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\begin{document}

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\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam (Part II of Written Exam)\\
Friday, January 5, 1996

\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
%Huffer -- Inference
Let $X_1, X_2, \ldots, X_n$ be i.i.d.\ from the density $f(x) = 
\lambda e^{-\lambda (x - \mu)}$ for $x \geq \mu$ (and $f(x) = 0$
otherwise).  

\begin{parts}

\PP Find a sufficient statistic for $(\lambda, \mu)$.

\PP Show that your statistic is complete.

\PP Let $\overline{X}$ and $S^2$ denote the sample mean and variance
respectively.  To be precise, we define $S^2 = \sum_{i=1}^n (X_i -
\overline{X})^2/(n-1)$.  Let $X_{(1)} = \min X_i$.  Suppose you are given that
\[ E(\overline{X} - X_{(1)})^2 = \frac{n-1}{n \lambda^2}\,.\]
Use this fact and the results of parts (a) and (b) to compute the value
of
\[ E \left( \frac{S^2}{(\overline{X} - X_{(1)})^2} \right) \,.\]
Justify each step of your calculation.  (If you could not prove the
completeness in (b), simply assume it now.)

\end{parts}





\QQ
%Meeter -- Applied Stat
The problem of bias in judging Olympic diving and ice skating seemed 
to be particularly noticeable during the Cold War, and 
there is usually a {\it perception} of bias indicated by some noisy 
members of the audience at every Olympics.\\  

Suppose you have seven judges and ten ice skaters in the Olympics; each judge
observes a skater's performance and assigns a score to him/her, for 70 scores
in all.

\begin{parts}
\PP
State a model which allows for systematic differences among judges and among skaters.  Identify symbols; give the ANOVA table.\\

Suppose that Skater 1 is Nancy Kerrigan from the USA and Skater 10 is Oksana Baiul from the former USSR; Judge 1 is from the USA and Judge 7 is from the former USSR.

\PP
How would you graphically analyze the data in order to see whether (say) Judge 1 favors Skater 1 and/or downrates Skater 10, and/or Judge 7 downrates Skater 1 and/or favors Skater 10?


\PP
We know from the work of Cuthbert Daniel that an outlier in cell $i,j$ of a two-way layout affects the predictions for other cells in Row $i$ and  Column $j$
in the obvious direction.  How does this knowledge affect your opinion about the sensitivity of the analysis in b)?

\PP
Modify the model of a) to allow for any or all of the biases mentioned in b).

\PP
Describe how you would test the null hypothesis that Judge 1 is not biased in favor of Nancy Kerrigan, using the model of d).  (Identify the things you need to calculate and give the test statistic and the df.  You may assume, if you need to, that you have a standard regression program available.)  

\end{parts}
\QQ
%Sethuraman -- Probability 
\begin{parts}

\PP
Show that if a random variable $X$ is independent of itself, 
then $X$ is degenerate.  

\PP
Let $X$ and $Y$ be two random variables with $X>0, Y>0$ and such that 
$E(X|Y) = Y$ and $E(Y|X) = X$. Show that $X=Y$ with probability $1$.\\ 
Hint: use the fact that $\frac{x}{y} + \frac{y}{x} \ge 2$ if $x>0, y>0$.   
\end{parts}

\QQ
% Niu -- Applied Statistics
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:
$\xi_i= \epsilon_i-0.70\epsilon_{i-1},$
where $\{\epsilon_i, i=0,1, \ldots, n\}$ are iid $N(0,\sigma^2)$ variables.

\begin{description}
\item[\bf (a)] Calculate the covariance matrix $\Gamma= \mbox{Cov}(\xi)$

\item[\bf (b)] Let $\hat{\beta}$ be the least squares estimate of $\beta$,
 $\hat{Y}=X\hat{\beta}$ and $\hat{\xi} = Y-\hat{Y}$. Are the following
 two statements valid? Why or why not? 

i) $\hat{\beta}$ is an unbiased estimate of $\beta$; \hspace{0.2in}
ii) $\hat{Y}$ and $\hat{\xi}$ are independent.

\item[\bf (c)] 
Find the maximum likelihood estimate $\tilde{\beta}$ of  $\beta$.
Define  $\tilde{Y}=X\tilde{\beta}$ and $\tilde{\xi} = Y-\tilde{Y}$.
Are the two statements in (b) valid for $\tilde{\beta}, \tilde{Y}$ and $\tilde{\xi}$?

