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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.\ D.\ Qualifying Exam\\
  Saturday, January 5, 2013

\end{center}

\vspace{.1in}

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

\vspace{.2in}


%%%%%%%%%%%%%%%% Part I %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\QQ{5166}
 Three treatments, A, B, and C, are compared in an experimental design.
For each treatment, measurements are recorded successively in time.
Suppose that the measurements follow the model:
\[ X_i(t) = \mu_i + \epsilon_i(t) + \theta \epsilon_i(t-1),\]
where  $\{\epsilon_i(t):  i=1,2,3; t=0, 1,\ldots, n\}$
are independent and identically distributed random variables
with mean zero and variance $\sigma^2$. Let
$\bar{X}_i = \sum_{t=1}^nX_i(t)/n$ and
$\bar{X} = \sum_{i=1}^3\sum_{t=1}^nX_i(t)/(3n)$.


\begin{parts}
\PP
 Calculate the means and variances of $\bar{X}_i$ and $\bar{X}$.

\PP
Let $s^2=\sum_{i=1}^3\sum_{t=1}^{n}(X_i(t)-\bar{X}_i)^2/[3(n-1)]$.
Calculate the mean value $\mbox{E}(s^2)$.  Find the range of $\theta$
for which $s^2$ over-estimates $\sigma^2$.

\PP
How do you test the null hypothesis $H_0: \mu_1=\mu_2=\mu_3$ in this experiment?
What cautions should you make about the conclusions in your analysis?
\end{parts}

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


\QQ{5167}
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
Find the distributions of  $\hat{\vecc{\beta}}$ and $(\hat{\vecc{\beta}}-\vecc{\beta})'(X'X)(\hat{\vecc{\beta}}-\vecc{\beta})$.

\PP
Define  $\hat{Y}=X\hat{\vecc{\beta}}$ and $\hat{\vecc{\epsilon}} = Y-\hat{Y}$.
Find the distribution of $\hat{\vecc{\epsilon}} $.

\PP
 Show that the statistic
\[ F = \frac{(\hat{\vecc{\beta}}-\vecc{\beta})'(X'X)(\hat{\vecc{\beta}}-\vecc{\beta})/p}
{\hat{\vecc{\epsilon}}'\hat{\vecc{\epsilon}}/(n-p)}\]
has an F-distribution.

\end{parts}


\QQ{5326} A particular parallel system has $k$ independent components,
where $k$ is a given positive integer, $ k\ge 2,$ the i-th component
having a lifetime with an exponential distribution of mean $\lambda$,
for all $i = 1, \dots, k.$ The lifetime of this parallel system is the
maximum of the individual lifetimes of its components. What is the
probability that the lifetime of this parallel system will be at least
${\lambda}$?  Fully justify your answer.

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


\QQ{5327}
Consider the parameter space ${\pmb \Theta} = \R^p \times
Sym_+(p,\R),$ and the observation space ${\pmb\cX} = (\R^p)^n,$
where  ${\bf X} =(X_1,\dots,X_n) \in {\pmb\cX} $  are
independent identically distributed random vectors, $X_1$ having a multivariate $\mathcal N_p(\mu, \Sigma)$ distribution. Given the
action space $\cA = \R^p$ and the loss function $L(\theta, a ) =
\|\mu - a\|^2,$ where $\theta = (\mu, \Sigma),$ compute the risk function $R(\theta, d)$ for the decision 
rule $d,$ given by $d({\bf
X}) = \bar{\bf X}.$ Fully justify your answer.


\QQ{6346}  Let $X$ be a Poisson random variable with mean $\lambda>0$.

\begin{parts}
  
\PP  Find the characteristic function of $X$.

\PP  Show $X$ is infinitely divisible.

\PP A random variable $Y$ is of the Poisson type if $Y=aX+b$ where $X$
is Poisson and $a\neq 0$.  Show $Y$ is infinitely divisible.

\end{parts}


\QQ{5106} Let $X_1, X_2, \cdots, X_n$ be a sequence of $i.i.d.$
observations from a logistic distribution with the probability density
function
$$ f(x|\theta) = \frac{\exp(-(x-\theta))}{(1+ \exp(-(x-\theta)))^2}.$$
Our goal is to find the maximum likelihood estimate (MLE) of $\theta$
with the observations $\{X_i\}_{i=1}^n$.

\begin{parts}

  \PP Derive an expression for the log-likelihood function
$$ l(\theta) = \sum_{i=1}^n \log(f(X_i|\theta)), $$
such that the MLE is given by 
$$ \hat \theta = \mbox{argmax}_\theta l(\theta).$$

\PP Find the expressions for $\dot{l}(\theta)$ and $\ddot{l}(\theta)$,
the first and second derivatives of $l$ with respect to $\theta$.
Verify that $\ddot{l}(\theta) < 0$.

  \PP Write out the Newton-Raphson algorithm to find the root of
  $\dot{l}(\theta)$.
  
