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\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam (Part II of Written Exam)\\
Saturday, January 28, 1995

\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.}




% submitted by Niu.
% Applied Statistics


\QQ
In a Balanced Incomplete Block (BIB) design, $r$ replications of $t$ treatments
are arranged in $b$ blocks of $k$ experimental units with $k<t$.
The total number of observations  is $n=rt=bk$ with each treatment pair appearing 
together in the same block $\lambda=r(k-1)/(t-1)$ times someplace in the experiment.
The linear model for a BIB design has the form:
\begin{eqnarray*}
y_{ij} & = & \mu + \tau_i + \beta_j + \epsilon_{ij}, \hspace{0.2in} i=1, \ldots, t; 
\hspace{0.1in} j=1, \ldots, b,
\end{eqnarray*}
where the $ \epsilon_{ij}$'s are i.i.d.\ $N(0, \sigma^2)$
and the parameters satisfy the constraints 
$\sum_{i=1}^t\tau_i=0$ and  $\sum_{j=1}^b\beta_j=0$.

\begin{parts}

\PP Given  the observations $\{y_{ij}\}$,  find the least squares
estimators for the parameters  $\mu$, $\{\tau_i\}$ and  $\{\beta_j\}$.

\PP Let $\hat{\mu}_i = \hat{\mu} + \hat{\tau}_i$. Show that
\[E\hat{\mu}_i = \mu +\tau_i \;\;\mbox{ and } \;\;\ \mbox{Var}(\hat{\mu}_i) =
\frac{\lambda t+kr(t-1)}{r\lambda t^2}\sigma^2\,.\]

\end{parts}


%Problem 2 Advanced Prob.

\QQ
Let $\mu$ be a probability
measure on the Borel subsets of the interval $(0,1]$ that
has a density $f$ with respect to Lebesgue measure $\lambda$.  
Let ${\cal B}_n$ be the
$\sigma$-field generated by the dyadic
intervals $(k2^{-n}, (k+1)2^{-n}]$, $k=0,1,\ldots, 2^n-1$. Denote
the restrictions of $\mu$ and
$\lambda$ to ${\cal B}_n$ by $\mu_n$ and $\lambda_n$, respectively.
\begin{parts}
\PP  Show that $\mu_n$ is absolutely continuous
with respect to $\lambda_n$.

\PP  Show that 
$d\mu_n/d\lambda_n$ 
converges Lebesgue a.e.\ to $f$ as $n\to \infty$.

\end{parts}






\QQ
Let $X_1, X_2, \ldots, X_n$ be i.i.d.\ $N(\theta, \theta^2)$.

\begin{parts}

\PP What is the Cram\'{e}r-Rao lower bound for the variance of
an unbiased estimator of $\theta^{\mbox{\bf 2}}$

\PP Find an unbiased estimator of $\theta^{\mbox{\bf 2}}$ which is asymptotically
efficient.

\end{parts}

\pagebreak

\QQ
Let $ X_1, X_2, \ldots$
be independent normal random variables such that $ EX_i = 0$, 
$ E X_i^2 = \sigma_i^2$, $i = 1, 2, \ldots$.  Show that 
\[ X_i \to 0 \;
\mbox{ a.s.\,\,\, if and only if } \;\; \sum_{i=1}^{\infty}
e^{-x^2/\sigma_i^2} < \infty \;\; \mbox{\ for all }\; x > 0\,.
\]
(You may find the following fact useful:
\[1 - \Phi(z) \sim \frac{\phi(z)}{z}
\;\;\mbox{as}\;\; z \rightarrow \infty,\]
that is,
the ratio of the two sides goes to 1 as $z \rightarrow \infty$.
Here $\Phi(z)$ and $\phi(z)$ are the c.d.f.\ and density of the
standard normal distribution.)


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

\QQ
\def\gop{{\buildrel P \over \longrightarrow}}
For fixed $0<p<1$, let $\{ A_n, n\ge 1\}$ be a sequence of events
such that $$n^{-1/2}\sum_{i=1}^n|P(A_i|{\cal F}_{i-1})-p| \gop 0$$ 
as $n\to \infty$, where ${\cal F}_n$ is the $\sigma$-field
generated by $A_1,\ldots,A_n$, and ${\cal F}_0$ is the trivial $\sigma$-field.
Let $S_n$ be the number of $A_1,\ldots,A_n$ that occur. 
\begin{parts}
\PP Show that 
$n^{-1/2}(S_n-np)$ converges in distribution to $N(0,p(1-p))$. 

