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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.\ D.\ Qualifying Exam\\
  Thursday, August 23, 2007

\end{center}

\vspace{.1in}

Please submit solutions to at most {\bf seven} problems.  You have
four 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 Consider the following linear model for a randomized block design:
\[y_{ti} = \mu + \beta_i + \tau_t +\epsilon_{ti}, t=1, \ldots, k; i=1, \ldots, n,\]
where $\mu$ is an overall mean, $\tau_t$ is the effect of $t$th
treatment, $\beta_i$ is the effect of $i$th block,
$\{\epsilon_{ti}: t=1, \ldots, k; i=1, \ldots, n\}$ are assumed
to be iid $N(0, \sigma^2)$.

\begin{parts}
\PP
 The least squares estimate of $\tau_t$ is $\hat{\tau}_t = \bar{y}_{t\cdot}-\bar{y}_{\cdot\cdot}$.
 Find the expectation and variance of $\hat{\tau}_t$.

\PP Show the decomposition of variation for the experiment: $S_D = S_B+ S_T  + S_R$ where
\begin{itemize}
\item $S_D$: Total Variation of the observations,
\item $S_B$: Sum of Squares for Blocks,
\item $S_T$: Sum of Squares for Treatments,
\item $S_R$: Sum of Squares for Experimental Errors.
\end{itemize}

\PP Find the expectation of $S_T$ and $S_R$.


\end{parts}

% \pagebreak


\QQ Consider an $I\times J$ contingency table that cross-classifies a
multinomial sample of $n$ subjects on two categorical responses. Let
$X$ denote the row variable and $Y$ the column variable.  The cell
probabilities are $\{\pi_{ij}, i=1, \cdots, I; j=1, \cdots, J\}$ and
the expected frequencies are $\{\mu_{ij} = n\pi_{ij}, i=1, \cdots, I;
j=1, \cdots, J\}$.

\begin{parts}

\PP
 Define $\theta_{ij} = \log(\pi_{ij}/\pi_{1,1})$ for $(i,j) \neq (1, 1)$. Show that the multinomial
 distribution with cell probabilities $\{\pi_{ij}, i=1, \cdots, I; j=1, \cdots, J\}$ belongs to a multiple
 parameter exponential family with $\{\theta_{ij}\}$ as the natural parameters. Find the maximum likelihood
 estimates for the cell probabilities $\{\pi_{ij}, i=1, \cdots, I; j=1, \cdots, J\}$.

 \PP
Show that the independence of $X$ and $Y$, i.e., $\pi_{ij} = \pi_{i+}\pi_{+j}$,  is equivalent to the expected
frequencies $\{\mu_{ij} = n\pi_{ij}, i=1, \cdots, I; j=1, \cdots, J\}$ following the loglinear model:
\begin{eqnarray*}
\log(\mu_{ij}) = \lambda + \lambda_i^{X} + \lambda_{j}^Y.
\end{eqnarray*}

\end{parts}


\QQ Suppose that $\{(x_1, y_1), \ldots, (x_n, y_n)\}$ is a sample from
a bivariate normal distribution, i.e.,
\[ \left( \begin{array}{c} x_i \\ y_i \end{array} \right) \sim
N\left(\left(\begin{array}{c} \mu_1 \\  \mu_2 \end{array} \right),
\left(\begin{array}{cc} \sigma^2_1 & \rho \sigma_1\sigma_2\\
\rho \sigma_1\sigma_2 & \sigma_2^2 \end{array} \right)\right), \hspace{0.05in} i=1, \cdots, n.\]

\begin{parts}

\PP
 Show that the conditional distribution of $y_i$ given $x_i$ is normal and
\[ y_i | x_i \sim
N\left(\mu_2+\rho \frac{\sigma_2}{\sigma_1}(x_i -\mu_1), \hspace{0.05in} \sigma_2^2(1-\rho^2)
\right), \hspace{0.05in} i=1, \cdots, n.\]

\PP
  Define
  \begin{eqnarray}\label{param}
  \beta_1 = \rho \frac{\sigma_2}{\sigma_1}, & \beta_0 = \mu_2 - \beta_1 \mu_1, & \sigma^2 =\sigma_2^2(1-\rho^2).
  \end{eqnarray}
  Then $y_i$ given $x_i$ follows the simple regression model:
  \[ y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \hspace{0.05in} i=1, \cdots, n.\]
where $\{\epsilon_i, \hspace{0.05in} i=1, \cdots, n\}$ are iid N(0, $\sigma^2$). The moment estimates of $\beta_0$,
$\beta_1$, and $\sigma^2$, denoted by $\tilde{\beta}_0$, $\tilde{\beta}_1$, and $\tilde{\sigma}^2$, respectively,
are obtained by simply substituting the sample means ($\bar{x}$ and $\bar{y}$), sample variances ($\hat{\sigma}^2_1$
and $\hat{\sigma}^2_2$), and sample correlation $\hat{\rho}$ into (\ref{param}). Are the moment estimates the same as
the least squares estimates $\hat{\beta}_0$, $\hat{\beta}_1$, and $\hat{\sigma}^2$?

