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\begin{center}
\bf \large

Ph.\ D.\ Preliminary Exam, Part II (Advanced Exam)\\
Monday, January 4, 1993

\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.}


\vsp\noindent {\bf Problem 1} \hspace{.2cm}
Let $X_1, X_2, \ldots$ be i.i.d.\ nonnegative random variables with a
finite second moment.  Show that 
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1] 
$n^{-{1 \over 2}} \max_{1 \le i \le n} X_i \to 0$ almost surely.

\item[2] 
$ n^{-{3 \over 2}} \sum_{i=1}^nX_i^3 \to 0$ almost surely.
\end{list}



\vsp\noindent {\bf Problem 2} \hspace{.2cm}
Let $ X_1, X_2, \ldots $ be a sequence of i.i.d.\ random variables with
the $\Gamma(2,1)$ distribution, i.e.\ the $X_i$'s have density 
$$
f(x) = x \exp(-x) \quad \mbox{for } x > 0
$$
Show that 
$$
\limsup_{n \to \infty} { X_n \over \log n } = 1 \quad \hbox{a.s.}
$$

\vspace{2ex} \noindent
Hint: write down the expression for $P(X_i > a)$. 


\vsp\noindent {\bf Problem 3} \hspace{.2cm}
Suppose that $X_1, \ldots, X_n$ are i.i.d.\ random variables with the
exponential distribution with unknown parameter $\lambda$.  Suppose
that $\lambda$ has as prior distribution the Gamma distribution with
parameters $\alpha$ and $\beta$. Recall that this distribution has
density
$$
f(\lambda) =  \frac{\beta^{\alpha}}{\Gamma(\alpha)}
\lambda^{\alpha -1} \exp(-\beta \lambda)
\quad \mbox{for $\lambda > 0$}.
$$
Consider the censoring model in which the $X_i$'s are (possibly)
censored by constants $c_i$. That is, we do not necessarily observe the
$X_i$'s; rather, our data consists of pairs $(Z_i, \delta_i); \; i = 1,
\ldots, n$, where
$$
Z_i = \min(X_i, c_i) \qquad \mbox{and} \qquad \delta_i = I(X_i \le c_i).
$$
Find the posterior distribution of $\lambda$.

\vspace{1ex} \noindent
Let $t_0$ be a fixed constant.  Find a $95\%$ probability interval for
$\exp(-\lambda t_0)$.



\vsp\noindent {\bf Problem 4} \hspace{.2cm}
Let $X$ be a Poisson random variable with mean $\theta$ and let
$F_{\theta}$ be the conditional distribution of $X$ 
given that $X > 0$.
Let $Z_1, Z_2, Z_3, \ldots$ be i.i.d.\ from $F_{\theta}$.
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1]
Suppose you observe $Z_1 = Z_2 = 1,\, Z_3 = 2$.
What is the likelihood equation for $\theta$?
Use the likelihood equation to calculate the MLE of $\theta$
to the nearest tenth.

\item[2]
Suppose you observe $Z_1 = Z_2 = Z_3 = 1$.  What can you say about the
MLE of $\theta$?

\item[3]
Does there exist an unbiased estimator of $\theta$ based on a single
observation $Z_1$?  If so, what is this estimator?

\item[4]
What is the asymptotic variance of the MLE of $\theta$ based on a sample $Z_1,
Z_2, \ldots, Z_n$ as $n \rightarrow \infty$? 
(Be explicit.)

\end{list}



\vsp\noindent {\bf Problem 5} \hspace{.2cm}
Let $X_1, X_2, \ldots$ be i.i.d.\ random variables with common normal
distribution $N(\mu, \sigma^2)$ with $\sigma^2 > 0$. Obtain the asymptotic
distribution of $T_n = \frac{s_n^2}{\bar X_n}$ where
$\bar X_n = \frac{1}{n}\sum_1^n X_i$ and $s_n^2 = \frac{1}{n}\sum_1^n
X_i^2 - {\bar X_n}^2$.

\vspace{2ex}
\noindent
(Hint: Consider separately the cases $\mu = 0$ and $\mu \not= 0$).



