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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.\ D.\ Qualifying Exam\\
  Monday, January 3, 2011

\end{center}

\vspace{.1in}

{\bf Put your solution to each problem on a separate sheet of paper.}

\vspace{.2in}


% xxx Note: Need an F-table for this problem.
\QQ {5166}  An
experiment was planned to compare three different fertilizers (A, B,
C) on watermelon yields. The treatments were randomly assigned
according to a Latin square design conducted over a large farm plot
which was divided into rows and columns. The watermelon yields (in
tons per acre) were recorded below after the growing season. \\ [3pt]

\begin{tabular}{cccc} \hline
 & \multicolumn{3}{c}{Column} \\ \cline{2-4}
Row & 1 & 2& 3  \\ \hline
1  & 6.5 (B) &  9.8 (C)  & 5.0 (A)   \\
2  & 7.2 (A) &  8.1 (B)  & 8.6 (C) \\
3  & 10.5 (C) &  6.6  (A) & 6.7 (B) \\ \hline
\end{tabular}

%\vspace{0.1in}

\begin{parts}
\PP
 If you were the experimenter, how would you randomize the design? Write
an appropriate additive model for the data and state your assumptions on the model.


\PP Conduct an analysis of variance ($\alpha=0.05$) based on the
data and draw your conclusions.
\end{parts}


\QQ {5166} Consider the following two $2^{7-2}$ fractional factorial
designs: \\ [3pt]

\begin{tabular}{cccc}\hline
$d_1$: & with & 6=1234, &7=1235 \\
$d_2$: & with & 6=123,  & 7=145 \\ \hline
\end{tabular}


\begin{parts}

  \PP What is the defining relation for each of the two designs?  What
  is the resolution of each design?  For each design, write out the
  confounding pattern of the main effects $\{1, 2, 3, 4, 5, 6, 7\}$
  with other effects.

  \PP Write out the word-length pattern for each design.  Which design
  is better?  Justify your choice.

\end{parts}


\QQ {5106} Let $X$ be a real-valued, continuous random variable.
Assuming that you are given a program for simulating $U$, the uniform
random variable on $[0,1]$:

\begin{parts}

  \PP Describe the inverse transform method for generating random
  samples of $X$.

  \PP Prove that this method works.

  \PP Write a program to simulate the exponential random variable with
  mean $\lambda$.

\end{parts}


\mbox{}

\vspace{-.75in}

\QQ {5327} Let $X_1, \ldots, X_n$ be a random sample from the density (pdf)
\begin{displaymath}
  f(x|\theta) = \frac{2\theta x e^{\theta x^2}}{e^{\theta}-1}\,,
\quad 0 \leq x \leq 1\,.
\end{displaymath}

Answer the following. Justify your answers in detail.  Show all
calculations.

\begin{parts}
  
\PP Find a complete sufficient statistic for $\theta$.

\PP Is there a function of $\theta$, say $g(\theta)$, for which there
exists an unbiased estimator whose variance attains the Cram\'er-Rao
Lower Bound?  If so, find it.  If not, show why not.

\PP Find $I(\theta)$, the Fisher information in a {\bf single} observation
from $f(x|\theta)$.

\end{parts}


\QQ {5327} Suppose you observe $X_1,\ldots, X_n$ iid
$\text{Uniform}(\theta,\theta+1)$. Answer the following.  Justify your
answers.

\begin{parts}
  
\PP Find a minimal sufficient statistic for $\theta$.

\PP Is the statistic you found above complete?

\PP Suppose you are a Bayesian and your prior distribution for
$\theta$ is exponential with mean 1.  Find the posterior density for
$\theta$.

\end{parts}


\QQ {6346} Suppose $X \sim \text{Geometric}(p)$ with pmf 
\begin{displaymath}
P(X=x) =
(1-p)^{x-1}p
\quad\text{for $x=1, 2, 3, \ldots$}
\end{displaymath}
and $X_1, X_2, X_3, \ldots$ is an arbitrary sequence of random
variables.

For each of the functions $g$ below, answer the following question: If
$X_n \Rightarrow X$, is it necessarily true that $Eg(X_n) \rightarrow
Eg(X)$?

If your answer is ``yes'', state your reason(s).  If it is
``no'', give a counterexample.\\ \mbox{} [Note: $\Rightarrow$ and
$\stackrel{\cal L}{\rightarrow}$ and $\stackrel{d}{\rightarrow}$
all mean the same thing: convergence in distribution (or weak
convergence).] \mbox{}

\begin{parts}
  
\PP $g(x) =I_{(-1,5/2)}(x) = I(-1<x<5/2)$

\PP $g(x) = I(\sin(\pi x)>0)$

\PP $g(x) = e^{-|x|}$

\PP $g(x) = \frac{1}{x+1}$

\PP $g(x) = e^{itx}$ where $i =\sqrt{-1}$ and $t \in \mathbb{R}$

\end{parts}



\QQ {5167} In a complete factorial experiment, consider the following
random-effects model:
\begin{eqnarray*}
Y_{ijk} & = & \mu +\alpha_i + \beta_j  + \epsilon_{ijk}, \\
& & \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1,\ldots, r,
\end{eqnarray*}
where $\mu$ is the overall mean, the random effects $\alpha_i$ and
$\beta_j$, are assumed to be independent and normally distributed with
means of zero and variances $\sigma^2_{\alpha}$ and
$\sigma^2_{\beta}$, respectively, and the random errors
$\{\epsilon_{ijk}\}$ are assumed to be independent and normally
distributed with mean zero and variance $\sigma^2$.  Furthermore,
$\{\alpha_i\}$, $\{\beta_j\}$, and $\{\epsilon_{ijk}\}$ are assumed to
be independent of one another.


\begin{parts}

  \PP What are the distributions of $\bar{Y}_{ij\cdot}$,
  $\bar{Y}_{i\cdot\cdot}$, and $\bar{Y}_{\cdot j \cdot}$?  Find the
  covariance Cov($\bar{Y}_{ij\cdot}, \bar{Y}_{uv\cdot}$).

