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\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam\\
Tuesday, August 20, 2002

\end{center}

\vsp Please submit solutions to at most {\bf seven} problems.
You have four hours.  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 complete factorial experiment, consider the following  mixed-effects model:
\begin{eqnarray*}
Y_{ij} & = & \mu +\alpha_i + \beta_j + \epsilon_{ij}, \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b,
\end{eqnarray*}
where  $\mu$ is the overall mean, $\alpha_i$ is the fixed effect 
corresponding to the $i$th level of factor $A$, and $\beta_j$ is 
the random effect due to the $j$th level of factor $B$.
The  $\beta_j$'s are iid $N(0, \sigma^2_{\beta})$ variables
  and the $\epsilon_{ij}$'s are iid
 $N(0, \sigma^2_{\epsilon})$ variables.
Furthermore, $\{\beta_j\}$ and $\{\epsilon_{ij}\}$ are assumed to be independent.
 Let $MSA$ and $MSB$ denote the mean squares for factors $A$ and $B$,
respectively, and let $MSE$ be the mean square for error.



\begin{parts}
\PP
What are the differences between a random-effect design and a fixed-effect design?
What hypotheses do you test in the mixed-effects model?

\PP
Calculate the expected values E($MSA$) and E($MSB$).

\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:
\[\xi_i + \theta \xi_{i-1}= \epsilon_i,\]
where $\{\epsilon_i, i=0, \pm 1,  \pm 2 \ldots,\}$ are iid $N(0,\sigma^2)$ variables.

\begin{parts}
\PP
Show that $\{\xi_i, i=1,2,\ldots,\}$ is a second-order stationary process
when $|\theta|<1$.
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 mean, covariance matrix, and distribution of $\hat{\beta}$.

\end{parts}


\QQ
  A $2^{5-2}$ design is defined by $4=12$ and $5=13$.

\begin{parts}
\PP
Find the defining relation and resolution of this design. What are the advantages
and disadvantages of this kind of design compared with complete factorial designs?

\PP
After the experiment, factor 5 turns out to be  inert. It is assumed that 
all two-factor interactions involving factor 5 and all higher order interactions
are negligible. In addition to estimating the four main effects, there are
still three degrees of freedom left. What two-factor interactions can be 
estimated with these three degrees of freedom?

\end{parts}


\QQ 
Let $Y_1, \ldots, Y_n$ be independent random variables such that
$Y_i \sim \mbox{Binomial}(m, \pi_i)$ where $0<\pi_i<1$. Let
$Y=\sum_{i=1}^n Y_i$.

\begin{parts}
\PP
Show that, given $\pi_1, \ldots, \pi_n$, E$(Y) = nm \bar{\pi}$ and 
Var$(Y) = nm \bar{\pi}(1-\bar{\pi}) - m (n-1) k(\pi)$. Give the expression
for $k(\pi)$ in terms of $\pi_1, \ldots, \pi_n$.

\PP
Assume that $\pi_1, \ldots, \pi_n$ are independent random variables
with common mean $\pi$ and common variance $\tau^2\pi(1-\pi)$. Show that,
unconditionally, E$(Y) = nm \pi$ and
Var$(Y) = nm \pi(1-\pi)[1+(m-1)\tau^2]$. 

\end{parts}

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



\QQ
\begin{parts}
\PP
Define the notion of product measurability.
\PP
Let $f\colon \Omega \to \RR$ be a measurable function, where
$(\Omega, {\cal F})$ is a measurable space.  
Show that the set
$${\rm Graph}(f)= \{(\omega,f(\omega))\colon \omega \in \Omega\}$$
is product measurable in a suitable sense.  Hint: 
the graph of $f$ is the inverse image under $(\omega, y)\mapsto
(f(\omega),y)$
of a certain Borel
measurable subset of $\RR^2$.
\end{parts}


\QQ
\begin{parts}
\PP State Markov's inequality.
\PP 
State H\"older's inequality and use it to derive
Lyapounov's inequality.  
\PP 
What can you say about a random variable
$X$ such that $E|X|^n\to 0$ as $n\to\infty$?
\PP 
Does there exist a random variable $X$ for which
$E|X|^n=1/n$ for all $n\ge 2$?

\end{parts}



\QQ Let $X_1,\dots,X_n$ be i.i.d random variables with pdf $1$ on
[0,1] (i.e. uniform) and let $Y_1,\dots,Y_m$ be i.i.d. with pdf
$2x$ on $[0,1]$, and let the $X$'s be independent of the $Y$'s.
Let $\phi(x;y) = I(x\le y)$ and let $\phi^*(x;y) = \phi(x;y) -
\frac{2}{3}$. Let $N = n+m$ and let $\frac{n}{N} \rightarrow
\lambda$ where $0< \lambda <1$. Define
\[U_N = \frac{1}{nm} \sum_{i=1}^n \sum_{j=1}^m \phi^*(X_i;Y_j)\,,\]
the two-sample $U$-statistic based on $\phi^*$.  State any
requisite result on such $U$-statistics and obtain the asymptotic
distribution of $U_N$. 

\QQ
Let $X_1,X_2,\dots$ be random variables lying in $[0,1]$. Let
${\cal F}_0$ be the $\sigma$-field containing the empty set and
the whole space, and let ${\cal F}_n = \sigma(X_1,\dots,X_n)$,
the smallest $\sigma$-field generated by $X_1,\dots,X_n$. Show
that the series
\[\sum_{i=1}^\infty X_i \]
and the series
\[\sum_{i=1}^{\infty} E(X_i|{\cal F}_{i-1}) \]
converge or diverge together, with probability one.

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


\QQ Suppose $Y \sim (X\beta,\sigma^2I)$ where $X$ has {\bf less
  than} full rank.  Let $M$ denote the orthogonal projection
matrix onto the range of $X$.  Prove the following: If $d'Y$ is
an unbiased estimate of $\ell'\beta$, then $(Md)'Y$ is the BLUE
of $\ell'\beta$.

\QQ Let $X_1, X_2, \ldots, X_n$ be iid Uniform($\theta,2\theta$)
where $\theta > 0$.

\begin{parts}

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

\PP Show that the statistic you found in part (a) is {\bf not}
complete.
  
\end{parts}



\QQ Suppose that the observed failure times of ball bearings 
$Y_1, \ldots, Y_n$ are a random sample from the lognormal 
distribution with p.d.f. 
\begin{displaymath}
 f(y)=\frac{1}{y\sqrt{2\pi}}
\exp\{-(\log y-\mu)^2/2\} \;\;\; \mbox{for $y >0$}
\end{displaymath}
 where the parameter $\mu$ is unknown. 

\begin{parts}
  

\PP  Find the MLE of the mean failure time (i.e., $E(Y)$) of 
the ball bearings and its asymptotic distribution. 

\item[] [Note: In your calculations, you may find it useful to
  recall the mgf of the Normal distribution: if $X \sim
  N(\mu,\sigma^2)$, then $E e^{tX} = e^{\mu t + \sigma^2
    t^2/2}$\,.]



\PP Another natural estimator of the mean failure time would be
the average of the $Y's$: $\bar Y_n=\frac{1}{n}\sum_{i=1}^nY_i$.
Is this estimator a good estimator relative to the MLE? \\

 
%If not,
%use $\bar Y_n$ to construct a better estimator.

\PP Find $E(\bar Y_n \,|\, \bar X_n)$ where we define $\bar X_n =
\frac{1}{n}\sum_{i=1}^n X_i$ with $X_i = \log Y_i$\,.
  
\end{parts}


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