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\begin{document}


\begin{center}
\bf \large
 M.\ S.\ Comprehensive Exam\\
Saturday, August 27, 2005
\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 We consider measurements on rock samples from a petroleum
reservoir ({\tt rock} data).  Each sample was measured for
permeability ({\tt perm}, in milli-Darcies), total area of pores
({\tt area}, in pixels out of 256 by 256), total perimeter of
pores ({\tt peri}, in pixels) and shape ({\tt shape},
perimeter/sqrt(area)).

We fitted a linear regression with {\tt perm} as the dependent
variable and {\tt area} and {\tt peri} as the independent
variables: 
\vspace{-.25cm}\begin{verbatim}
g<-lm(perm~area+peri,data=rock)
\end{verbatim}\vspace{-.25cm}
The anova table of {\tt rock} is as follows:
\vspace{-.25cm}\begin{verbatim}
> anova(g)

Analysis of Variance Table

Response: perm
          Df  Sum Sq Mean Sq F value    Pr(>F)   
area       1 1417333 1417333  22.352 2.263e-05 ***
peri       1 4738469 4738469  74.729 4.052e-11 ***
Residuals 45 2853383   63409                     
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
\end{verbatim}\vspace{-.25cm}
\begin{parts}

\PP How many observations does the data set contain?

\PP Explain (1) in words what the number 4738469 under ``Sum Sq'' means 
and (2) how do you compute it?

\PP Compute the regression sum of squares.

\PP Next we fit the regression in a different order:
\vspace{-.25cm}\begin{verbatim}
gg<-lm(perm~peri+area,data=rock)
\end{verbatim}\vspace{-.25cm}
Describe the changes in the anova table given by
\vspace{-.25cm}\begin{verbatim}
>anova(gg)
\end{verbatim}\vspace{-.25cm}

\end{parts}



\QQ Consider the linear regression model
\begin{displaymath}
Y=X_1 \beta_1 + X_2 \beta_2 + \epsilon
\end{displaymath}
with $X_1^T X_2 =0$.  Find the least squares estimates of
$\beta_1$ and $\beta_2$.  Justify your answer. \\ [6pt]
[Note: $Y$, $X_1$, $X_2$, $\epsilon$\, have dimensions $n \times
1$, $n \times p_1$, $n \times p_2$, $n \times 1$, respectively,
with $n \geq p_1 + p_2$\,.  $X_1$ and $X_2$ have full rank.]



\QQ Consider a Factorial design with two factors A and B. Suppose
the factors A and B have $a$ and $b$ levels, respectively and
there are $r$ replications for each treatment.
\begin{parts}
\PP Write the ANOVA model for the above factorial design {\it with
  interaction\/}. Explain the meaning of each symbol that you use. Write
  down the distributional assumptions about the random variables in
  the model.  If any constraints or assumptions regarding the 
  parameters are used, also write them down. 
\PP Write down the null and alternative hypotheses in terms of
  the model parameters for testing no interaction between the two
  factors. How would you test this by using the ANOVA table?
\PP State a graphical method for testing no interaction between
  the factors. What are the advantages and disadvantages of the
  graphical method compared to the method based on the ANOVA table?
\end{parts}



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


\QQ Let $X_1, X_2,\dots $ be a sequence of independent and
identically distributed random variables, each with mean $\mu$
and variance $\sigma^2$. Let $N$ be a nonnegative integer-valued
random variable with finite variance that is independent of the
$X_i$'s. Find the variance of $\sum_{i=1}^N X_i$.



\QQ Let $X$ be distributed as a Poisson random variable with mean
$\lambda$ and $Y$ be distributed as a Poisson random variable
with mean $\mu$.  Assume $X$ and $Y$ are independent.

\vspace{.2cm}

\noindent For a fixed positive integer $w$, find the distribution of $X$ given $X+Y=w$.


\QQ For a continuous random variable $Y$, show that
\begin{displaymath} 
E[Y]=\int_0^\infty P\{Y>y\}dy - \int_0^\infty P\{Y<-y\}dy\,. 
\end{displaymath}


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


\pagebreak


\QQ Suppose a random sample of size $n$ is taken from a
population with density
$$
f(x)=\left\{\begin{array}{ll} \frac{2x}{2\theta+1} & \mbox{for $\theta \leq x \leq \theta+1$}\\
0 & \mbox{otherwise} \end{array}\right.
$$
where $\theta>1$. 
Find the maximum likelihood estimate of $\theta$.  Is it unique?
\vspace{.2in}


\QQ Let $X_1, \ldots, X_n$ be an i.i.d.\ sample from an $N(\mu,
\sigma^2)$ distribution. Let $\bar{X} = \frac{1}{n}\sum_{i=1}^{n}
X_i$. Let $S^2 = \frac{1}{n-1}\sum_{i=1}^{n} (X_i - \bar{X})^2$.
\begin{parts}
\PP Find the maximum likelihood estimators of $\mu$ and $\sigma^2$, respectively.
\PP State the distribution of 
\[ \frac{\sqrt{n}(\bar{X} - \mu)}{S}.\]
\noindent (You do {\bf not} need to derive this distribution.)
\PP Use (b) to {\it derive} a $(1 - \alpha)\%$ confidence
interval for $\mu$.
\end{parts}

\QQ Suppose that $X_1$ and $X_2$ are iid observations from the
pdf 
\begin{displaymath}
f(x|\theta) = \theta x^{\theta-1}e^{-x^{\theta}}
\;\;\;\mbox{for $x>0$ and $\theta>0$.}  
\end{displaymath}
Show that $(\log X_1)/(\log X_2)$ is an ancillary statistic.

\end{document}

