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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.\ D.\ Qualifying Exam\\
  Monday, January 5, 2009

\end{center}

\vspace{.1in}

Please submit solutions to at most {\bf eight} problems.  You have
five hours.  No one is expected to answer all the problems correctly.
Partial credit will be given.  All problems are worth an equal amount
of credit.

\vspace{.2in}

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

\vspace{.2in}

\topic{Applied Statistics}


\QQ {5166} Consider the following linear model involving two factors A
and B in a split-plot experiment:
\begin{eqnarray*}
Y_{ijk} & = & \mu +\alpha_i +e_{ij} + \beta_j + \epsilon_{ijk}, \hspace{0.1in}
\hspace{0.2in} i=1,\ldots, a; \hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1, \ldots, r,
\end{eqnarray*}
where $\mu$, $\{\alpha_{i}, i=1, \ldots, a\}$,
and $\{\beta_{j}, j=1, \ldots, b\}$ are unknown constants. The random errors
 $\{e_{ij}\}$ are iid  $N(0,\sigma_e^2)$, $\{\epsilon_{ijk}\}$ are iid
 $N(0, \sigma_\epsilon^2)$, and the two groups $\{e_{ij}\}$ and $\{\epsilon_{ijk}\}$ are
 independent. Define
 \[ Y = (Y_{111}, Y_{112}, \ldots, Y_{11r}, Y_{121}, Y_{122}, \ldots, Y_{12r},
 \ldots, Y_{ab1}, Y_{ab2}, \ldots, Y_{abr})'.\]


\begin{parts}
\PP
 Find the expectation $\vecc{\mu} = E(Y)$ and the variance-covariance matrix $V= \mbox{Cov}(Y)$.
 What is the distribution of Y?

\PP Write an ANOVA table for this experiment, including the Sources of Variation, Degrees of Freedom (df),
Sum of Squares, Mean Squares. Derive the expectations for the mean squares.

\PP Construct a test statistic for testing $H_0: \hspace{0.05in}
\alpha_1=\alpha_2=\cdots=\alpha_a = 0$.  Give your justification for
the test statistic.


\end{parts}

\QQ {5167} Consider the linear regression model:
\[Y = X\vecc{\beta}+\xi,\]
 where $Y =(y_{1},  \ldots, y_{n})'$,  $\vecc{\xi}=(\xi_{1}, \ldots, \xi_n)'$,
$\vecc{\beta}=(\beta_1, \ldots, \beta_p)'$ and
  $X$  is an $n\times p$ full-rank matrix.
The process $\{\xi_i\}$ is generated by the  moving-average model:
\[\xi_i= \epsilon_i + \theta_1\epsilon_{i-1}+\theta_2\epsilon_{i-2},\]
where $\{\epsilon_i, i=-1, 0, 1, \ldots, n\}$ are iid $N(0,\sigma^2)$
variables. Let $\hat{\vecc{\beta}}$ be the least squares estimate of
$\vecc{\beta}$. Define
 $\hat{Y}=X\hat{\vecc{\beta}}$,  and $\hat{\vecc{\xi}} = Y-\hat{Y}$.

\begin{parts}
  \PP What are the means and covariance matrices of $\hat{Y}$ and
  $\hat{\vecc{\xi}}$?  What are the distributions of $\hat{Y}$ and
  $\hat{\vecc{\xi}}$?

\PP Are $\hat{Y}$ and $\hat{\vecc{\xi}}$ independent? Show your reasons.

\PP Can you find an estimate for $\vecc{\beta}$, say $\tilde{\vecc{\beta}}$,
such that $\tilde{Y}=X\tilde{\vecc{\beta}}$ and $\tilde{\vecc{\xi}} = Y-\tilde{Y}$ are independent?

