\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{\QQ}{\vsp \stepcounter{qnum} 
%                {\noindent \bf Problem \theqnum.} \hspace{.2cm} }

\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}}

\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam (Part II of Written Exam)\\
Friday, January 5, 2001

\end{center}

\vsp
You have four hours.  Do as many problems as you can.  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



\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 Suppose that an experiment involves 6 factors each with two
levels.  Consider the following two fractional factorial designs:
\begin{itemize}
\item[(i).] $2^{6-2}$ with $5=1234$ and $6=134$
\item[(ii.)] $2^{6-2}$ with $5=123$ and $6=124$
\end{itemize}
\begin{parts}
  \PP What is the resolution for each of the fractional factorial
  designs?  Which design do you prefer? Justify your answers.
  
  \PP For the design in (ii), if we further know that all
  two-factor interactions involving factor 6 (i.e., 16, 26, 36,
  46, 56) are negligible, which two-factor interactions in the
  design are estimable under the assumption that three-factor and
  higher interactions are negligible?
\end{parts}

\pagebreak


\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  moving-average model:
\[\xi_i= \epsilon_i + \theta\epsilon_{i-1},\]
where $\{\epsilon_i, i=0,1, \ldots, n\}$ are iid $N(0,\sigma^2)$
variables.
Let $\hat{\beta}$ be the least squares estimate of $\beta$,
 $\hat{Y}=X\hat{\beta}$,  and $\hat{\xi} = Y-\hat{Y}$.
\begin{parts}
  \PP What are means and covariance matrices of $\hat{Y}$ and
  $\hat{\xi}$?  What are the distributions of $\hat{Y}$ and
  $\hat{\xi}$?

\PP
Are $\hat{Y}$ and $\hat{\xi}$ independent?
\end{parts}


\QQ A social scientist was studying students' admiration for
world leaders.  She selected a random sample of 10 seniors from
each of three majors at FSU.  Each senior was asked to name eight
world leaders.  She then scored the results by counting, for each
of $30 \times 29/2 = 435$ pairs of students, the number of world
leaders they agreed on (X = 0, 1, ..., 8).  Her hypothesis is
that students from the same major will name more leaders in
common than students from different majors.

How would you test this hypothesis, given data like this?



\QQ Suppose one assumes the multiple linear regression model 
$$\bf {Y=X_1\beta_1+X_2\beta_2+\epsilon }$$
when in fact $\bf
\beta_2=0$, in other words, the model fitted by the experimenter
is an overfitted model and the true model is $\bf
{Y=X_1\beta_1+\epsilon }$, where $\bf Y$ is a column vector of
dimension $n \times 1$, $\bf X_1$ and $\bf X_2$ are design
matrices of dimensions $n \times p$ and $n \times (m-p)$,
respectively, $\bf \beta_1 $ and $\bf \beta_2 $ are $p \times 1$
and $(m-p) \times 1$ parameter vectors, respectively, and $\bf
\epsilon \sim N(0, \sigma^2 I)$.  Let $s^2_m$ denote the residual
mean square based on the overfitted model, i.e., $ s^2_m={\bf
  Y^t[I-X(X^tX)^{-1}X^t]Y}/(n-m)$, where $\bf X=[X_1\vdots X_2]$.
Show that the residual mean square $s^2_m$ is unbiased for
$\sigma^2$ even if one overfits.



\QQ
 Let $\{ X_n\}$ be 
a sequence of identically distributed random variables with finite mean,
and write
 $M_n= \max_{1\le i\le  n}|X_i|$. 
\begin{parts}
\PP Show that $E(M_n)/n \to 0$.  
\PP Under the additional assumption that
the $X_n$ are independent show that
$M_n/n \to 0$ almost surely.
\end{parts}


\QQ Let $P$ be a probability measure on the real line having a
density $f$ with respect to Lebesgue measure $\lambda$.
\begin{parts}
\PP 
Show that $\lambda(A_n) \to 0$
implies $P(A_n)\to 0$ for 
every sequence of Lebesgue measurable sets $A_n$.
\PP Show that 
$$P(A) = \lambda_2\{(x,y) \colon x\in A, 0\le y\le f(x)\}$$ 
for any Lebesgue measurable set $A$,
where $\lambda_2$ is two-dimensional Lebesgue
measure. 
\end{parts}



\QQ Let $X$ be a random variable taking values in
$\{-2,-1,0,1,2\}$ with mass function given by $P(X=0) =
(1-\theta)^2$, $P(X=1) = P(X=-1) = \theta (1-\theta)$, and
$P(X=2)=P(X=-2)=\theta^2/2$ where $ \theta$ is unknown and $0 <
\theta < 1$.  Suppose you observe data $X_1, X_2, \ldots, X_n$
which are i.i.d.\ with the same distribution as $X$.  The
following two parts are unrelated and can be worked independently
of each other.
\begin{parts}
  
  \PP Find a complete sufficient statistic for $\theta$ (and show
  that your statistic has these properties).
  
  \PP Find the Cram\'er-Rao lower bound for the variance of an
  unbiased estimator of $\theta^{\mbox{\bf 3}}$.

\end{parts}



\QQ \mbox{}

\begin{parts}
  \PP Prove the following fact:
  \begin{eqnarray*}
&& \mbox{If} \;\;
\left(
      \begin{array}{c}
        X_1 \\ X_2
      \end{array}
   \right)
\sim N \left(
\left(
  \begin{array}{c}
\mu_1 \\ \mu_2
  \end{array}
\right),
\left(
  \begin{array}{cc}
\Sigma_{11} & \Sigma_{12}\\
\Sigma_{21} & \Sigma_{22}
  \end{array}
\right)
\right)\,,
\\ [.2cm]
&& \mbox{then} \;\;
X_1 \mid X_2 \sim N(\mu^{*}, \Sigma^{*})\\ [.2cm]
&& \mbox{with} \;\;
\mu^{*} = \mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(X_2 - \mu_2) \;\;
\mbox{and} \;\; 
\Sigma^{*} = \Sigma_{11}- \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\,.
\end{eqnarray*}
(Assume the matrices and vectors above are conformable and that
$\Sigma_{22}^{-1}$ exists.)

({\bf Hint:} Show that $X_1 - \mu^{*}$ and $X_2$ are independent.)
  
\PP Let $X$ be a known $n \times p$ matrix.  Consider the
Bayesian linear model where $Y \sim N(X\beta,I)$ and $\beta$ has
the prior distribution $\beta \sim N(\xi, I)$.  Find the
posterior distribution of $\beta$.
  
({\bf Hint:} Use the result of part (a). You do {\bf not} have to
write the answer in the simplest possible form; just leave it the
way you get it upon application of part (a).)

\end{parts}


\end{document}