\end{parts}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\topic{Probability}


\QQ  Let $\{X_n\}_{n \geq 1}$ be a sequence of discrete random variables such that $P(X_n = 0) = 1 - \frac{1}{n^2}$ and $P(X_n = n^2) = \frac{1}{n^2}$, for each 
$n$.

\begin{parts}

\PP  Show that $E X_n \rightarrow 1$, as $n \rightarrow \infty$. 

\PP Show that $X_n \rightarrow 0$  a.s. as $ n \rightarrow \infty$. 

\PP State Vitali's Theorem.  Does the sequence $X_n$ converge in $L_1$
to zero?
  
\end{parts}



\QQ Show that if $X$ is a Poisson random variable with parameter
$\lambda$ then
\[ \frac{X - \lambda}{\sqrt{\lambda}}\, \stackrel{d}{\longrightarrow}\, N(0, 1), 
\;\;\;
\mbox{as $\lambda \rightarrow \infty$.} \]

\vspace{.1in}

\noindent {\it Hint.} You can use any method of proof. However, you
may find it easier to use characteristic functions. Recall that the
characteristic function of a Poisson $(\lambda)$ random variable is
$\exp\left(\lambda(e^{it} - 1)\right)$ and that of a $N(0, 1)$ random
variable is $\exp(-t^2/2)$.


\QQ Let $p_\theta(x)$ be a {\em Hellinger} differentiable density in
$\RR$ with score function $\tau_\theta(x)$.

\begin{parts}

\PP Show that $\int \tau_\theta(x) p_\theta(x)\, dx=0$.

\PP Show that the product density $p_\theta(x,y):= p_\theta(x) \times
p_\theta(y)$ in $\RR^2$ is Hellinger differentiable with score
function $\tau_\theta(x,y)=\tau_\theta(x)+\tau_\theta(y)$.

\end{parts}


\QQ Consider a branching process in which each individual lives
exactly one unit of time and then gives birth to either 0, 1, or 2
offspring with probabilities 1/3, 1/6, and 1/2, respectively.

\begin{parts}
  \PP If you start with exactly {\large \bf 5} organisms at time zero,
  what is the probability this population will eventually become
  extinct?

  \PP What are the mean and variance of the total number of organisms
  born in the third generation (that is, at time 3)?

\end{parts}

\topic{Theoretical Statistics}

\QQ Suppose $X_1, \ldots, X_n$ are iid with pdf
\begin{displaymath}
f(x \,|\, \alpha, \beta) = \frac{\alpha x^{\alpha -1}}{\beta^{\alpha}}
\;\;\;\mbox{for $0 \leq x \leq \beta$}  
\end{displaymath}
where $\alpha$ and $\beta$ are positive.

\begin{parts}
  
\PP Find a two-dimensional sufficient statistic for
  $(\alpha,\beta)$.

\PP Find the MLEs of $\alpha$ and $\beta$.

\PP Find an ancillary statistic.

\end{parts}

\QQ Let $X_1, \ldots, X_n$ be iid $N(\mu,\sigma^2)$.

\begin{parts}

\PP Find the best unbiased estimator of $\mu^2$.

\PP Find the best unbiased estimator of $\sigma^4$.

\item[Hint:] You may find it useful to know that the $\chi^2_k$
distribution has mean $k$ and variance $2k$.

\end{parts}


\QQ Consider the model
\begin{displaymath}
  y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_{11} x_{i1}^2 
+ \beta_{22} x_{i2}^2 + \beta_{12} x_{i1} x_{i2} + e_i\,,
\end{displaymath}
where the predictor variables take on the following values
\begin{center}
\begin{tabular}{c|rrrrrrr}
$i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline
$x_{i1}$ & 1 & 1 & $-1$ & $-1$ & 0 & 0 & 0\\
$x_{i2}$ & 1 & $-1$ & 1 & $-1$ & 0 & 0 & 0   
\end{tabular}  
\end{center}
and the errors $e_i$ are iid $N(0,\sigma^2)$.

\begin{parts}

\PP Show that $\beta_1$ and $\beta_{11}+\beta_{22}$ are estimable.  

\PP Find (nonmatrix) algebraic expressions for the best linear
unbiased estimators (BLUEs) of $\beta_1$ and $\beta_{11}+\beta_{22}$.

% \PP Find algebraic expressions for the variance of each estimator you
% found above.

\PP Find an algebraic expression for the MSE for this model.

\end{parts}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\topic{Biostatistics}


\QQ For a single covariate $x$, the proportional hazards model specifies that:
\begin{displaymath}
\lambda(t|x)=\lambda_0(t)\exp(\beta x)
\end{displaymath}

\begin{parts}
\PP Consider the following  observations:

\begin{tabular}{ccc}
time & $x$ & fail\\
2 & 2 & 1\\
4 & 1 & 0 \\
6 & 3 & 1 \\
10 & 5 & 0 \\
\end{tabular}

Write out the partial likelihood associated with the data.