\end{parts}


\QQ {5168}
 In a generalized linear model (GLM), the response variable $Y$ is assumed
 to have a density function with the exponential dispersion form:
\begin{eqnarray*}
f(y; \theta, \phi) & = & \exp\{[y\theta-b(\theta)]/a(\phi)+c(y,\phi)\}.
\end{eqnarray*}

\begin{parts}

  \PP Suppose a random variable $X \sim$ Binomial($m, \pi$) and define
  $Y=X/m$.  Express the binomial mass function in the exponential form
  in terms of the canonical parameter $\theta =
  \mbox{logit}(\pi)$. Derive the deviance $D(y, \pi)$ for the binomial
  model.

  \PP Suppose that $\{X_i \sim \mbox{Binomial}(m_i, \pi_i), i=1,
  \ldots, n\}$ are independent random variables and the probabilities
  $\vecc{\pi}=(\pi_1, \pi_2, \ldots, \pi_n)'$ follow a linear logistic
  model.  Let $H_0 \subset H_1$ be nested hypotheses (fewer covariates
  under $H_0$). Show that the corresponding model deviances satisfy
  the Pythagorean relationship
\[ D(\vecc{y}, \hat{\vecc{\pi}}^{(0)}) = D(\vecc{y}, \hat{\vecc{\pi}}^{(1)})
+ D(\hat{\vecc{\pi}}^{(1)}, \hat{\vecc{\pi}}^{(0)}),\]
where $\vecc{y} = (y_1, \ldots, y_{n})'$  is the vector of observed proportions, and $\hat{\vecc{\pi}}^{(0)}$
and $\hat{\vecc{\pi}}^{(1)}$ are the vectors of fitted proportions under $H_0$ and $H_1$, respectively.

\end{parts}


\QQ {5507} Suppose $X$ and $Y$ are random samples of size $m$ and $n$,
respectively, from a continuous distribution.  Assume that the $X$'s
and $Y$'s are independent.  Let $W$ be the Wilcoxon rank-sum statistic
for the sample $Y$, i.e., $W$ is the sum of the ranks of the sample
$Y$ with respect to the combined sample consisting of $X$ and $Y$.

\begin{parts}

\PP Find the distribution of $W$ when $m=3$ and $n=2$.

\PP For general integers $n\geq 1$ and $m \geq 1$, find the range of
$W$ and evaluate $P(W = n(n+1)/2)$.
  
\end{parts}




\QQ {5707} \mbox{} Suppose $\sigma^2 >0$.  Consider the matrix
\[ \Sigma =
  \sigma^2 \left(
\begin{array}{cccc}
1 & \rho_1 & \rho_2 & \rho_1 \\
\rho_1 & 1 & \rho_1 & \rho_2\\
\rho_2 & \rho_1 & 1 & \rho_1 \\
\rho_1 & \rho_2 & \rho_1 &  1
\end{array}
\right)\,.
\]

\begin{parts}

  \PP For what values of $\rho_1, \rho_2$ is $\Sigma$ a covariance
  matrix?

  \PP Suppose that $\Sigma$ is a covariance matrix and that $\rho_1,
  \rho_2$ are positive.  Find the proportion of the total variance
  explained by the first principal component.
  
\end{parts}



\topic{Probability}


\QQ {5446} Let $X_1, X_2, \ldots$ be independent random variables with common
continuous distribution function $F(x)$.  Let
\[ R_n=\sum_{j=1}^n 1_{\{ X_j \ge X_n\}} \] be the (relative) rank of
$X_n$ among $X_1,\ldots,X_n$. Here $1_A$ is the indicator function of
the set $A$. For example, $R_n=1$ means that $X_n$ is the largest of
$X_1,\ldots,X_n$ (we say $X_n$ is a record) and $R_n=2$ means that
$X_n$ is the second largest of $X_1,\ldots,X_n$.


\begin{parts}

\PP Show that $P(X_i=X_j)=0$ for $i\ne j$.

\PP Conclude that $P\left (\bigcup_{i\ne j} \{ X_i=X_j\} \right) =0$.

\PP Using the fact that all possible $n!$ orderings of
$X_1,\ldots,X_n$ have the same probability, show that $$P(R_n=k)=
\frac1n ,\ \text{ for } k=1,\ldots,n.$$

\PP Show that the random variables $R_1, \ldots, R_n$ are independent.

