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\begin{document}


\begin{center}
  \bf \large
  M.\ S.\ Comprehensive Exam\\
  Friday, August 25, 2006
\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.}


\vsph

%%%%%%%%%%%%%%%%%%%% Applied Statistics %%%%%%%%%%%%%%%%%%%%%%


\QQ In a regression setting, there are $p-1$ explanatory variables
$X_1, X_2, \ldots, X_{p-1}$ and $n$ observations.  Let SSE($X_{i_1},
X_{i_2}, \ldots, X_{i_k} | R$) be the sum of squares error when
regressing some response function $R$ on $k\leq p-1$ variables and the
intercept.  For example, the error when regressing $Y$ on the
intercept and $X_1$ is SSE($X_1 | Y$).  For each of the following
hypotheses, give the full and reduced model; the test statistic using
SSE notation; and the distribution of the test statistic.

\begin{parts}
  \PP $H_0: \beta_1=\beta_2=\beta_3=0$, $H_a: $ not $H_0$

  \PP $H_0: \beta_1=\beta_2$, $H_a: \beta_1\neq\beta_2$

  \PP $H_0: \beta_1=2, \beta_2=-3$, $H_a: $ not $H_0$

\end{parts}


\QQ Suppose it is desired to conduct a fractional factorial experiment
involving five factors ($A$, $B$, $C$, $D$, and $E$) each set at two
levels.  An eight run, one-fourth fraction is selected, that is, a
$2^{5-2}$ design with generators $I=ABD$ and $I=ACE$.

\begin{parts}

  \PP Construct the appropriate design for the above situation
  indicating how each of the of the factors should be set for each of
  the eight runs. (Use a `--' for a low setting and a `+' for a high
  setting.)

  \PP Show how each main effect and two-factor interaction is confounded.

  \PP Give the resolution of this design and provide a brief explanation
  on how it is determined.

\end{parts}

\pagebreak


\QQ 100 patients were randomly
assigned to 5 methods of treating breast cancer.  A numerical score
measuring the recovery of each patient was calculated.  The table
below summarizes the recovery scores for each method:

\begin{center}
  \begin{tabular}{|l|l|r|r|r|} \hline Group & Method & n & mean & s.d.
    \\ \hline
    1 & chemotherapy & 20 & 30.57 & 7.06 \\
    2 & surgery and radiation & 20 & 31.17 & 7.49 \\
    3 & radiation and chemotherapy & 20 & 26.47 & 6.82 \\
    4 & hormonal therapy and radiation & 20 & 29.48 & 7.63 \\
    5 & surgery and chemotherapy & 20 & 30.82 & 7.30 \\
    \hline
  \end{tabular}\end{center}

We assume the model $Y_{ij} = \tau_i + \epsilon_{ij}$, $i=1, \ldots,
5$, $j=1, \ldots, 20$, where $\epsilon_{ij} \sim N(0,
\sigma_\epsilon^2)$.  Here $Y_{ij}$ is the observed response for the
$j$th subject in the $i$th group, $\tau_i$ is the fixed effect for the
$i$th group, and $\epsilon_{ij}$ is the error in the measurement for
the $j$th subject in the $i$th group.

\begin{parts}

\PP Identify the weights for a linear contrast comparing the
  methods with chemotherapy to those methods without chemotherapy.

\PP Estimate the contrast and compute a 95 percent confidence
  interval for the contrast. Would you reject the null hypothesis that
  the contrast is zero?

\PP Explain what rejecting the hypothesis that the contrast is
  zero tells us about the group means. Explain what a failure to
  reject that hypothesis (i.e., accepting the hypothesis that the
  contrast is zero) tells us about the group means.


\PP A researcher notices that Group 2 has the highest mean and
  Group 3 has the lowest mean. To explore he carries out a two-sample
  test and finds the group 2 population mean is significantly higher
  than the group 3 population mean at the usual .05 error rate.
  Explain why the researcher's examining the data first before
  carrying out a test causes an interpretation problem.

\end{parts}



%%%%%%%%%%%%%%%%%%%% Probability %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\QQ Let $X$ be distributed as a Poisson random variable with mean
$\lambda$.  

\begin{parts}

\PP Let $g$ be any function such that $Eg(X)$ is finite.  Show that
\begin{displaymath}
E(\lambda g(X))=E(Xg(X-1)).
\end{displaymath}

\PP Use the result of part (a) to find $E(X^3)$.

\end{parts}


\QQ Let $X$ be an exponential random variable with mean $1$.  Find the
expected value of the distance between $X$ and the nearest integer to
$X$.


\pagebreak


\QQ Ninety-six percent of all babies survive delivery. However, 20
percent of all births involve Cesarean (C) sections, and when a C
section is performed the baby survives 90 percent of the time. If a
randomly chosen pregnant woman does {\bf not} have a C section, what
is the probability that her baby survives ?




%%%%%%%%%%%%%%%%%%% Theoretical Statistics %%%%%%%%%%%%%%%%%%%%


\QQ Let $X_1, \ldots, X_n$ be i.i.d. random variables with common
expected value $\mu$ and $\sigma^2 = 1$. Find $n$ such that we can be
90 percent certain that $ \bar{X} = \frac{1}{n}\sum_{i = 1}^{n} X_i$
is an accurate estimate of $\mu$, within an error margin of plus/minus
0.25 units. State the approximation result you are using.



\QQ Let $X_1,\ldots,X_n$ be independent trials with probability of
success\\ $p\in(0,1/2]$.

\begin{parts}
  \PP Compute the maximum likelihood estimator of $p$.

  Define
  \begin{eqnarray*}
\bar X =\frac{1}{n} \sum_{i=1}^n X_i
 \;\;\;\;\mbox{and}\;\;\;\;
 \widehat p& =&\begin{cases}
    \bar X & \text{if } \bar X<1/2,\\
    1/2 & \text{if } \bar X\ge 1/2.
  \end{cases} 
  \end{eqnarray*}

  \PP Is $\widehat p$ biased?

  \PP Show that the mean squared error of $\widehat p$ is smaller than
  the mean squared error of $\bar
  X$.

  \PP Is $\widehat p$ consistent?
\end{parts}


\QQ \mbox{}

\begin{parts}
  
\PP Let $X_1, X_2, \ldots, X_n$ be iid with density 
\begin{displaymath}
  f(x\,|\,\theta) =
  \begin{cases}
 e^{-(x-\theta)} & \text{for $x \geq \theta$}\\
0 & \text{otherwise}.    
  \end{cases}
\end{displaymath}
Find the method of moments (MOM) estimator for $\theta$.

\PP Let $X_1, X_2, \ldots, X_n$ be iid with density 
\begin{displaymath}
  f(x\,|\,\theta,\psi)=
  \begin{cases}
    \frac{1}{\psi}e^{-(x-\theta)/\psi} &  \text{for $x \geq \theta$}\\
0 & \text{otherwise} 
  \end{cases}
\end{displaymath}
where $\psi>0$.  Find the method of moments (MOM) estimator for the
pair $(\theta,\psi)$.


\item[\bf Hint for both parts:] You may use without proof the fact that
  the exponential distribution with density
  $\frac{1}{\beta}e^{-x/\beta}$ for $x \geq 0$ has mean $\beta$ and
  variance $\beta^2$.

\end{parts}

\end{document}