\PP The above contained no ties.  Suppose the data were the following:

\begin{tabular}{ccc}
time & $x$ & fail\\
4 & 2 & 1 \\
4 & 1 & 1 \\
6 & 3 & 0 \\
10 & 5 & 1 \\
\end{tabular}

Sketch at least two ways we could take the tie into account in
forming the partial likelihood contribution at time 4.



\end{parts}

\QQ The following table presents the results of four logistic models
relating the specified covariates to death from coronary heart disease
(CHD).  Age is age of the participant in years and sbp is the
participant's systolic blood pressure in mmHg.\\

% Table generated by Excel2LaTeX from sheet 'Sheet1'
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
 Covariate & \multicolumn{2}{c|}{\bf Null Model}            & \multicolumn{2}{c|}{\bf Univariate}             & \multicolumn{2}{c|}{\bf Univariate}             &  \multicolumn{2}{c|}{\bf Bivariate}       \\
\cline{2-9}
           &   $\beta$ & $se(\beta)$ &   $\beta$ & $se(\beta)$ &   $\beta$ & $se(\beta)$ &   $\beta$ & $se(\beta)$ \\
\hline
       age &            &            &      0.038 &      0.006 &            &            &      0.029 &      0.007 \\

       sbp &            &            &            &            &      0.019 &      0.003 &      0.016 &      0.003 \\

 $\alpha$ &     -1.237 &      0.053 &     -3.180 &      0.328 &     -3.796 &      0.350 &     -4.895 &      0.433 \\
\hline
log likelihood &    -1071.1 &            &    -1052.3 &            &    -1042.9 &            &    -1032.9 &            \\
\hline
\end{tabular}

\begin{parts}
\PP How do you decide which of these four models best describes the data?
\PP For the covariates age and sbp, provide an interpretation of
the estimated coefficients in the bivariate model.
\PP For the covariates age and sbp in the bivariate model, describe two ways to test
whether they are significantly different from zero.
\PP In the {\bf Null Model}, what does:
$$
\frac{\exp(\alpha)}{1+\exp(\alpha)}
$$
estimate?

\PP In the data from which these coefficients were estimated, the average sbp
for those 50 years of age or less is 128.8 mmHg, among those
greater than 50 years of age, the average sbp is 137.1 mmHg.
Similarly, among those with sbp $\le$ 130 mmHg the average age is
48.4 years, among those with sbp $>130$ mmHg the average age is
51.3 years.

 Provide a possible explanation for the change in the
coefficients in the bivariate model as compared to the univariate
models.
\end{parts}



\QQ

An investigator wants to do a clinical trial.  The primary
outcome will involve testing the difference between two normal
means.

\begin{description}
\item She is willing to assume that the variance is equal for the
two groups and is $\sigma^2$.
\item She specifies a minimum
difference that she thinks is clinically important:
$\Delta=\mu_1-\mu_2$
\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{parts}
\PP 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?

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

\PP Because of cost and practical considerations she can randomize $n$ participants.
\begin{description}
\item $n_1$ is the number of patients in the treatment group.
\item $n_0$ is the number of patients in the control group.
\item Then, $n=n_1+n_0$ is a fixed constant.
\item She asks you what values of $n_1$ and $n_2$ provide the
maximum power.
\end{description}
Assuming fixed costs, i.e., the costs for all patients is the
same regardless of treatment group, derive the values of $n_1$
and $n_0$ that maximize the power of the study.


\end{parts}

\pagebreak

\topic{Computational Statistics}


\QQ For a given probability density function $f(x)$,
we are interested in estimating its right tail:
$$
\theta = \int_a^{\infty} f(x) dx,\ \ \ a \gg
\int_{-\infty}^{\infty} x f(x) dx\ .
$$

\begin{parts}
\PP Show how you can use importance sampling to estimate
$\theta$, when the samples are obtained from the tilted density:
$$
f_t(x) = {e^{tx} f(x) \over M(t)},\ \ M(t) =
\int_{-\infty}^{\infty} e^{tx} f(x) dx,\ \ t > 0\ .
$$

\PP If $f(x)$ is the standard normal density, what form does the
tilted density take?

\PP Write an algorithm to estimate $\theta$ for the standard
normal density, with $a = 3$. What is the optimal choice of $t$
for this problem?
\end{parts}

\QQ  Let $X_t$ be a discrete-time, continuous-state
Markov chain with the transition kernel $\Pi(x,y)$. For a
probability density function $f$:
\begin{parts}
\PP State the detailed balance condition involving $\Pi$ and $f$.

\PP Show that if $f$ satisfies the detailed balance condition, then
$f$ is a stationary probability density associated with the Markov
chain.

\PP Assuming that it is computationally feasible to perform
transitions under $\Pi$, suggest a Monte Carlo Markov Chain (MCMC)
algorithm to estimate the mean of $f$. You can also assume that the
variance under $f$ is finite.

\end{parts}

\end{document}