\end{parts}


\QQ {5447} (This exercise uses the notation and situation of the
previous exercise, but does {\bf not} require you to have solved that
exercise.) Define
\begin{eqnarray*}
Y_k=1_{\{ R_k=1\}}= \begin{cases}1  & \text{ if } R_k=1\ (X_k \text{ is a record})\\
0& \text{ otherwise}
\end{cases}\\
\end{eqnarray*}
According to the results of the previous exercise $Y_1, Y_2, Y_3,
\ldots$ are independent with $P(Y_k = 1) = \displaystyle \frac{1}{k}$.
Using these facts, do the following:

\begin{parts}

\PP Show that \[ \sum_{n=2}^\infty \text{Var} \left( \frac{Y_n}{
      \log (n)}\right) <\infty.\]


  \PP One can show (and you may use without proof) that \[
  \frac{1}{\log (n)} \sum_{k=1}^n E (Y_k) \to 1\ \text{ as
    $n\to\infty$.}\] Prove that \[ \frac{1}{\log(n) } \sum_{k=1}^n Y_k
  \to 1\] almost surely, as $n\to\infty$. (Here $\sum_{k\le n} Y_k$ is
  the number of records in the first $n$ observations.) Carefully
  state which results you use.

\end{parts}



\QQ {5334}
Let the population proportions $p_1, p_2, p_3$ of three genotypes of
a single gene with two alleles be given by the Hardy-Weinberg
formula
\begin{displaymath}
  p_1 = (1-\theta)^2\,,\;\;\; p_2 = 2\theta(1 - \theta)\,,\;\;\; 
p_3 = \theta^2\,,\;\;\; 0<\theta<1.
\end{displaymath}

\begin{parts}

\PP Find the MLE of $\theta$ based on a large sample of size $n$ in
which the observed frequencies of the three genotypes
are $n_1, n_2, n_3.$
\PP Construct an asymptotically optimal confidence interval for
$\theta$ having asymptotic confidence coefficient $1 - \alpha.$
  
\end{parts}



\QQ {5326} Suppose that tosses of a coin are independent with probability $p$
of heads.  Two people each toss this coin $n$ times.

\begin{parts}
  
  \PP What is the probability they will toss the same number of heads?
  Give an {\bf exact} formula.  (The formula may be left as a
  summation.)

  \PP Give a normal approximation for the probability of tossing the
  same number of heads.  (Assume here that $n$ is large.)  Evaluate
  this normal approximation explicitly for $n=\mbox{100,000}$ and $p=0.25$\,.


\comment{
> p<-0.25; n<-100000; sig2<-n*p*(1-p); sigD<-sqrt(2*sig2); zdiff<-0.5/sigD;
> zdiff
[1] 0.002581989
> pnorm(zdiff)-pnorm(-zdiff)
[1] 0.002060127
> 2*zdiff/sqrt(2*pi);
[1] 0.002060129
}

\end{parts}

\QQ {5807} At a certain location, cars pass by according to a Poisson process
with rate $\alpha$ cars per minute, and bicycles pass by according to
a Poisson process with rate $\beta$ bicycles per minute.  The cars and
bicycles are independent.  Both cars and bicycles are considered to be
``vehicles''.  Answer the following.  ({\bf No} proofs are required.)

\begin{parts}

  \PP Standing at this location, what is the probability that exactly
  $k$ vehicles will pass by in the next minute?

  \PP What is the probability that the first vehicle to pass you will
  be a bicycle?

  \PP What is the distribution of the length of time you must wait for
  the first vehicle to pass?  (State the density.)

  \PP Let $W(t)$ be the total number of {\bf wheels} which have passed
  you by time $t$. (Assume that all cars have 4 wheels, and all
  bicycles have 2 wheels.)  What kind of process is $W(\cdot)$?  (Be
  as detailed as possible).  What are the mean and variance of $W(t)$?

