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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.\ D.\ Qualifying Exam\\
  Tuesday, January 3, 2012

\end{center}

\vspace{.1in}

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

\vspace{.2in}

\QQ     Applied Statistics (STA5166) \hspace{0.2in} \newline ({\bf Note: Need a
Chi-Square table for this problem.})

\vspace{0.1in}

  A study was conducted to determine whether the age of customers is related to the type
  of movie he or she watches. A sample is shown in the following table.

\vspace{0.1in}

\begin{tabular}{cccc} \hline
Age            &  Documentary & Comedy  &  Mystery\\ \hline
12-20          & 24 & 19 & 18 \\
21-40          & 35 & 46 & 58 \\
41 and over    & 23 & 60 & 49 \\ \hline
\end{tabular}

\begin{parts}
\PP
 Given that the total sample size $n=332$ is fixed, what is the distribution of the nine categories?
 What are the mean value and variance of each cell frequency?
Run a chi-square test of independence and draw your conclusion. Use
$\alpha=0.05$.

\vspace{0.05in}

\PP
 Run a chi-square test of  "Comedy" verse "Mystery".
 Use $\alpha=0.05$.


\end{parts}

\vspace{.2in}

\QQ    Applied Statistics  (STA5166)

\vspace{0.1in}

In a complete factorial experiment, consider the following  random-effects model:
\begin{eqnarray*}
Y_{ijk} & = & \mu +\alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}, \\
& & \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1,\ldots, r,
\end{eqnarray*}
where  $\mu$ is the overall mean, the random effects $\alpha_i$, $\beta_j$,
and $(\alpha\beta)_{ij}$ are assumed to be independent and normally
distributed with means of zero and variances $\sigma^2_{\alpha}$, $\sigma^2_{\beta}$,
and $\sigma^2_{\alpha\beta}$, respectively. The random errors $\{\epsilon_{ijk}\}$ are assumed
to be independent and normally distributed with mean zero and variance $\sigma^2$.
Furthermore, $\{\alpha_i\}$, $\{\beta_j\}$, $\{\alpha\beta)_{ij}\}$,  and $\{\epsilon_{ijk}\}$
are assumed to be independent of one another.

\vspace{0.1in}

Define
\[SS(AB) = r \sum_{i=1}^a \sum_{j=1}^b(Y_{ij\cdot}- \bar{Y}_{i\cdot\cdot}-\bar{Y}_{\cdot j\cdot}+\bar{Y}_{\cdots})^2\]
 and
\[SSE = \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^r (Y_{ijk}- \bar{Y}_{ij\cdot})^2.\]

\begin{parts}
\PP
What are the distributions of $\bar{Y}_{ij\cdot}$, $\bar{Y}_{i\cdot\cdot}$,
and  $\bar{Y}_{\cdot j\cdot}$?
Find the covariance  Cov($\bar{Y}_{ij\cdot}, \bar{Y}_{uv\cdot}$).


\PP
Calculate the mean values of $SS(AB)$ and $SSE$.
Find unbiased estimates for the variances  $\sigma^2_{\alpha\beta}$
and $\sigma^2$ based on the observations $\{Y_{ijk}, \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1, \ldots,, r\}$.

\end{parts}

\vspace{.2in}

\QQ    Applied Statistics  (STA5167)

\vspace{0.1in}

Consider the linear regression model:
\[Y = X\vecc{\beta}+\vecc{\epsilon},\]
 where $Y =(y_{1},  \ldots, y_{n})'$,  $\vecc{\epsilon}=(\epsilon_{1}, \ldots, \epsilon _n)'$,
$\vecc{\beta}=(\beta_1, \ldots, \beta_p)'$,  and
  $X$  is an $n\times p$ full-rank matrix.
 $\{\epsilon_i, i=1, \ldots, n\}$  are assumed to be iid $N(0,\sigma^2)$ variables.
 Let $\hat{\vecc{\beta}}$ be the least squares estimate of $\vecc{\beta}$.


\begin{parts}
\PP
If $\vecc{\xi}=(\xi_1, \ldots, \xi_n)'$  has a multivariate normal  distribution
with mean zero and nonsingular covariance matrix $V=\mbox{Cov}(\vecc{\xi})$,  and if $A$ is
a $n\times n$  symmetric matrix with rank $r$, show that $\vecc{\xi}'A\vecc{\xi}$ has a
$\chi^2$ distribution with $df=r$ if $AV$ is idempotent.
(Hint: If $AV$ is idempotent and $V=C'C$ where $C$ is nonsigular, then $B=CAC'$ is idempotent with rank $r$).