\end{description}

\QQ
%McKeague -- Probability
An engineer who studies traffic congestion comes to 
you with the problem of finding the expectation of a 
waiting time $T\ge 0$.  The tail probabilities
of $T$ are given by the mean of 
a certain nonnegative stochastic process
$\{ X(t), t\ge 0\}$:
$$P(T>t)=E[ X(t)]\quad {\rm for\ all\ } t\ge 0.$$
\begin{parts}
\PP
Let $L$ be a mean-1 exponential random variable which is independent 
of the process $\{ X(t), t\ge 0\}$.
Show that
$$Y=\int_0^L e^t X(t)\, dt$$
is an unbiased estimator of $E(T)$. 
\PP
Express the condition of independence that is used in a)
in terms of independence between events.

\PP
State any extra assumptions ({\it if} any) 
that are needed for your solution to a).
\end{parts}


\QQ
%Lin -- Applied Stat
\begin{parts}

\PP
Let $X$ be a binomial random variable with
parameters $n$ and $p$, and set $\hat p=X/n$.  Show that
$$2\sqrt n\left(\sin^{-1}\sqrt{\hat p} -\sin^{-1}\sqrt{ p}\right)\god N(0,1).$$

\PP
Researchers in a school of medicine plan to conduct a 
clinical experiment comparing the infection rates of minor
cut and injury when the fresh wound is cleansed either with saline solution
or with tap water.  Let $p_1$ and $p_2$ be the infection rates 
using saline solution and tap water, respectively.\\

The researchers wish to know how large the sample sizes $n_1$ and $n_2$
need to be to test the hypothesis
$$H_0\colon p_1=p_2\quad {\rm vs}\quad H_1\colon p_1<p_2$$
at the 5\% level with 80\% power.
(Preliminary results indicate that $p_1\approx .03$ and
$p_2\approx .08$.) Using a) or otherwise,
determine suitable $n_1$ and $n_2$.  Express your
answer in a form that could be numerically evaluated using Splus.  
Make all necessary assumptions.
\end{parts}

\QQ
%Li-Shan Huang -- Inference
Let $X_1,  X_2, \dots, X_n$ be i.i.d.\ with  uniform distribution on
$(0, \theta), \theta>0$.
\begin{parts}
\PP
Show that $T=\max_i X_i$ is sufficient for $\theta$.
\PP
Show that $T$ is complete for $\theta$.
\PP
Find  $c$ such that $cT$ is unbiased for $\theta$ and deduce that
 $cT$ is the  uniformly minimum variance unbiased (UMVU) estimator for $\theta$.
\PP
Consider all estimators of the form $cT$. Show that the UMVU estimator is 
inadmissible with respect to mean square error.
\end{parts}


\QQ
%McKeague -- Probability
Let $X_n $, $Y_n$, $n\ge 1$, be 
random variables such that the conditional
distribution of $Y_n$ given $X_n$ is Bernoulli with
parameter $p_n(X_n)$, where $p_n\colon \RR \to [0,1]$ is non-random.

\begin{parts}
\PP
Suppose that $X_1,X_2,\ldots$ are i.i.d.\ Bernoulli with
parameter $0<p<1$ and $p_n(0) = n^{-2}= 1-p_n(1)$. Show that
$$n^{-1/2}\sum_{i=1}^n (Y_i-p)$$
 converges 
in distribution as $n\to \infty$.
\PP  Suppose that $X_n$ converges
in distribution to a continuous random variable $X$,
and $p_n=F$, where $F$ is a fixed distribution function.
Show that $Y_n$ converges in distribution as $n\to \infty$.
\PP 
Identify the limits established in a) and b). 
\end{parts}



%Kaisheng Song -- Inference
\QQ
Let $X_1, ..., X_n$ be i.i.d. with pdf $f(x,\theta )=\theta
x^{-(\theta +1)}I_{(1,\infty )}(x)$ and suppose that $\theta >2$.
Let $\hat \theta_n$ be the method of moments estimator of
$\theta $.
\begin{parts}
\PP
Determine which of the following is true (you need to show your
work):
\begin{enumerate}
\item
$\hat \theta_n$ is an unbiased estimator of $\theta$;
\item
$\hat \theta_n$ has an upward bias (i.e.,
$E_\theta(\hat \theta_n)>\theta$);
\item
$\hat \theta_n$ has a downward bias.
\end{enumerate}
\PP
Find the asymptotic distribution of $\hat \theta_n$.
\PP
Calculate the asymptotic relative efficiency of $\hat \theta_n$ when
$\theta=3$.
\end{parts}

\end{document}