\vsp\noindent {\bf Problem 6} \hspace{.2cm}
Suppose that $\epsilon_{1}, \epsilon_{2}, \ldots, \epsilon_{n}$ are
i.i.d.\ with density $\frac{1}{\sigma}f(\frac{\epsilon}{\sigma})$,
$-\infty < \epsilon <\infty$, and $\mbox{Var}(\epsilon_i) = \sigma^{2}
< \infty$.  Suppose also that the vector of observations $\vecc{Y} =
(y_{1}, y_{2}, \ldots, y_{n})$ satisfies the linear model
$$
\vecc{Y} = \vecc{X}\vecc{\beta} + \vecc{Z}\vecc{\gamma} + \vecc{\epsilon},
$$
where $\vecc{X}$, $\vecc{Z}$ are known $n \times p$ and $n \times q$
full-rank matrices and $\vecc{\beta}$, $\vecc{\gamma}$ are unknown
parameters.  Let $\hat{\vecc{\beta}}$ and $\hat{\vecc{\gamma}}$ be the
least squares estimates of $\vecc{\beta}$ and $\vecc{\gamma}$,
respectively.  Define \[\hat{\vecc{\epsilon}}=\vecc{Y} -
\vecc{X}\hat{\vecc{\beta}} +\vecc{Z}\hat{\vecc{\gamma}}, \hspace{0.2in}
\hat{\sigma}^{2} =
\frac{\hat{\vecc{\epsilon}}'\hat{\vecc{\epsilon}}}{n-p-q}\] and
$$
\vecc{P}_{X} = \vecc{X}(\vecc{X}'\vecc{X})^{-1}\vecc{X}'.
$$
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1]
Show that the distribution of $\vecc{R}=(I-\vecc{P}_{X})\vecc{Y}$ does
not depend on $\vecc{\beta}$.

\item[2]
If the $\epsilon_{i}$'s have normal distribution with mean equal to 0,
show that the estimates $\hat{\vecc{\beta}}$ and $\hat{\sigma}^{2}$ are
independent.

\item[3]
If the $\epsilon_{i}$'s have normal distribution with mean equal to 0,
show that the likelihood for $(\vecc{\gamma}, \sigma^{2})$ based on
$\vecc{R}$ is identical (up to a multiplicative constant) to the
conditional likelihood of $(\vecc{\gamma}, \sigma^{2})$ given
$\vecc{P}_{X}\vecc{Y}$.

\end{list}



\pagebreak[2]
\vsp\noindent {\bf Problem 7} \hspace{.2cm}
A researcher locates four nests of birds of the same species, each of
which contains four eggs.  For each nest, the four eggs it contains are
transferred to the other nests so that each nest ends up with one egg
each from nests A, B, C, and D.  Thus, each nest contains one egg which
was originally laid in it.  Suppose that all the eggs hatch, and the
response $Y$ = weight gain after 14 days is observed.  The origin of
the chicks is known.  The researcher wants to test the hypothesis that
chicks that are hatched and grown in their own nests gain more than
chicks from eggs which have been transferred from another nest.

Describe a test of this hypothesis.  The researcher feels that it is
reasonable to assume that the weight gain of chicks does not depend on
which nest they are taken from.  However, it is preferable to not assume
that different nests produce the same mean weight gain.



\vsp\noindent {\bf Problem 8} \hspace{.2cm}
An investigator collects data $(Y_i, x_i); \; i = 1, \ldots, 10$, where
the $Y_i$'s are response variables and the $x_i$'s are the observed
values of some univariate covariate.  The least squares linear
regression of $Y_i$ on $x_i$ has residual sum of squares equal to 27.
Later, the investigator collects 20 more observations $(Y^{'}_i,
x^{'}_i); \; i = 1, \ldots, 20$.  The least squares linear fit now has
residual sum of squares equal to 64.  When all 30 points are fitted by
a single linear regression the residual sum of squares equals 99.  Test
the hypothesis that the two sets of data follow the same linear
regression.



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