^2$?


\PP
In practice, $\alpha$ is usually unknown. If $\hat{\sigma}_i^2$ and
$\hat{\mu}_i$ are available, give an empirical way to  find the transformation
$f(X_i)$.

\end{parts}

\QQ In a complete factorial experiment, consider the following
random-effects model:
\begin{eqnarray*}
Y_{ij} & = & \mu +\alpha_i + \beta_j + \epsilon_{ij}, \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b,
\end{eqnarray*}
where  $\mu$ is the overall mean, $\alpha_i$ is the random effect
corresponding to the $i$th level of factor $A$, and $\beta_j$ is
the random effect due to the $j$th level of factor $B$.
The  $\alpha_i$'s are iid $N(0, \sigma^2_{\alpha})$ variables,
the  $\beta_j$'s are iid $N(0, \sigma^2_{\beta})$ variables,
  and the $\epsilon_{ij}$'s are iid
 $N(0, \sigma^2_{\epsilon})$ variables.
Furthermore, $\{\alpha_i\}$, $\{\beta_j\}$, and $\{\epsilon_{ij}\}$
are assumed to be independent.




\begin{parts}
\PP
What are the distributions of $Y_{ij}$, $Y_{i\cdot}$,  and  $Y_{\cdot j}$?
Find the covariance  Cov($Y_{ij}, Y_{kl}$).


\PP
Find unbiased estimates for the variances $\sigma^2_{\alpha}$, $\sigma^2_{\beta}$,
and $\sigma^2_{\epsilon}$ based on the observations $\{Y_{ij}, \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b\}$.


\end{parts}


\newpage
\vspace{0.5in}

\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.
The process $\{\xi_i\}$ is generated by the  model:
\[\xi_i - \phi \xi_{i-1} = a_i,\;\;\;\;\;\;\;(**)\]
where $|\phi| < 1$ and $\{a_i, i=0, \pm 1,  \pm 2 \ldots,\}$
 are iid $N(0,\sigma^2)$ variables.


\begin{parts}
  
  \PP Show that $\xi_i = \sum_{j=0}^{\infty} \phi^j a_{i-j}$ is a
  solution of equation (**) above.

\PP
  Based on the expression in (a) for $\xi_i$,  calculate  autocorrelations
$\gamma_k = \mbox{Cov}(\xi_i, \xi_{i+k})$ for $k \geq 0$.  Show that
$\gamma_k = \phi \gamma_{k-1}$ for any $k\geq 0$.

\PP
Suppose that $\phi$ is known.
How do you estimate $\beta$ in this setting?
Discuss the properties of your estimate such as mean, covariance matrix, and
distribution of $\hat{\beta}$. Compare your estimate with the least squares
estimate of $\beta$.

\end{parts}



\QQ In a generalized linear model, the response variable $Y$ is
assumed to have a density function with the form:
\begin{eqnarray*}
f(y; \theta, \phi) & = & \exp\{[y\theta-b(\theta)]/a(\phi)+c(y,\phi)\}.
\end{eqnarray*}


\begin{parts}
  \PP Identify $\theta$, $b(\theta)$, $a(\phi)$, and $c(y,\phi)$
  when $Y$ has a binomial distribution and when $Y$ has a Poisson
  distribution.

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

\end{parts}



\topic{Probability}


\QQ Let $X_1, X_2, \ldots$ be independent. Show that $\sup X_n <
\infty$ a.s.  if and only if $\sum_n P(X_n > M) < \infty$ for
some $M < \infty$.

\noindent {\it Hint : Use Borel-Cantelli Lemmas.}


\QQ Let $X_n$ be a sequence of random variables and let $a_n
\rightarrow \infty$ be an increasing sequence of constants such
that $a_nX_n \rightarrow_d X$, as $n \rightarrow \infty$. Let $g$
be a continuously differentiable function at 0. Show that
\[a_n\left(g(X_n) - g(0)\right) \rightarrow_d\, g^{\prime}(0)X.\]


\QQ Let $X_1,X_2,\dots$ be i.i.d.\ with a uniform distribution on
$(0,1)$. Consider the statistic
\[U_n = \frac{1}{{{n}\choose{2}}} \sum_{1 \leq i<j \leq n} \min(X_i,X_j).\]
Obtain the asymptotic distribution of $U_n$.

