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\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam (Part II of Written Exam)\\
Tuesday, January 5, 1999

\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 $F_n, n\ge 1$ and $F$ be distribution functions. 

\begin{parts}
\PP Suppose  $F_n(x)\to F(x)$ for all $x$ in some dense 
subset of the real line.
Show that $F_n$ converges in distribution to $F$.

\PP Is it possible for $F_n$ to converge in distribution to $F$
but $F_n(x)\not\to F(x)$ for all $x$ in  some dense 
subset of the real line?     




\end{parts}

\QQ
Let $\Omega_n$ be the the space of
all counting measures on $[0,1]$ of total mass $n$:
$$\Omega_n=\Big\{ \sum_{i=1}^n \delta_{x_i}\colon x_i \in [0,1], \ i=1,\ldots,n
\Big\},$$
where $\delta_{x_i}$ denotes the Dirac measure with mass 1 at
$x_i$.  Define the projection $\pi_n\colon [0,1]^n \to \Omega_n$
by $\pi_n(x_1,\ldots,x_n)=\sum_{i=1}^n \delta_{x_i}$.

\begin{parts}
\PP  Show that ${\cal F}_n =\{ \pi_n(B)\colon B\in {\cal B}^n\}$
is a $\sigma$-field of subsets of $\Omega_n$ 
for each $n\ge 1$, where ${\cal B}^n$ is the 
$\sigma$-field of $n$-dimensional Borel subsets of $[0,1]^n$.


\PP  How would you define $\Omega_0$ and ${\cal F}_0$?


\PP Let $\cal{F}$ be the smallest $\sigma$-field of subsets of
$\Omega= \cup_{n=0}^\infty \Omega_n$ containing
${\cal F}_0, {\cal F}_1, \ldots$.  Show that all sets
in $\cal{F}$ can be explicitly presented
in terms of sets in ${\cal F}_0, {\cal F}_1, \ldots$.
 


\PP Define
$$P (A)=\frac{1}{e} \left(1\{A\cap \Omega_0\not= \emptyset\}
+ \sum_{n=1}^\infty {\lambda^n(\pi_n^{-1} (A\cap \Omega_n))\over n!}\right)$$
for $A\in \cal{F}$, where $\lambda^n$ is Lebesgue measure
on $[0,1]^n$ and $1\{\cdot \}$ denotes the indicator
function.  Show that $P$ is a probability measure on
$(\Omega,\cal{F})$.  

\end{parts}

\pagebreak

\QQ Let $X_1, X_2, \ldots, X_n$ be i.i.d.\ Bernoulli($p$).

\begin{parts}
\PP What is the MLE of $p^2$?

\PP What is the best unbiased estimator of $p^2$?

\PP What is the best unbiased estimator of $p^n$?

Hint for (b) and (c): Use the Rao-Blackwell procedure.

\PP Do the estimators in (b) and (c) achieve the Cramer-Rao lower
bound? (This can be answered with little or no calculation.)

\end{parts}


\QQ
Suppose that you are helping a client to  carry out 
 an experimental study to determine the effects
of 6  factors on  the yield of a chemical reaction. Called the
 6 factors A, B, C, D, E, and F.

\begin{parts}
\PP
Construct a $2^{6-2}$ fractional factorial design with as high a resolution as possible.

\PP
What are the generators of your design? What is the definition relation of your design?

\PP
Which effects are confounded with the main effect of factor A in your design?
 Which effects are confounded with the two-factor interaction BC?
\end{parts}


\QQ
Consider the following regression model:
\begin{eqnarray*}
\vecc{Y} = X\vecc{\beta}+Z\vecc{\gamma}+\vecc{\epsilon},
\end{eqnarray*}
where $X$ and $Z$ are known $n\times p$ and $n\times q$ full-rank
matrices and $X'Z=0$, and $\vecc{\epsilon}$ is normally distributed
with $\mbox{E}\vecc{\epsilon}=\vecc{0}$ and
$\mbox{Var}(\vecc{\epsilon}) = \sigma^2 I$.

\begin{parts}
\PP
Find the maximum likelihood estimates of $\vecc{\beta}$, $\vecc{\gamma}$, and $\sigma^2$.
Denote these estimates by $\hat{\vecc{\beta}}$, $\hat{\vecc{\gamma}}$, and $\hat{\sigma}^2$.

