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\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam (Part II of Written Exam)\\
Friday, January 4, 2002

\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 In a Balanced Incomplete Block (BIB) design, the block size
$k$ is less than the number of treatments $t$. There are $b$
blocks and each treatment is replicated $r$ times.  Furthermore,
each pair of treatments is compared in $\lambda$ blocks.

Consider the following linear model for a BIB design:
\begin{eqnarray*}
Y_{ij} & = & \mu +\alpha_i + \beta_j + \epsilon_{ij}, \hspace{0.2in}
i=1,\ldots, b;
\hspace{0.1in} j=1, \ldots, t,
\end{eqnarray*}
where $\alpha_i$ is the $i$th block effect, $\beta_j$ is the $j$th
treatment effect,
and the $\epsilon_{ij}$'s are independent $N(0, \sigma^2)$.  Let
$\hat{\beta}_j$
be the least squares estimate of $\beta_j$. It is known that 
Var($\hat{\beta}_j-\hat{\beta}_l) = \frac{2k}{\lambda t}\sigma^2$ for
$j\neq l$.


\begin{parts}
\PP
Show that $\lambda (t-1) = r(k-1)$.

\PP
If the experiment were run as a randomized complete block design, show
that
Var($\hat{\beta}_j-\hat{\beta}_k) = \frac{2}{b}\sigma^2$.

\PP
Show that $\frac{k}{\lambda t} > \frac{1}{b}$ and give it a statistical
interpretation.
\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.
The process $\{\xi_i\}$ is generated by the  autoregressive  model:
\[\xi_i - 0.6 \xi_{i-1}= \epsilon_i,\]
where $\{\epsilon_i, i=0,1, \ldots, n\}$ are iid $N(0,\sigma^2)$
variables.

\begin{parts}
\PP
Show that $\{\xi_i, i=1,2,\ldots,\}$ is a second-order stationary
process.
Calculate the autocovariance function  $\gamma_k = \mbox{Cov}(\xi_i,
\xi_{i+k})$.

\PP How do you estimate $\beta$ in this setting? Why do you
choose this method?  Discuss the properties of your estimate such
as the mean, covariance matrix, and distribution of
$\hat{\beta}$.

\end{parts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\QQ In evaluating candidates for a three-year space voyage, a
psychologist evaluated four candidates, who were from Asia,
Africa, South America, and Europe.

Each trainee was shown twelve pictures, in random order, and
asked to rate each person on a scale of 1 to 10 on the question:
{\it How suitable would this person be to accompany you on a long
  space voyage?}  Naturally, the twelve pictures showed persons
from Asia, Africa, South America, and Europe, three people from
each group.

\begin{parts}
  
  \PP Give an appropriate model for $y$, the score assigned to a
  picture by a candidate.
  
  \PP Give the ANOVA table (just Source and d.f.) for this model.
  
  \PP The psychologist thinks that trainees will tend to rate
  people from their own continent higher than people from other
  continents.  How would you test this hypothesis?

\PP Show the revised Anova table.

\end{parts}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\QQ
Let $\{ X_n\}$ be a sequence of 
random variables, $X$ a random variable
with distribution function $F$,  and 
$\alpha $ a real number. 
Suppose 
$X_n\nearrow X$ on $\{ X>\alpha\}$, and  $X_n\le X$
almost surely.
Show that
$$\lim_{n\to \infty}E[\max (X_n,\alpha)]$$ exists
and express the limit in terms of $F$ and $\alpha$.


\QQ
\begin{parts}
\PP 
State the L\'evy continuity theorem.  
\PP 
Show that if $\varphi(t)$ is a characteristic function, then
so is $|\varphi(t)|^2$.
\PP 
Suppose that the distribution function $F$ is 
infinitely divisible, i.e., for each
$n\ge 1$ there exists a distribution function $F_n$
such that $F$ coincides with the $n$-fold convolution $F_n^{*n}$.
Show that the characteristic function of $F$ never vanishes.
HINT:  Apply (b) to the characteristic function of $F_n$ and take 
limits.
\end{parts}

\QQ Let $Y_n,M_n,Z_n$ be the $1$-st quartile, median and $3$-rd
quartile, respectively of $X_1,\dots,X_n$ which are i.i.d. with
common df $F$ and pdf $f(x) = \frac{1}{\pi}
\frac{1}{1+(x-\theta)^2}$.  Suppose that one proposes to use
$M_n$ and $W_n = \frac{1}{2}(Y_n+Z_n)$ as estimates of $\theta$.
State the required results and obtain the asymptotic
distributions of these two estimates and compute the efficiency
by comparing their asymptotic variances. \textit{Note that
  $F(\theta)=\frac{1}{2}, F(\theta+ 1) = \frac{3}{4}$.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\QQ Suppose that $X$ has a Poisson distribution with parameter
$\theta$

\begin{parts}
  
  \PP Find the UMVUE of $\exp\{-3\theta\}$ and its corresponding
  variance. [Hint: evaluate $E(u^X)$ for arbitrary $u$.]
  
  \PP Is the estimator found in (a) a good estimator. If not,
  find a better estimator.
  
  \PP Prove that there is no unbiased estimator of $1/\theta$.
  
  \PP Let $Y=\sum_{i=1}^nX_i$, where $X_1, \cdots, X_n$ is a
  random sample from the above Poisson distribution. We wish to
  estimate $1/\theta$. Consider the following estimator
  $T_n(Y)=n/(Y+1)$. Show that the bias of this estimator tends to
  0 as $n\rightarrow \infty$ and find the asymptotic distribution
  of $T_n(Y)$.

\end{parts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\QQ Suppose $X_1, \ldots, X_n$ are iid $N(\mu,\sigma^2)$.

\begin{parts}
  
  \PP Show that $(\bar{X},S^2)$, the sample mean and variance, is
  a complete sufficient statistic for $(\mu,\sigma^2)$.
  
  \PP Show that $\displaystyle \sum_{i=1}^n
  \left(\frac{X_i-\bar{X}}{S}\right)^3$ is independent of
  $\bar{X}/S$.

\end{parts}

\QQ Suppose $Y \sim N_p(a,I)$ where $a$ is a $p \times 1$ unit
vector ($a'a=1$).  Let $\Gamma = [a, \Gamma_1]$ be an orthogonal
matrix (that is, $\Gamma$ is a $p \times p$ orthogonal matrix
whose first column is $a$).

\begin{parts}
  
  \PP What is the distribution of $\Gamma'Y$? (Be specific.)

  \PP What is the distribution of $Y'(I-aa')Y$? (Be specific.)
  
  \PP Are $\Gamma'Y$ and $Y'(I-aa')Y$ independent? (Prove your
  answer.)

\end{parts}

\end{document}

