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\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam\\
Thursday, August 25, 2005

\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 simple linear regression model:
\[y_i = \beta_0 + \beta_1 x_i + e_i\,, \hspace{0.1in} i=1, \ldots, n,\]
 where $\{e_i, i=1, \ldots, n\}$ are i.i.d.\ $N(0, \sigma^2)$ variables.
 Let $\{\hat{\beta}_0, \hat{\beta}_1, \hat{\sigma}^2\}$ denote the least-squares
 estimates of the unknown parameters $\{\beta_0, \beta_1, \sigma^2\}$

\begin{parts}

\PP
 What are the distributions of the least-squares estimates $\{\hat{\beta}_0, \hat{\beta}_1, \hat{\sigma}^2\}$?

\PP
  In the model, suppose the predictor values are replaced by $\{cx_i, \hspace{0.1in} i=1, \ldots, n\}$
  where $c$ is a non-zero constant. How are $\hat{\beta}_0, \hat{\beta}_1, \hat{\sigma}^2, R^2$ and the $t$-test
  of ``$H_0: \beta_1 = 0$" affected by the change?

\PP
Suppose the response values are replaced by $\{ay_i, \hspace{0.1in} i=1, \ldots, n\}$
where $a$ is a non-zero constant. How are $\hat{\beta}_0, \hat{\beta}_1, \hat{\sigma}^2, R^2$ and the $t$-test
  of ``$H_0: \beta_1 = 0$" affected by the change?

\end{parts}



\QQ In a generalized linear model, the response variable $Y_i$ is
assumed to have a density function with the form:
\begin{eqnarray*}
f(y_i; \theta_i, \phi) & = & \exp\{[y_i\theta_i-b(\theta_i)]/a_i(\phi)+c(y_i,\phi)\}, \hspace{0.1in}
i=1, \ldots, n.
\end{eqnarray*}
Denote $\mu_i = E(Y_i)$, $\vecc{\mu}= (\mu_1, \ldots, \mu_n)'$,
and $\vecc{y} = (y_1, \ldots, y_n)'$.  Let $l(\vecc{\mu};
\vecc{y})$ be the log-likelihood function. For a given model with
$p$ unknown parameters, let $l(\hat{\vecc{\mu}}; \vecc{y})$
denote the maximum of the log-likelihood function.


\begin{parts}
  
  \PP Suppose that $\{Y_i, i=1, \ldots, n\}$ are independent
  Poisson random variables.  Show that $l(\vecc{\mu}; \vecc{y})$
  reaches its maximum when $\mu_i = y_i$. Letting $a_i(\phi) =
  \phi/w_i$\,, write out the scaled deviance $2[l(\vecc{y};
  \vecc{y})-l(\hat{\vecc{\mu}}; \vecc{y})]$ in this case.

 \PP
Suppose that $\{n_iY_i, i=1, \ldots, n\}$ are independent with 
$n_iY_i \sim \mbox{Binomial}(n_i, \pi_i)$.
Repeat the questions in Part (a).


\end{parts}



\QQ  Consider the following linear model for a one-way layout design:
\begin{eqnarray*}
Y_{ij} & = & \mu +\alpha_i +\epsilon_{ij}, \hspace{0.1in}
\hspace{0.2in} i=1,\ldots, k; \hspace{0.1in} j=1, \ldots, n_i\,,
\end{eqnarray*}
where for each $i$,  the random errors $\{\epsilon_{ij}, j=1, \ldots, n_i\}$ are assumed to be
i.i.d.\ $N(0, \sigma_i^2)$. The sample variances for the $k$ treatments
are denoted by $s_i^2, \hspace{0.1in} i=1,\ldots, k$.

\begin{parts}
\PP
 It is well known that
 \[ (n_i -1)s_i^2 = \sum_{j=1}^{n_i}(Y_{ij} - \bar{Y}_i)^2 \sim \sigma_i^2\chi_{n_i-1}^2.\]
 Based on this result, show that the distribution of $\log(s_i^2)$ can be approximated
 by $N(\log(\sigma_i^2), 2(n_i-1)^{-1})$.

\PP
In a one-way layout design with $k=10$ and $n_1=\cdots=n_{10} = 6$, the $s_i^2$'s and $\log(s_i^2)$'s
are given in the following table:

\vspace{0.1in}

\begin{tabular}{rrrrrrrrrrr}\hline
$s_i^2$ & 0.56 & 0.82 & 0.45 & 1.02 & 0.34 & 0.77 & 0.52 & 0.88 & 0.31 & 1.20 \\
$\log(s_i^2)$ & $-$0.58 & $-$0.20 & $-$0.80 & 0.02 & $-$1.08 & $-$0.26 & $-$0.65 & $-$0.13 & $-$1.17 & 0.18 \\ \hline
\end{tabular}

\vspace{0.1in}

Based on the result in part (a), perform a test on ``$H_0: \hspace{0.05in} \sigma_1^2 = \cdots = \sigma_{10}^2$"
against the alternative hypothesis ``$H_a:$ \hspace{0.05in}  some of the variances are not equal."

