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\begin{center}
\bf \large
Ph.\ D.\ Preliminary Exam, Part I (Elementary Exam)\\
Tuesday, January 5, 1993
\end{center}

\vsp
You have three 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}

Suppose  that  $\{X_{1}, X_{2}, \ldots, X_{n}\}$ is a sample from a
population with Weibull density
$$
p(x, \lambda) = \lambda c x^{c-1}\exp(-\lambda x^{c}) \quad
x > 0, \hspace{0.1cm} \lambda > 0,
$$
where $c$ is a known positive number.
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1]
Find a sufficient statistic for $\lambda$.  Is the statistic complete?
\item[2]
Find a uniformly minimum variance unbiased estimate for
$\frac{1}{\lambda}$. What is the variance of this estimate?
\end{list}



\vsp\noindent {\bf Problem 2} \hspace{.2cm}
Let $\alpha_1, \alpha_2, \ldots$ be a sequence of independent random
variables with
$$
\alpha_j \sim \hbox{Poisson$(z^j/j)$}
$$
where $0 < z < 1$ is a fixed parameter.  Let 
$$
\nu = \sum_{j=1}^\infty j\alpha_j
$$
Show that $ \nu \sim \hbox{Geometric($1 - z$)}$.

\vspace{2ex} \noindent
Hint: For $0 < u < 1, \; \sum_{j=1}^\infty u^j/j = -\log(1 - u)$.



\vsp\noindent {\bf Problem 3} \hspace{.2cm}
Suppose that $Y_1, \ldots, Y_n$ are independent random variables with
means $\alpha + \beta x_1, \ldots, \alpha + \beta x_n$, respectively,
and common variance $\sigma^2 > 0$, where $x_1, \ldots , x_n$ are known
constants, but $\alpha$, $\beta$, $\sigma^2$ are unknown.  From many
previous experiments at a single fixed covariate value, say $x_0$, the
average value of the response variable $Y$ was $y_0$.  The individual
data were not saved, but from the previous work it may be assumed
that the point $(x_0, y_0)$ lies on the line $y = \alpha + \beta x$.
Find the least-squares estimators for $\alpha$ and $\beta$.



\eject
\vsp\noindent {\bf Problem 4} \hspace{.2cm}
Suppose $X$ and $Y$ are i.i.d.\ positive random variables, and
$Z = X/(X + Y)$.
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1]
Find the mean $\mu_Z$ of $Z$.
\item[2]
If the variance $\sigma_Z^2$ of $Z$ is known, place a lower
bound on $E(1/Z)$.

Hint:  use Taylor series.
\end{list}



\vsp\noindent {\bf Problem 5} \hspace{.2cm}
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1]
Why is it important to have all important predictor variables in a
multiple regression analysis?   Compare the consequences of including
worthless predictors in the model vs.\ excluding important predictors
from the model.
\item[2]
Discuss some aspects of how multicollinearity complicates multiple
regression analyses.
\item[3]
Present one assumption that is commonly made in the multiple regression
model.  How can you check to see if the data are consistent with this
assumption?  What remedial measures might be used if the assumption
were violated in the data set?
\end{list}



\vsp\noindent {\bf Problem 6} \hspace{.2cm}
Consider the simple linear regression model with intercept
$$
Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad i = 1, \ldots, n.
$$
Assume that the $\epsilon$'s have mean 0 and are uncorrelated.

In model diagnosis a common approach to evaluate the adequacy of a
model would be to plot the residuals against the predicted values.  The
model is judged to be adequate if there is no visible apparent pattern
in the scatterplot.
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1]
Give a mathematical justification for this approach.
\item[2] 
Is this approach still valid in the evaluation of model adequacy if
$\beta_0$ is known to be 0?  Why or why not?
\end{list}



\vsp\noindent {\bf Problem 7} \hspace{.2cm}
Let $X$ and $Y$ be independent random variables uniformly distributed
on $[0,1 ]$. Find the distribution of $\min(X, Y) / \max(X, Y)$.



\eject
\vsp\noindent {\bf Problem 8} \hspace{.2cm}
Let $(X_1, Y_1), \ldots, (X_n, Y_n)$ be i.i.d.\ pairs of points from a
bivariate normal distribution.  Denote the mean of $X_1$ by $\mu_X$,
the mean of $Y_1$ by $\mu_Y$, the variance of $X_1$ by $\sigma^2_X$,
the variance of $Y_1$ by $\sigma^2_Y$, and the correlation coefficient
between $X_1$ and $Y_1$ by $\rho$.  Suppose that $n \ge 2$ and that
there are no ties among the $X$'s or the $Y$'s.
\begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
\setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
\item[1]
Write the formula for ${\hat \rho}$, the Pearson (i.e.\ sample)
correlation coefficient.
\item[2]
Prove or disprove the following statement.
The distribution of ${\hat \rho}$ is symmetric about $\rho$.
\item[3]
Show that the distribution of ${\hat \rho}$ does not depend on
$\mu_X, \; \mu_Y, \; \sigma^2_X$, or $\sigma^2_Y$.

%\item[4]
%If $n = 2$, what can you say about the value of ${\hat \rho}$? 
\end{list}


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