{\noindent \it Hint: The df and pdf of $X_1$ are easy. So it will
  be easy to compute all moments etc. that you may require.}



\topic{Theoretical Statistics}


\QQ Consider the following model:
\begin{eqnarray*}
 Y_1 & = & \alpha_1+\alpha_2+e_1 \\
 Y_2 & = & 2\alpha_2+e_2 \\
 Y_3 & = & -\alpha_1+\alpha_2+e_3   
\end{eqnarray*}
where $e_i\sim $ independent $ N(0, \sigma^2) $.  Derive the
F-test for testing $ H_0: \alpha_1=2\alpha_2 $.

\QQ Let $X_1, X_2, \ldots, X_n$ be iid with a {\bf shifted}
geometric distribution
\begin{displaymath}
  P_{\theta}(X=x) = \left(\frac{1}{2}\right)^{x-\theta+1},
\;\;\; x= \theta, \theta+1, \theta+2, \ldots, \;\;\; 
-\infty < \theta < \infty\,.
\end{displaymath}

\begin{parts}
  \PP Find a minimal sufficient statistic for $\theta$ (and prove
  that your statistic has both these properties).
  
  \PP Find the MLE for $\theta$.
  
  \PP Show that the statistic you found in part (a) is complete.
\end{parts}

\QQ Suppose that we have two independent random samples: $X_1,
X_2, \ldots, X_n$ are iid $N(0,\sigma^2)$, and $Y_1, Y_2, \ldots,
Y_m$ are iid $N(0,\tau^2)$.

\begin{parts}
  \PP Find the likelihood ratio test statistic (LRT) for
  $H_0:\sigma^2 = \tau^2$ versus $H_1: \sigma^2 \neq \tau^2$.
  
  \PP Show that the test in part (a) can be based on the
  statistic 
  \begin{displaymath}
T = \frac{\sum_{i=1}^n X_i^2}{\sum_{i=1}^m Y_i^2}\,.    
  \end{displaymath}

  
\end{parts}



\topic{Computational Statistics}

\QQ For a given $\theta \in \Re$, let $f(x|\theta)$ be a density
function given by:
$$
f(x|\theta) = { \exp(-x + \theta) \over (1 + \exp(-x +
  \theta))^2} \ , \ \ \mbox{for}\ \ x \in \Re.
$$
Assuming that a uniform random number generator between 0 and 1 is
available, suggest a method for generating exact samples from $f$.
Write an algorithm (e.g. in matlab code) to implement this method.

\QQ Let $X$ be a random variable whose mean we are interested in
estimating using Monte Carlo methods. Let $Y$ be another random
variable and define $Z = E[X|Y]$.

\begin{parts}
  
  \PP Show that $E[Z] = E[X]$, but $\mbox{variance}(Z) \leq
  \mbox{variance}(X)$.
  
  \PP Suppose that $Y$ is an exponential random variable with
  mean one, and given $Y=y$, $X$ is an exponential random
  variable with mean $y$. Use variance reduction by conditioning
  to set up an efficient Monte Carlo method to estimate $Pr\{XY
  \leq 3\}$. State the procedure algorithmically.

\end{parts}



\topic{Biostatistics}

\QQ Assume that $Y$ is a random variable taking values 0,1 and
that $X$ is a random variable such that:

If $Y=0$, $X$ has density $f_0$, and

If $Y=1$, $X$ has density $f_1$.

\begin{parts}

\PP  Assume that the proportion of people in the population for
whom $Y=1$ is $p$.  Derive a general expression for
$$\Pr(Y=1 | X=x)$$

\PP Show that if $f_i$ is the $N(\mu_i,\sigma^2)$ density for
$i=0,1$, then
$$\Pr(Y=1|X=x)= \frac{1}{1+\exp\{-(\alpha+\beta x)\}}$$
and give
expressions for $\alpha$ and $\beta$.  That is, the logistic
model arises naturally.

\PP Assume you have a random sample of individuals, $n_0$ of them have $Y=0$, and $n_1$ have $Y=1$.
What are the maximum likelihood estimates of $\alpha$ and $\beta$?
\end{parts}

\QQ Suppose that an investigator wants to do a clinical trial
testing the difference between two normal means. She is willing
to assume that the variance is equal for the two groups and is
$\sigma^2$.  She specifies a minimum difference that she thinks
is clinically important: $\Delta=\mu_1-\mu_2$, and specifies the
level of type I error she is willing to accept, $\alpha$.
Because of cost considerations $n=n_1+n_2$ is fixed ($n_1=$ the
number to be randomized in the first group, $n_2=$ the number to
be randomized in the second group.)  What are the values of $n_1$
and $n_2$ that maximize the power of the study?





\end{document}