\end{parts}


\topic{Probability}


\QQ \mbox{}

\begin{parts}
  
  \PP Suppose that $P, Q$ are two probability measures on the
  same measurable space $(\Omega, \mathcal{A})$. Suppose that $P$
  and $Q$ are both absolutely continuous with respect to the
  measure $\mu$ with densities (Radon-Nikodym derivatives) $p$
  and $q$, respectively. Thus $P(A) = \int_{A}p\,d\mu$ and $Q(A) =
  \int_{A}q\,d\mu$, for $A \in \mathcal{A}$. Show that
\[\sup_{A \in \mathcal{A}}|P(A) - Q(A)| = \frac{1}{2}\int|p -
  q|d\mu.\]

\PP Suppose that $f_0, f_1, \ldots$ are $ \geq 0$, defined on a
sigma-finite measure space $(\Omega, \mathcal{A}, \mu)$, and
satisfy $\int_{\Omega}f_n\,d\mu = 1$, for all $n = 0, 1, 2,
\ldots$. Suppose also that $f_n \rightarrow_{\mbox{a.e.}}f_0$
with respect to $\mu$. Use (a) above to show that
\[ \sup_{A \in \mathcal{A}}\left|\int_{A} f_n\,d\mu - \int_A f_0\,d\mu\right| \rightarrow 0, \  \ \ \mbox{as} \ \ n \rightarrow \infty.\]

\end{parts}


\QQ \mbox{}
\begin{parts}
  \PP Assume that $X_1, \ldots, X_n, \ldots$ are uncorrelated and
  that $EX_j^2 \leq M < \infty$ for all $j \geq 1$. Let
  $\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i$. Show that
\[\bar{X}_n - E\bar{X}_n \rightarrow_{\mbox{L}_{2}} 0, \ \ \mbox{as} \ \ n \rightarrow \infty.\]
\PP Assume now that $X_1, \ldots, X_n, \ldots$ are i.i.d.\ with
$E|X_1| \leq M < \infty$ and $EX_1 = m$. Use Vitali's theorem to
show that
\[\bar{X}_n \rightarrow_{\mbox{L}_{1}} m, \ \ \mbox{as} \ \ n \rightarrow \infty.\]


\end{parts}


\QQ The median of a distribution $F$ on the real line is defined as the
smallest value $m$ for which $F(m)\ge1/2$.
\begin{parts}

  \PP Assume $F(t)>1/2$ for each $t>m$.  Show that for every
  $\eps>0$ there is a $\delta>0$ such that
  $|m(F)-m(G)|\le\varepsilon$ whenever $\sup_t | F(t)-G(t)|\le
  \delta$.\\ \mbox{} [{{\em Hint:} Choose $\delta>0$ such that
    $F(m-\eps)<\frac12-\delta$ and $F(m+\eps)>\frac12+\delta$.}\,]

\PP
Assume $F(t)>1/2$ for each $t>m$. Let $X_1,\ldots,X_n$ be an
independent sample from $F$. Let $F_n(x)=\frac1n\sum_{i=1}^n I_{\{
X_i\le x\}}$ be the empirical distribution function. Using part (a),
show that the sample median $m(F_n)$ is strongly consistent.

\PP Assume that $f(x)=F^\prime(x)$ exists and is continuous and
positive at $m$.  Show that the standardized sample median
$\sqrt{n}(m(F_n)-m)$ is asymptotically normal and compute
the asymptotic variance. 
\end{parts}


\QQ Consider a Markov chain with 9 states arranged in a 3 by 3
rectangular lattice.  If the chain is in a particular state at
time $n$, then at time $n+1$ it moves to one of the neighboring
states to the north, south, east, or west, choosing among them
with equal probability.  (The center state has 4 neighbors, edge
states have 3 neighbors, and corner states have only 2
neighbors.)  Answer the following.  Clearly state any results you
use in your answer.

\begin{parts}
  \PP If the chain starts from the southwest corner, what is the
  expected length of time until it returns to this state?
  
  \PP If the chain starts from the southwest corner, what is the
  expected number of times it will visit the center state before
  first visiting the northeast corner?  (Your answer may be given
  as a number or as a matrix expression.  All matrices which
  appear in your answer must be given explicitly.)

\end{parts}

\topic{Theoretical Statistics}



\QQ Consider the linear model
$Y_i=\beta_0+\sum_{j=1}^p\beta_jX_{ij}+e_i$, where $e_i$ are
i.i.d.\ $N(0,\sigma^2)$. Let $\{\hat Y_i\}$ denote the fitted
values obtained by the least squares method.

\begin{parts}

\PP Show that $\sum_{i=1}^nY_i=\sum_{i=1}^n \hat Y_i$


\PP In particular, let $p=2, n=4, X_{i1}=1, X_{i2}=0 $ for $i=1,2$, and 
$X_{i1}=0, X_{i2}=1$ for $i=3,4$. Is $\beta_2$ estimable? 
(Justify your answer.)