\end{parts}

\topic{Theoretical Statistics}



\QQ {5327} Let $X_1, \ldots, X_n$ be iid with the geometric distribution
\begin{displaymath}
  P_{\theta}(X=x) = \theta (1-\theta)^{x-1}, 
\;\;\;x=1, 2, \ldots,
\;\;\;0 < \theta < 1.
\end{displaymath}

\begin{parts}
  
  \PP Show that $T = \sum X_i$ is a complete and sufficient statistic
  for $\theta$.

  \PP Show that the level $\alpha$ likelihood ratio test (LRT) of
  \begin{displaymath}
    H_0 \,:\, \theta= 0.5
\;\;\;\mbox{versus}\;\;\;
H_1 \,:\, \theta \neq 0.5
  \end{displaymath}
  rejects when $T \leq c_1$ or $T \geq
    c_2$ for some choice of constants $c_1$ and $c_2$.

\end{parts}



\QQ {5208} Answer the following.  Parts (a) and (b) are unrelated.
\begin{parts}

\PP  Assume
\(\vecc{Y}=\begin{pmatrix}
  Y_{1} \\
  Y_{2} \\
  Y_{3}
\end{pmatrix} \sim N  (\vecc{\mu},\vecc{\Sigma})\)
\,where\,
\(\vecc{\mu}=\begin{pmatrix}
  3 \\
  2 \\
  1
\end{pmatrix} \hbox{ and } \vecc{\Sigma}= \begin{pmatrix}
  4 & 0 & 0 \\
  0 & 9 & 0 \\
  0 & 0 & 25
\end{pmatrix}\,.
\)

\begin{enumerate}
\item  What is the distribution of
\(\vecc{W}=\begin{pmatrix}
  \frac{Y_{1}-3}{2} \\[3pt]
 \frac{Y_{2}-2}{3} \\[3pt]
  \frac{Y_{3}-1}{5}
\end{pmatrix}
\)?

\item What is the distribution of $\vecc{W}'\vecc{W}$?

\end{enumerate}

\PP Consider $n$ independent Bernoulli random variables, $Y_1, \dots,
Y_n$ with $\Pr(Y_i=1)=\pi_i \hbox{ and } \Pr(Y_i=0)=1-\pi_i$.
The joint distribution in this case may be written:
$$\prod \pi_i^{Y_i} (1-\pi_i)^{1-Y_i}\,.$$

\begin{enumerate}
\item Show that this is a member of the exponential family.
\item If the link is
$g(\pi)=\log(\frac{\pi}{1-\pi})=\vecc{x'\beta}$, show that this is
equivalent to
$$\pi= \frac{\exp(\vecc{x'\beta})}{1+\exp(\vecc{x'\beta})}\,.$$
\end{enumerate}

\end{parts}


\topic{Biostatistics}



\QQ {5172} Answer the following.  Parts (a) and (b) are unrelated.

\begin{parts}
  \PP The results of a {\bf cohort} study examining the relationship
  between gender and coronary heart disease (CHD) are summarized in the
  following table. In this study, 4,541 individuals were followed 10
  years and it was determined which of these individuals actually
  experienced CHD.  For example, 236 out of 2,009 males developed the
  disease, 1,773 males did not develop the disease, etc.

\begin{center}
\begin{tabular}{llccc}
\hline && \multicolumn{2}{c}{CHD} &  \\
& & Yes & No & Total \\
\hline & Yes & 236 & 1,773 & 2,009\\
Male &&&&\\
& No &142 & 2,390 & 2,532\\
\hline & Total& 378 & 4,163 & 4,541\\
\end{tabular}
\end{center}

Provide definitions and estimates for each of the following
statistics.

\begin{enumerate}
\item The relative risk of CHD for males.
\item the odds ratio of CHD for males.
\item Recall that there are several types (definitions) of attributable risk.
Define any one and provide the correct estimator.
\end{enumerate}



\PP The following table provides the results of submitting urine
samples with known status as to amphetamine use to a commercial
laboratory.

\begin{center}
\begin{tabular}{llccc}
\hline && \multicolumn{2}{c}{True Status} & \\
&& Present & Absent & Total\\
\hline Lab Determination & Present & 180 & 20 & 200\\

& Absent & 400 & 600 & 1,000\\
& Total & 580 & 620 & 1,200\\
\hline
\end{tabular}
\end{center}


\begin{enumerate}

\item What is the sensitivity of this test?