\PP
Find the expectations  and  covariance matrices of  $\hat{\vecc{\beta}}$ and  $\hat{\vecc{\gamma}}$.

\PP
Show that $\hat{\vecc{\beta}}$ and  $\hat{\vecc{\gamma}}$ are independent.
Show that ($\hat{\vecc{\beta}}$, $\hat{\vecc{\gamma}}$) is independent of $\hat{\sigma}^2$.
\end{parts}

\pagebreak

\QQ
Suppose that, under $P_{\theta}$, the random variables $X_1,X_2,
  \dots, X_n$ are i.i.d. with distribution $N(\theta,1)$.

\begin{parts}
\PP Write down the asymptotic distribution of 
$\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$
under $P_{\theta}$.

\PP Define 
$$
Y_n = \Big\{
     \begin{array}{ll}
     1 & \mbox{if \ \ } |\bar{X}_n| \le n^{-\frac{1}{4}},\\
     0 & \mbox{if \ \ } |\bar{X}_n| > n^{-\frac{1}{4}}.\\
     \end{array}
$$
 For each $\theta$, show that
$$
Y_n \p \Big\{ 
    \begin{array}{ll}
    1 & \mbox{if\ \ } \theta =  0\\
    0 & \mbox{if\ \ } \theta \not= 0\\
    \end{array}
$$
under $P_{\theta}$, where $\p$ stands for convergence in probability.
\end{parts}


\QQ
There are five workers selling food from little carts at five locations on
our campus. The owner keeps getting complaints from her workers that they
could sell more if they were at some other location which (they claim) is a
better one than theirs. 

\begin{parts}
\PP Set up an experiment so that, in one week (MTuWThF) she can look at the
daily sales in such a way that 

-each worker works one day in each location;\\
-she can find out whether some locations are better than others.

Defend your choice of design.

\PP What other questions can the owner answer?  Give a model, and use a table
format to help describe your answers, but please do not include any
formulas in the table. 

\PP The owner wants to know whether (taking all workers as a group) workers
who work at their customary location get more sales than when they are
working at a different location.  How would you answer this question?   Give
a model and a description.
\end{parts}



\QQ Let $X$ be an $n \times p$ (with $p \leq n$) matrix having rank $r < p$,
and assume the $n \times 1$ random vector $Y$ follows the linear model
$Y \sim N(X\beta, \sigma^2 I)$.

\begin{parts}
\PP Let $L$ be an $m \times p$ matrix. Consider the hypothesis 
$H_0 : L\beta= 0$ versus $H_1 : L\beta \neq 0$.  State conditions
on $L$ which guarantee that this hypothesis is {\it testable\/},
that is, state the definition of a {\it testable\/} hypothesis.

\PP Let $V = \mbox{range}(X)$ and let $W$ be any subspace of $V$.
Define $\mu = X\beta$.
Show that the hypothesis $H_0 : \mu \in W$ versus $H_1 : \mu \notin W \;
\mbox{(but  $\in V$)}$ can be formulated as a testable hypothesis.

\end{parts}

\QQ
Suppose that $Y_1, \cdots, Y_n$ are independent 
Bernoulli random variables 
with the corresponding covariates $x_1, \cdots, x_n$ such that 
$$ \log \left\{ \frac{P(Y_i=1)}{P(Y_i=0)}\right\}=\gamma +\beta x_i$$

\begin{parts}
\PP Find a minimal sufficient statistic for $(\gamma, \beta)$.\\ \mbox{}
[Recall that a complete sufficient statistic is automatically minimal.]

\PP Let $(R, T)$ be the minimal sufficient statistic for $(\gamma, \beta)$ 
found in part (a).  Derive the conditional distribution of $T$ given $R$. 


\PP Treating $\gamma $ as a nuisance parameter, develop a size $\alpha $ test 
for testing the null hypothesis $\beta=0$ (no dependence on the covariates) 
versus the one-sided alternative $\beta>0$.

\end{parts}


\end{document}

