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\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam (Part II of Written Exam)\\
Saturday, January 4, 1997

\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.}


\QQ
%McKeague -- Probability
Let $X$ be a random variable such that $0\le X\le 1$,
and let $X_1,X_2,\ldots$ be i.i.d.\ copies of $X$.
Denote $\mu=EX$ and let $\psi(t)=Ee^{tX}$ be the moment generating
function of
$X$.
\begin{parts}
\PP
Show that
$$\psi(t)\le 1-\mu +\mu e^t\ \ \ {\rm for\ all\ }t.$$

\PP
Show that
$$P\left( \sum_{i=1}^n X_i\ge n(\mu +\epsilon)\right)
\le \left\{ e^{-t(\mu +\epsilon)}\left(1-\mu +\mu e^t\right)\right\}^n$$
for all $t\ge 0$ and $\epsilon >0$.

\PP
Use  (b) to
conclude that
$$\limsup_{n\to \infty} {1\over n} \sum_{i=1}^n X_i\le \mu$$
almost surely.
HINT:  Regard the expression inside the
braces as a function of $t$ and consider its behavior
close to $t=0$.

\end{parts}

\QQ
%Sethu-- Ph.~D. limit theory problem.
  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
$$ T_n =\big\{
\begin{array}{ll}
V_n  & {\rm if\ \ } |V_n| \le n^{-\frac{1}{4}}\\
M_n  & {\rm if\ \ } |V_n| >  n^{-\frac{1}{4}}.\\
\end{array} $$
Obtain the limiting distribution of
$\sqrt{n} (T_n-1) $. {\it Hint:  First compute the limit of
  $P(|V_n| \le n^{-\frac{1}{4}})$ and consider the two cases of
  the definition of $T_n$ separately.}
\end{parts}

\newpage
\QQ
%McKeague -- Probability
Let $N$ be a positive integer-valued random variable
and let $X_1,X_2,\ldots $ be random variables with finite
expectations.
\begin{parts}
\PP
 Define $X_N(\omega)=X_{N(\omega)}(\omega)$.
Show that $X_N$ is a random variable.

\PP
Suppose that $N$ is
bounded and let $\{N_n$, $n=1,2,\ldots\}$ be a  sequence of
positive integer-valued random variables  such that
$N_n\nearrow N$ almost surely.   Show that
$$\lim _{n\to \infty} EX_{N_n} =EX_N.$$
\end{parts}

\QQ
%Niu -- Applied Statistics
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, Box-Cox power transformations
are often used to stabilize the variances of $\{X_i\}$.

\begin{parts}
\PP
If $\sigma_i=\alpha \mu_i^\beta$ and  the transformation $Y_i=X_i^p$
is used to stabilize the variances, find an approximate  relationship
between $\beta$ and $p$ using the Taylor expansion.

\PP
If $\{X_i, i=1, \ldots, n\}$ are independent Poisson random variables,
which transformation will stabilize the variances of $\{X_i\}$?

\PP
In practice, $\beta$ is usually unknown. If $\hat{\sigma}_i^2$ and
$\hat{\mu}_i$ are available, give an empirical way to  estimate the power
$p$.
\end{parts}

\QQ
%Niu -- Applied Statistics
Consider the following linear regression model:
\begin{eqnarray*}
y_i = \beta_0 + \beta_1 x_{1i} + \cdots + \beta_px_{pi} +\epsilon_i, \hspace{0.1in} i=1, \ldots, n,
\end{eqnarray*}
where the $\epsilon_i$'s are iid normally distributed random variables with mean
zero and variance $\sigma^2$.

\begin{parts}
\PP
If $\vecc{\xi}=(\xi_1, \ldots, \xi_n)'$  has a multivariate normal  distribution
with mean zero and nonsingular covariance matrix $V=\mbox{Cov}(\vecc{\xi})$,  and if $A$ is
a $n\times n$  symmetric matrix with rank $r$, show that $\vecc{\xi}'A\vecc{\xi}$ has a
$\chi^2$ distribution with $df=r$ if $AV$ is idempotent.
(Hint: If $AV$ is idempotent and $V=C'C$, then $B=CAC'$ is idempotent with rank $r$).