\PP
Using the result in (a), find the distribution
of $(\hat{\vecc{\beta}}-\vecc{\beta})'(X'X)(\hat{\vecc{\beta}}-\vecc{\beta})$.
\end{parts}

\vspace{.2in}

\QQ     Applied Statistics (STA5167)

\vspace{0.1in}

 Consider the linear regression model:
\[Y = X\vecc{\beta}+ \vecc{\xi},\]
 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  model:
\[\xi_i + 0.5 \xi_{i-1} = a_i,\]
where $\{a_i, i=0, \pm 1,  \pm 2 \ldots,\}$  are iid $N(0,\sigma^2)$ variables.


\begin{parts}

\PP
 Show that  $\xi_i = \sum_{j=0}^{\infty} (-0.5)^j a_{i-j}$ is a
solution of the equation $\xi_i - 0.5 \xi_{i-1} = a_i$.

\PP
  Based on the expression in (a) for $\xi_i$,  calculate  autocorrelations
$\gamma_k = \mbox{Cov}(\xi_i, \xi_{i+k})$ for $k \geq 0$.  Show that
$\gamma_k = -0.5 \gamma_{k-1}$ for any $k\geq 0$.

\PP
How do you estimate $\vecc{\beta}$ in this setting?
Discuss the properties of your estimate such as mean, covariance matrix, and
distribution of $\hat{\vecc{\beta}}$. Compare your estimate with the least squares
estimate of $\vecc{\beta}$.

\end{parts}

\vspace{.2in}

\QQ  Advanced Probability and Inference (STA6346) 

\vspace{0.1in}

Let $N(t)$ be a homogeneous Poisson process with rate $\lambda$.  Find $E[N(t)N(s)]$ for $s\neq t$.

\vspace{.2in}

\QQ Advanced Probability and Inference (STA6346) 
\vspace{0.1in}

 Let $X_1, X_2, \ldots$ be the positions of a random walk on the integers starting from $0$.  That is, $X_1$ is the position of the random walk after the first step, $X_2$ is the position after the second step, and so on.  Let $P_{i, i + 1} = p \in (0, 1)$ be the probability the walk moves from $i$ to $i + 1$ for any integer $i$.

\begin{parts}
\PP
Find $E(X_n)$.

\PP
For what values of $p$ is $X_1, X_2, \ldots$ a martingale?  Prove your answer.  You may assume that $E|X_i|$ is finite.

\PP

Create a new process $Y_n$ from $X_1, X_2, \ldots, X_n$ such that $Y_1, Y_2, \ldots, Y_n$ is a martingale for ANY fixed $p \in (0, 1)$.

\end{parts}

\vspace{.2in}

\QQ  Advanced Probability and Inference (STA6346) 

\vspace{0.1in}

Suppose $\mu, \nu$ and $\lambda$ are three $\sigma$-finite measures on a measureable space $(\Omega, \cal{F})$.

\begin{parts}
\PP
Find $\frac{d\mu}{d\mu}$.

\PP
Suppose all three measures are absolutely continuous to each other.  Show
$\frac{d\mu}{d\nu}=\left[\frac{d\nu}{d\mu}\right]^{-1}.$

\end{parts}
\vspace{.2in}

\pagebreak

\QQ  Computational Methods in Statistics I (STA5106) 

\vspace{0.1in}

Let $H$ be an $n \times n$ householder matrix given by
$$H=I_n-2\frac{vv^T}{v^Tv},$$ for any non-zero $n$-length column vector $v(\neq 0)$. 
Show that $H$ is a symmetric, orthogonal, and reflection matrix.  That is, $H$ satisfies  i) $H = H^T$, ii) $HH^T = I_n$, iii) det($H$) = -1.  

\vspace{.2in}

\QQ  Computational Methods in Statistics I (STA5106) 

\vspace{0.1in}

Derive an EM algorithm to find the maximum likelihood estimate of $\theta$ where $\theta$ is a parameter in the multinomial distribution:
 $$(x_1,x_2,x_3,x_4) \sim M(n; 0.2\theta, 0.2(2+\theta), 0.6(1-2\theta),0.8\theta)$$
Choose a variable for the missing data and derive the EM algorithm for iteratively estimating $\theta$. 