\item What is the specificity of this test?

\item
Suppose at a particular company, .1\% (0.001) of employees at the
company actually use amphetamines and the company orders a drug
test on a random employee.  If the test is positive, what is the
probability that the employee actually uses drugs.
\end{enumerate}

\end{parts}



\QQ {5244}  Answer the following.

\begin{parts}
\PP Explain the ``phases'' of clinical trials, that is, their
names and what the main question being addressed is.
\PP Explain what a ``run-in'' period is and what it is meant to
accomplish.
\PP Explain what a ``non-inferiority'' trial is and how its
design differs from a clinical trial that is attempting to
demonstrate treatment efficacy.

\PP
An investigator wants to conduct a study examining whether the
correlation between two laboratory measures differs in the two
treatment arms. The primary outcome will involve testing the
difference between two normal means.

\begin{description}
\item She is willing to assume the variance is equal for the
two groups and is $\sigma^2$.
\item She specifies a minimum
difference that she thinks is clinically important:
$\Delta=\mu_1-\mu_0$
\item She specifies the level of type I
error she is willing to accept, $\alpha$.
\item Finally, she specifies the level of the type II error she is
  willing to accept (or alternatively the power she wishes her study
  to have).
\end{description}

\begin{enumerate}

\item After you have done the sample size calculation based on the
data she provides, she says that after thinking about it, the
minimum difference that she thinks is clinically important is
really:

$$\Delta_1=.5*\Delta$$

What effect does this have on the estimated sample size?

\item She later comes backs with yet another value:
$$\Delta_2=2*\Delta$$
What effect does this have on the original sample size
calculation?

\item Let $n_1$ and $n_0$ denote the number of patients in the
  treatment and control groups, respectively, and let $n=n_1 + n_0$.
  Because of cost and practical considerations she can randomize only
  a fixed total number $n$ of participants.  Derive the values of
  $n_1$ and $n_0$ that maximize the power of the study. (Assume fixed
  costs, i.e., the costs for all patients is the same regardless of
  treatment group.)

\end{enumerate}

\end{parts}

\QQ {5179} Let the survival time $T>0$ be an integer-valued (discrete)
random variable with finite mean residual lifetime $\mu_k = E[T-k \G
T>k]$ and discrete hazard rate $h_k = P[T=k \G T \geq k]$ for
$k=0,1,2,\ldots$

\begin{parts}
  
\PP Show that $\mu_0 - (1-h_1)\mu_1 = 1$.

Hint: One approach is to write down the terms in the sums for $\mu_0$
and $\mu_1$ in two rows, multiply each term of $\mu_1$ by $(1-h_1)$,
and subtract the rows.

\PP Can you generalize the result of (a) to $\mu_k -
(1-h_{k+1})\mu_{k+1}$?

\PP Use the results of (a) and (b) to show that the Geometric
distribution is the only discrete distribution with constant mean
residual lifetime.

\PP Show that the exponential distribution is the only continuous
distribution for which the mean residual lifetime $r(t)$ is constant
for all $t>0$.

\end{parts}


\topic{Computational Statistics}

\QQ {5106} Let $V = (V_1, V_2)$ where $V_1, V_2$ are iid uniform
random variables in the interval $(-1, 1)$. Our goal is to develop a
Monte Carlo technique to estimate the quantity
$$
\theta  =P(V \in C)  = E[I]
$$
where $C$ is the unit disk in $\real^2$ and $I$ is the indicator
variable for $V$ being inside the disk.

\begin{parts}

\PP
Define a new random variable $Z$ such that 
$Z = E[I|Y]$ for some random variable $Y$. 
Show that by 
choosing an appropriate $Y$ we can get an explicit expression for $Z$.  

\PP Show that by averaging independent samples of $Z$, we can 
estimate $\theta$ and this estimator has a smaller variance than
the one that averages independent samples of $I$. 

\end{parts}


\end{document}