\end{parts}

\pagebreak


%%%%%%%%%%%%%%%%%%%%%%%%% Part II %%%%%%%%%%%%%%%%%%%%%%%%%%

\QQ{5166}
 Consider the following unbalanced random one-way model:
\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  $\{\alpha_{i}, i=1, \ldots, k\}$ are iid  $N(0,\sigma_\alpha^2)$,
$\{\epsilon_{ij}, i=1, \ldots, k; j=1, \ldots, n_i\}$ are iid
 $N(0, \sigma_\epsilon^2)$, and the $\alpha_{i}$'s and $\epsilon_{ij}$'s are
 independent. Define
 \[ \mbox{SSA} = \sum_{i=1}^kn_i(\bar{y}_{i\cdot}-\bar{y}_{\cdot\cdot})^2 = \sum_{i=1}^k\frac{y_{i\cdot}^2}{n_i}
 - \frac{y_{\cdot\cdot}^2}{\sum_{i=1}^kn_i},\]
\[ \mbox{SSE} = \sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y}_{i\cdot})^2,\]
where
\[y_{i\cdot}=\sum_{j=1}^{n_i}y_{ij}, \hspace{0.1in} \bar{y}_{i\cdot}=\frac{y_{i\cdot}}{n_{i}}; \hspace{0.4in}
y_{\cdot\cdot} =\sum_{i=1}^k\sum_{j=1}^{n_i}y_{ij}, \hspace{0.1in}\bar{y}_{\cdot\cdot}=
\frac{y_{\cdot\cdot}}{\sum_{i=1}^kn_i}.\]


\begin{parts}
\PP
 Find the expectations and variances of $\bar{y}_{i\cdot}$ and $\bar{y}_{\cdot\cdot}$.
 What are the distributions of $\bar{y}_{i\cdot}$ and $\bar{y}_{\cdot\cdot}$?

\PP Show that SSA and SSE are independent.

\PP Find the expectations of SSA and SSE.


\end{parts}

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

\QQ{5167}
 In the following 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.

\begin{parts}

\PP
Show that the simple regression model can be expressed in the form:
 \[y_i = \alpha + \beta_1 (x_i -\bar{x}) + e_i, \hspace{0.1in} i=1, \ldots, n,\]
 where $\alpha = \beta_0 + \beta_1 \bar{x}$.

\PP
Let $\{\hat{\alpha}, \hat{\beta}_1, \hat{\sigma}^2\}$ denote the least-squares
 estimates of the unknown parameters $\{\alpha, \beta_1, \sigma^2\}$.
 Give the expressions of  $\{\hat{\alpha}, \hat{\beta}_1, \hat{\sigma}^2\}$

\PP
Find the distributions of $\hat{\alpha}$ and  $\hat{\beta}_1$. Find the covariance between $\hat{\alpha}$ and  $\hat{\beta}_1$.

\end{parts}

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

\QQ{5326} Assume $\alpha \in (0,1)$ is given, and $x_1, x_2, \dots,
x_n$ is a random sample from a probability distribution on $\mathbb
R^m$ with finite mean vector $\mu,$ and positive definite covariance
matrix.  Derive a large sample $(1 - \alpha)100\%$ confidence region
$\mathcal C_\alpha(x_1,\dots,x_n)$ for $\mu.$ Fully justify your
answer.

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

\QQ{5327} Consider the parameter space $\Theta =[0,1] = \cA$,
${\pmb\cX} =\{0,1\}^n$, $P_\theta (\{{\bf x}\}) =
\theta^r(1-\theta)^{n-r}$ for ${\bf x} \in {\pmb\cX}$ where
$r=\sum^n_1 x_i$, and the loss function $L: \Theta \times \cA \to
\mathbb R, \ L(\theta,a) = c(\theta-a)^2$ $(c>0)$. Find a Bayes
estimator of $\theta$ for the prior on $\Theta$ having a beta
distribution ${\mathcal B} e(\alpha,\beta)$.  Fully justify your
answer.

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

\QQ{6346} $(\Omega, \mathcal{F}, \mu)$ is a probability space and
$(\mathcal{O}, \mathcal{G})$ is a measurable space (you may take them
to be the reals and the Borel sets).  Define $X:\Omega \rightarrow
\mathcal{O}$.  Let $B\in\mathcal{F}$ be a fixed set.  Let
$\mu_0(A)=\mu(\{X\in A\} \cap B), A\in\mathcal{G}$.

\begin{parts}

\PP Show $\mu_0$ is a measure.

\PP  Show $\mu_0 \ll \mu_X$, where  $\mu_X$ is the measure induced by $X$.

\PP Show there exists a function $g:\mathcal{O}\rightarrow \field{R}$
such that
$$
\mu(\{X\in A\} \cap B)=\int_Ag(x)d\mu_X(x), \hspace{.1in} A\in\mathcal{G}\,.
$$
  
\PP Suppose $\mathcal{O}=\{0, 1\}$, $\mathcal{G}=2^\mathcal{O}$ and
$X=0, 1$.  Let $\mu_X(1)=p$ and $\mu_X(0)=1-p$. Let $B$ be fixed. Give
an explicit form for $g$.  Writing $P$ in place of $\mu$, show this is
a conditional probability.

\end{parts}




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



\QQ{5106} Let $Y$ be a continuous random variable with probability
density function:
$$ Y \sim \alpha_1 f_1(y; \mu_1, \sigma_1^2) +  \alpha_2 f_2(y; \mu_2, \sigma_2^2),$$
where $f_1$ and $f_2$ are two Gaussian density functions with means
$\mu_1$, $\mu_2$ and variances $\sigma_1^2$, $\sigma_2^2$,
respectively. Also, $0 \le \alpha_1, \alpha_2 \le 1$, such that
$\alpha_1 + \alpha_2 = 1$. Given $n$ $i.i.d.$ observations
$\{Y_i\}_{i=1}^n$, our goal is to find the maximum likelihood estimate
of
$$\theta = (\alpha_1, \mu_1, \sigma_1, \alpha_2, \mu_2, \sigma_2).$$
Use an EM algorithm for this estimation.  Let $\theta^{(m)}$ be the
current values of the unknown.  Derive the mathematical formula to
update for $\theta^{(m+1)}$.



\end{document}