\PP 
For a fixed positive integer $k$, let
$T=\min \{ n\colon S_n \ge k\}$, where $\min\emptyset =\infty$. 
Deduce from (a) that $T<\infty$ a.s.

\end{parts}


\QQ Suppose we observe responses $Y_1, Y_2, \ldots, Y_n$ which follow
the model $Y_i = x_i^{\beta} U_i$ where $\beta$ is unknown, the values
$U_1, \ldots, U_n$ are unobserved but known to be i.i.d.\ Uniform(0,1),
and the values $x_1, \ldots x_n$ are known fixed (nonrandom) constants with
$x_i >1$ for all $i$.

\begin{parts}

\PP
Find a consistent, unbiased estimator of $\beta$ and verify that it
is consistent and unbiased.
(The estimator does not have to be optimal in any sense.)

\PP
What is (are) the sufficient statistic(s) in this problem?

\PP
Is your estimator in part (a) admissible under
squared error loss?

\PP
What is the maximum likelihood estimator of $\beta$?

\end{parts}

\pagebreak

\QQ
\begin{parts}

  \PP Show that
\[
\lim_{n \to\infty} e^{-nt} \sum_{k \le nT} { (nt)^k \over k!} =
\cases{ 1&$ t < T$\cr 0& $t >T$\cr}
\]
[{\sl HINT:\/} Let $S_n \sim \hbox{Poisson}(nt)$ and consider $P[ S_n
\leq nT ]$. ] 

\PP
Let $F$ be a c.d.f.\ such that $ F(0) = 0$.
Define the {\sl Laplace transform\/} of $F$ by 
\[
\phi(\lambda) = \int_0^\infty e^{-\lambda t} F(dt)\,\qquad \lambda \ge 0.
\]
For $T \ge 0$ such that $F$ is continuous at $T$, show that
\[
F(T) = \lim_{n\to\infty} \sum_{ k \le nT} { (-n)^k \over k!} [
D^k\phi](n),
\]
where $[ D^k \phi](n) $ is the $k^{th}$ derivative of $\phi$ evaluated
at $n$.  [{\sl HINT:\/} Compute the $k^{th}$ derivative of $\phi$ and
substitute.]

\end{parts}




% submitted by Meeter.
% Applied Statistics

\QQ An ecologist samples $n\/$ leaves; leaf $i\/$ has area $A_i\/$ and
has on it $k_i\/$ insects of the same species.  The objective is to see
whether insects prefer (say) large leaves.

\begin{parts}

  \PP Method 1 is to calculate the correlation between $k_i\/$ and
  $A_i\,$; if the correlation is positive and significant, the claim is
  made that the insects prefer large leaves.  Briefly criticize this
  approach.  (What null hypothesis are they implicitly making when they
  use this test?)

  \PP Method 2 is to assume that \[ k_i/A_i = \beta_0 A_i^{\theta}
  \;\;\;\;\;\;\; (\dag) \] at least in some average sense.  If there is no
  preference for size of leaf, what should the value of $\theta\/$ be?
  Why?  What did you assume to get your answer?

  \PP Using Method 2 described above, researchers wanted to regress
  $\log k_i\/$ on $\log A_i\,\/$.  Why?  What should the slope be, if
  leaf size does not attract insects?

  \PP Because some leaves contained no insects researchers used
  $\log(k_i + 1)\/$ as the dependent variable, instead of $\log
  k_i\,.\/$ They found by simulation that this reduced the slope of the
  line compared to that predicted by ($\dag$).  Give an heuristic
  argument why the slope should be reduced.

  \PP Suppose that the mean number of insects on a leaf of area $A$ is
  \(\lambda A^{\theta} \).  Suggest a reasonable probability model
  for this situation.   Assuming that $\theta\/$ is known, what is
  the maximum likelihood estimator of $\lambda\/$ for your model?

\end{parts}


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\end{document}