\PP
Let  $\hat{\vecc{\beta}}$ be the least squares estimate of
$\vecc{\beta}=(\beta_0, \beta_1, \ldots, \beta_p)'$ and let $\hat{\vecc{\epsilon}}$
be the residuals from the the fitted model.  Assuming  that  the rank of $X$ is
$p+1$, show that the statistic
\[ F = \frac{(\hat{\vecc{\beta}}-\vecc{\beta})'(X'X)(\hat{\vecc{\beta}}-\vecc{\beta})/(p+1)}
{\hat{\vecc{\epsilon}}'\hat{\vecc{\epsilon}}/(n-p-1)}\]
has an F-distribution.
\end{parts}

\QQ
%meeter -- Applied Statistics
Questionnaires about NAFTA, containing 20 multiple-choice questions,  were
sent to a random sample of executives in each of the three countries involved
in this important free-trade agreement.  Each questionnaire was categorized as
 ``high" or ``low"  based on the number of
questions correctly answered ($Y\geq 12$ or $Y<12$).
The results are listed in the following table:

\vspace{0.1in}

\begin{center}
\begin{tabular}{cccc}\hline\hline
      & USA   & Canada  &  Mexico \\ \hline
High  &  22   &  33     &   31    \\
Low   &  29   &  19     &   19    \\ \hline
\end{tabular}
\end{center}

\begin{parts}
\PP
A $\chi^2$ test of association for the table yielded  a value of 5.333 with the critical
value of $\chi^2_2(0.05) = 5.99$. In words, what hypothesis is being tested?
What is your  conclusion based on the test?

\PP
After examining  the table, do you think  your  conclusion in {\bf (a)} is reasonable?
If not, what further analysis  can you perform on the table?  What are the conclusions
from your analysis? ($\chi^2_1(0.05)=3.84$).

\PP
What simple alternative analysis would be appropriate if the number of correct answers (Y)
for each questionnaire is used directly? Include the analysis suggested in {\bf (b)}.
Which form of analysis would you prefer, and why?
\end{parts}



% Inference -- submitted by Huffer
\QQ Let $X_1, X_2, \ldots, X_n$ be i.i.d.\ with density
\[ f(x|\theta) = \frac{1}{2}\theta^3 x^2 e^{-\theta x}\,, \hspace{0.1in} x>0.\]

\begin{parts}

\PP What is the Cramer-Rao lower bound for the variance
of an unbiased estimator of $\theta$?

\PP Find a simple unbiased estimator of $\theta$.  (Hint:
evaluate $EX^k$ for arbitrary real values $k$.)

\PP Does the estimator you found in the previous part
achieve the Cramer-Rao lower bound?  Is it asymptotically
efficient?

% Optional extra part (if needed).
%\PP Suggest a (possibly biased) estimator of $\theta$ which you
%believe to be asymptotically efficient.  (Do {\em not} prove that this
%estimator is asymptotically efficient, just state your ``general''
%reason for believing it to be so.)

\PP Does there exist an unbiased estimator of $\theta$ which exactly
achieves the Cramer-Rao lower bound for finite $n$?  (Justify your
answer.)

\end{parts}

\newpage

% Inference -- submitted by Huffer
\QQ Let $X_1, X_2, \ldots, X_n$ be independent with probability
density functions given by
\[ f(x_i |\theta) = \left\{
  \begin{array}{ll}
  \mbox{\large
$\frac{1}{2i\theta}$
} & \mbox{for\, $-i\theta < x_i < i\theta$}\\
 & \\
  0 & \mbox{otherwise.}
  \end{array}
\right.
\]

\begin{parts}

\PP Find a one-dimensional sufficient statistic for $\theta$. (Your statistic
should {\bf also} be minimal, but you are not required to show this.)

\PP Find the minimum variance unbiased estimator of $\theta$. (Prove
that your estimator has this property.)

\end{parts}

\QQ
%Huang--Inference
 Suppose that $X_1, \dots, X_n$ are i.i.d. with the Poisson distribution,
\[ P[X_i=x|\lambda]= \lambda^x e^{-\lambda} / x !\]
for $x=0,1, \dots$ and for $\lambda >0$.

\begin{parts}
\PP
 Show that a conjugate family of prior distributions is given
by the Gamma distribution, with probability density:
\[f(\lambda, \alpha, \beta)= \Gamma(\alpha)^{-1} \beta^{-\alpha} \lambda^{\alpha-1}
  e^{-\lambda/\beta} I_{(0, \infty)}(\lambda), \]
and show that the posterior distribution of $\lambda|X_1, \dots, X_n$
is
\[\mbox{Gamma}(\sum_i X_i+ \alpha, (n+1/\beta)^{-1}).\]

\PP
 Show that the mean of the posterior distribution is
\[ \hat{\lambda}= (\sum_i X_i + \alpha)/ (n+1/\beta).\]

\PP
Find weights, so that $\hat{\lambda}$ is a weighted average
of $\bar{X}$ and the mean of the prior distribution.

\PP
Show that in the limit as $n \rightarrow \infty$, with
$n^{-1} \sum_i X_i= \lambda^* + O(n^{-1}) $ (some given value), if
$\Lambda$ is a random variable with the posterior distribution,
then \[ n^{1/2} (\Lambda- \lambda^*) \rightarrow N(0, \lambda^*) \]
in distribution. (Hint: recall that the Gamma$(\alpha, \beta)$
distribution has characteristic function $(1- \beta i t)^{-\alpha}$).
\end{parts}


\end{document}