  \PP Let $S_A =br \sum_{i=1}^a(\bar{Y}_{i\cdot\cdot}-\bar{Y})^2$,
  $S_B =ar \sum_{j=1}^b(\bar{Y}_{\cdot j\cdot}-\bar{Y})^2$, and
  \[S_R
  =\sum_{i=1}^a\sum_{j=1}^b\sum_{k=1}^r(Y_{ijk}-\bar{Y}_{i\cdot\cdot}-\bar{Y}_{\cdot
    j\cdot}+\bar{Y})^2.\] Find the mean values of $S_A$, $S_B$, and
  $S_R$.

  \PP Find unbiased estimates for the variances $\sigma^2_{\alpha}$
  and $\sigma^2$ based on the observations $\{Y_{ijk}, \quad
  i=1,\ldots, a; \quad j=1, \ldots, b; \quad k=1,
  \ldots,, r\}$.

\end{parts}



\QQ {5167} Consider a multiple linear regression model with the form:
\begin{eqnarray*}
 y_i = \beta_0 + \beta_1 x_{i1} + \cdots +\beta_px_{ip}+\xi_i\,,
\quad i=1,\ldots,n,
\end{eqnarray*}
where $\vecc{\xi}=(\xi_{1}, \ldots, \xi_n)'$ has a multivariate normal
distribution with mean zero and covariance matrix
$V=\sigma^2\Sigma(\vecc{\theta})$, i.e., each element of matrix
$\Sigma$ is a function of the vector parameter
$\vecc{\theta}=(\theta_1, \ldots,\theta_q)'$.  Let
$\vecc{\beta}=(\beta_0, \beta_1, \ldots, \beta_p)'$.


\begin{parts}

  \PP How do you estimate the parameters $\vecc{\beta}$,
  $\vecc{\theta}$, and $\sigma^2$ in this model?

  \PP How do you define outliers in this linear model?.

  \PP Give a procedure to identify and test outliers in this
  model and give your justifications for the procedure.

\end{parts}


\QQ {5106} Suppose ${\bf x} \in \real^n$ is a vector of random variables
and one is interested in finding a smaller vector ${\bf z} \in
\real^d$ ($d << n$) that represents ${\bf x}$ as closely as possible.
The vector ${\bf z}$ is constrained to be ${\bf z} = U^T {\bf x}$,
where $U \in \real^{n \times d}$ is a tall, skinny matrix.  The
criterion for selecting $U$ is such that $E[\| {\bf x} - U{\bf z}\|^2]
= E[\| {\bf x} - UU^T{\bf x}\|^2]$ is minimized.  (Here $\| \cdot \|$
denotes the 2-norm of a vector.)  Derive an expression for the optimal
$U$ and illustrate your derivation using clear steps. \\ [3pt]

\QQ {5326} Consider a Pareto distribution with density (pdf)
\begin{equation*}
  f(x) =
  \begin{cases}
{\displaystyle \frac{3}{x^4}} & \text{for $1 \leq x < \infty$,}\\[8pt]
0 & \text{otherwise.}
  \end{cases}
\end{equation*}

\begin{parts}
  
\PP For this density, calculate the mean, median, mode, and variance.

\PP Suppose $X$ and $Y$ are iid with this density.  Find the density
of their product $Z=XY$. (Make sure to specify the support of the
density.)

\end{parts}



\QQ {5326} A random sequence of men and women enter a random
sex-change clinic.  The people entering the clinic are independent
with probability 2/3 of being a man and 1/3 of being a woman.  In the
clinic, men are transformed into women with probability 1/4, and women
are transformed into men with probability 1/5, with these decisions
being made independently for each man or woman.  Assume that one
person arrives at the clinic each day, entering in the morning and
leaving in the evening.

Give a detailed argument and calculation for each of the following.

\begin{parts}

\PP A person is seen {\bf leaving} the clinic.  What is the probability this
person is a male?

\PP If a man is seen {\bf leaving} the clinic, what is the probability
that it was a man who {\bf entered} the clinic?

\PP If, over the course of two days, two men are seen {\bf leaving} the clinic,
what is the probability that it was two men who {\bf entered} the clinic?

\PP Over the course of two days, two people are observed {\bf leaving}
the clinic.  If at least one of the people leaving is a man, what is
the probability that both are men?
  
\end{parts}


\QQ {6346} \mbox{}

\begin{parts}
  
\PP State the Radon-Nikodym Theorem, and define the important terms used
in this theorem.  Be as detailed as you can.

\PP Show that if $\nu \ll \mu \ll m$, then $\displaystyle
\frac{d\nu}{d\mu} = \frac{\frac{d\nu}{dm}}{\frac{d\mu}{dm}}\,.$

\end{parts}



\end{document}