\vspace{.2in}

\QQ Distribution Theory and Inference (STA5326) 
\vspace{0.1in}

A small nation has 2,000,000 inhabitants.  The people live
in 1,00{\large \bf 1} communities.  1,000 of these communities are small
villages, each with a population of 1,000 people.  The last community
is a single large city with a population of 1,000,000.  In each
village 90\% of the residents are farmers.  In the city only 5\% of
the residents are farmers (who must walk to the countryside to till
their fields).

Consider two different methods of sampling a person from this nation.
(In the following, when we say that something is done ``at random'', we
mean that all the possibilities are equally likely.)
\begin{description}

\item[Method \#1:] Choose a person at random from census
records which list all 2,000,000 people.

\item [Method \#2:] Choose a community at random, and then choose a
  person at random from this community.

\end{description}
Answer the following.

\begin{parts}

\PP Suppose a person is sampled using Method \#1.  Given that this person
is {\bf not} a farmer, what is the probability he/she lives in the
city?

\PP Answer the question in (a) if the sampling is done using Method \#2.

\PP Suppose we sample people one by one by repeated independent use of
Method \#1, and continue until we have sampled exactly 2 farmers.  Let
$X$ be the total number of people sampled in this process.  Find
$P(X>k)$ for $k \geq 2$.

\PP Answer the question in (c) if the sampling is done by repeated
independent use of Method \#2.

[Note: In parts (c) and (d) it is possible to choose the same person
more than once.  In part (d) it is possible to choose the same
community more than once.]

\end{parts}

\vspace{.2in}

\QQ Distribution Theory and Inference (STA5326) 
\vspace{0.1in}

Suppose that $(X,Y)$ has the joint density
\begin{equation*}
  f(x,y)=
  \begin{cases}
\displaystyle \frac{\sqrt{3}}{\pi}\exp\left\{-(x^2-xy+y^2)\right\}
 & x>0,\\[10pt]
0    & x \leq 0
  \end{cases}
\end{equation*}

Find the following by direct calculation.

\begin{parts}

\PP The marginal density of $X$.

\PP The conditional density of $Y$ given $X=x$.

\PP The density of $Z = \displaystyle \frac{Y}{X}$.

\end{parts}

\vspace{.2in}

\QQ Statistical Inference (STA5327) 
\vspace{0.1in}

Suppose we observe data $X_1, \ldots, X_n$ which is a random
sample from the density
\begin{equation*}
  f(x|\theta)=\theta x^{\theta-1}e^{-x^{\theta}}\,,\quad
  x>0,\; \theta>0.
\end{equation*}

\begin{parts}

  \PP Find the MOM (method of moments) estimator of $\theta$. \,\,
  [Your answer may be written in terms of $\Gamma^{-1}$, the inverse
  of the gamma function defined by $\Gamma(a) \equiv \int_0^{\infty}
  x^{a-1}e^{-x}\,dx$.]

\PP Find an ancillary statistic. \,\, [Hint: Consider the density of
$\log(X_i)$.]

\PP Give a detailed statement of the most powerful test of level $\alpha$ for
\begin{equation*}
  H_0: \theta = 1 \quad \text{versus} \quad H_1: \theta=2\,.
\end{equation*}
[Do not try to find the critical value explicitly, but state the condition it must satisfy.]

\end{parts}

\vspace{.2in}

\QQ Statistical Inference (STA5327) 
\vspace{0.1in}

Suppose we observe $X_1,\ldots,X_n$ i.i.d.\ from the
density
\begin{align*}
  f(x|\alpha) &= \frac{c(\alpha) x^{\alpha}}{1+x^2}\,,
\quad 0<x<\infty\,,\; -1<\alpha<1\\
&\text{where} \quad c(\alpha)= 
\frac{2}{\pi}\cos\left(\frac{\pi \alpha}{2}\right)\,.
\end{align*}
Answer the following. Justify your answers.

\begin{parts}

\PP Find a complete sufficient statistic for $\alpha$.

\PP Find the MLE for $\alpha$.

\PP Is there a function of $\alpha$, say $g(\alpha)$, for which there
exists an unbiased estimator whose variance attains the Cram\'er-Rao
Lower Bound?  If so, find it.  If not, show why not.

\PP Let $T$ denote the complete sufficient statistic found in part
(a).  Is there a function of $\alpha$, say $h(\alpha)$, for which
there are two different unbiased estimators which are functions of
$T$?  Answer `Yes' or `No' and prove your answer.

\end{parts}


\end{document}