\PP Compute $E\left(\sum_{i=1}^n(Y_i-\hat Y_i)Y_i\right)$   

\end{parts}


\QQ Let $X$ be a random variable whose distribution depends on a
parameter $\theta$.  Consider the three different families of
distributions for $X$ described below.  The first two are
families of {\bf mass} functions in which $X \in \{0, 1, 2\}$ and
$0 < \theta < \infty$.  The third is a {\bf finite} family of
{\bf density} functions in which $0 < X < 1$ and $\theta \in \{0,
1, 2\}$.  In each case determine whether the family of
distributions of $X$ is complete.  In other words, in each case
determine whether $X$ is a complete statistic. \\ [3pt]

\begin{tabular}{c|cccc} \hline
& $P(X=0)$ &  $P(X=1)$ &  $P(X=2)$ & \\ \cline{2-4}
Family \#1 & $c\, \theta$ & $c\, 2 \theta$ & $c\, (1+\theta)$ &
for $\theta > 0$\\
Family \#2 & $c\, \theta$ & $c\,  \theta^2$ & $c\, \theta^3$ &
for $\theta > 0$ \\ \cline{2-4}
&&&&\\ [-6pt]
Family \#3 & \multicolumn{4}{c}{$f(x \,|\, \theta) = c\, x^{\theta}\;\;$
  for $0 < x < 1$ and $\theta \in \{0,1,2\}$.} \\ \hline
\end{tabular} \\ [3pt]

\noindent In the above table, $c$ is a normalizing constant whose value
depends on $\theta$. 

\QQ Let $X_1, X_2 \ldots, X_n$ be a random sample from a
$N(\theta,\sigma^2)$ distribution, and suppose that the prior
distribution on $\theta$ is $N(\mu,\tau^2)$.  Find the posterior
distribution of $\theta$ and give the mean and variance of this
distribution.

\vspace{.3cm}

 \noindent [ Hint: You may find it useful to use the following
fact: If $f(y)$ is a probability density function satisfying
\begin{displaymath}
  f(y) \propto e^{-\frac{1}{2} (a y^2 - 2 b y)}
\;\;\;\mbox{for $-\infty < y < \infty$,}
\end{displaymath}
then $f$ is the density of a normal distribution with mean $b/a$
and variance $1/a$. ] \mbox{}


\pagebreak

\topic{Biostatistics}


\QQ  Assume that we are interested in drawing a random sample of size $n$
to examine a random variable $X$.  We wish to test whether the mean in
the population is equal to some specified value, $\mu_0$, versus that
it is less than this value, that is:

$H_0: \mu=\mu_0$ vs $H_a:\mu < \mu_0$

For our test, we want the values of the Type I and Type II errors
to be .05 and .10 respectively, that is, $\alpha=0.05$ and
$\beta=0.10$\,.

You may assume that $X$ is normally distributed with variance
$\sigma^2$, and that $\sigma^2$ is known.


What is the minimum size of $n$ to assure the correct levels of
$\alpha$ and $\beta$?  That is, derive the formula for computing
the sample size, based on the probability statements involved and
the assumption of normality.



\QQ If $Y$ is a dichotomous random variable, and $X$ is a continuous
random variable, then the logistic model specifies that:


$$\Pr(Y=1)=\frac{\exp(\alpha+\beta X)}{1+\exp(\alpha+\beta X)}$$

\begin{parts}
\PP Sketch out a derivation of how, given a sample of size $n$ in
which we have measured $Y$ and $X$, we would establish estimates of
$\alpha$ and $\beta$.
\PP State two different ways we might test the hypothesis:  $H_0:\beta=0$
\end{parts}

\QQ Consider the results of a diagnostic test, $Y$, used to
classify disease status ($Y=0$ signifies that on the basis of the
test we call people non-diseased, and $Y=1$ signifies that on the
basis of the test results we call people diseased).  We denote
disease state by $D$ ($D=0$ signifies non-diseased and $D=1$
signifies diseased).

\begin{parts}
\PP  Provide a definition of each of the following in terms of probabilities
\begin{enumerate}
\item TPF=True Positive Fraction
\item FPF=False Positive Fraction
\item PPV=Positive Predictive Value
\item NPV=Negative Predictive Value
\end{enumerate}

\PP
If we also denote $\rho$=the prevalence of disease in the population
and $\tau=\Pr(Y=1)$, then we can examine misclassification problems
in terms of $(TPF,FPF,\rho)$ or in terms of $(PPV,NPV,\tau)$.  Are
these two parameterizations equivalent?  If they are not, provide a
counter example.  If they are, provide at least a sketch of a proof.
\end{parts}



\end{document}

