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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.D.\ Qualifying Exam\\
  Friday--Saturday, January 3--4, 2014

\end{center}

\vspace{.1in}

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

\vspace{.2in}

%%%%%%%%%%%%% xufeng %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\QQ{5166}
 Assume that  two random samples $\{x_i\}$ and $\{y_i\}$ are independently drawn from distributions with means $\mu_1$ and $\mu_2$ and the same variance $\sigma^2$.

 \begin{center}
\begin{tabular}{lcccccccccc} \hline
 Observation ID         & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9& 10\\ \hline
  $x$  & $x_1$ & $x_2$ & $x_3$ & $x_4$ &$x_5$ & $x_6$   & $x_7$  & $x_8$   & $x_9$ & $x_{10}$ \\ \hline
  $y$  & $y_1$ & $y_2$ & $y_3$ & $y_4$ &$y_5$ & $y_6$   & $y_7$  & $y_8$   & $y_9$ & $y_{10}$ \\ \hline
 \end{tabular}
 \end{center}

\begin{parts}

  \PP Describe how to test hypotheses on the means of the two
  distributions based on the two samples.  State any additional
  assumptions on the distributions you need for the test. Give your
  procedure and formulas.

  \PP Describe how to test the two distributions using a nonparametric
  method such as the Wilcoxon Rank-Sum test.  Give your procedure and
  formulas.

  \PP Describe how to test the mean difference ($\mu_1 -\mu_2$) using
  a randomization distribution method.  Give your procedure and
  compare this method with the other two methods in (a) and (b).
\end{parts}

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

\QQ{5166} In a two-way factorial experiment with replicates, consider
the following linear model:
\begin{equation*}
Y_{ijk}  =  \mu +\alpha_i + \beta_j  + \epsilon_{ijk}, \quad
  \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1,\ldots, r,
\end{equation*}
where $\mu$ is the overall mean, $\alpha_i$ is the effect for the
$i$th level of factor $A$, and $\beta_j$ is the effect for the $j$th
level of factor $B$. The random errors $\{\epsilon_{ijk}\}$ are
assumed to be independent and normally distributed with mean zero and
variance $\mbox{Var}(\epsilon_{ijk})=\sigma_j^2$.

\begin{parts}

  \PP What are the distributions of $\bar{Y}_{i\cdot\cdot}$ and
  $\bar{Y}_{\cdots}$?


  \PP Let $S_A =br
  \sum_{i=1}^a(\bar{Y}_{i\cdot\cdot}-\bar{Y}_{\cdots})^2$, $S_B
  =ar\sum_{j=1}^b(\bar{Y}_{\cdot j\cdot}-\bar{Y}_{\cdots})^2$, and
\[S_R =\sum_{i=1}^a\sum_{j=1}^b\sum_{k=1}^r(Y_{ijk}-\bar{Y}_{i\cdot\cdot}-\bar{Y}_{\cdot j\cdot}+\bar{Y}_{\cdots})^2.\]
Find the mean values of $S_A$ and $S_R$.

\PP Show that the decomposition of variation of this model is:
\[ S_D = S_A + S_{B} + S_R \]
where
\begin{equation*}
S_D = \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^r (Y_{ijk}- \bar{Y}_{\cdots})^2.
\end{equation*}

\end{parts}

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

\QQ{5167}
 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  moving-average model:
\[\xi_i= \epsilon_i - \theta_1\epsilon_{i-1},\]
where $\{\epsilon_i, i=0, 1, \ldots, n\}$ are iid $N(0,\sigma^2)$
variables.

\begin{parts}

  \PP Calculate the variance-covariance matrix of $\vecc{\xi}$.
  Describe how to use the least squares method, the weighted least
  squares method, and the maximum likelihood method to estimate the
  coefficients $\vecc{\beta}$ in the model.  Give details of the three
  procedures.

\PP Let $\hat{\vecc{\beta}}$ be the least squares estimate of
$\vecc{\beta}$. Define
 $\hat{Y}=X\hat{\vecc{\beta}}$,  and $\hat{\vecc{\xi}} = Y-\hat{Y}$.  Is $\hat{\vecc{\beta}}$ an unbiased estimate of $\vecc{\beta}$?
 What is the variance matrix of $\hat{\vecc{\beta}}$?
Are $\hat{Y}$ and $\hat{\vecc{\xi}}$ independent? Show your reasons.

\end{parts}


\QQ{5167}
 Consider a multiple linear regression model with the  form:
\begin{eqnarray*}
 y_i = \beta_0 + \beta_1 x_{i1} + \cdots +\beta_px_{ip}+\epsilon_i, \hspace{0.2in} i=1, \cdots, n,
\end{eqnarray*}
where $\{\epsilon_i, i=1, \ldots, n\}$ are independent random
variables with mean zero and variance $\sigma_i^2$.  Define
$\vecc{\beta}=(\beta_0, \beta_1, \ldots, \beta_p)'$.


\begin{parts}

  \PP Assume that all the observations $\{y_i; i=1, \cdots, n\}$ are
  positive values. Describe how to choose a transformation of the response $y$
  to make the distribution of $y_i$ approximately normal with constant
  variance. Give a procedure and reasons.

  \PP Describe how to select variables $\{x_1, \cdots, x_p\}$ in this linear
  model.  Give at least two procedures for model selection and compare
  the procedures.

  \PP Give a procedure to evaluate the influence of each case
  $\{y_i, x_{i1}, x_{i2}, \cdots, x_{ip}\}$ and give your
  justifications for the procedure.

\end{parts}

\pagebreak

%%%%%%%%%%%% wu %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\QQ{5106} Let $(X_1, Y_1), (X_2, Y_2), \cdots, (X_n, Y_n)$ be a
sequence of i.i.d.\ paired samples.  Conditioned on $X_i$, the value
$Y_i$ follows a Poisson distribution with parameter $\lambda_{X_i} =
e^{\beta X_i}$.  That is,
$$ Y_i | X_i, \beta \sim \text{Poisson}(\lambda_{X_i})$$
Our goal is to find the maximum likelihood estimate (MLE) of $\beta$
with the observations $\{(X_i, Y_i)\}_{i=1}^n$.

\begin{parts}

\PP 
Derive an expression for the log-likelihood function 
$$ l(\beta) = \sum_{i=1}^n \log(f(Y_i|X_i, \beta)), $$
such that the MLE is given by 
$$ \hat \beta = \mbox{argmax}_\beta\, l(\beta).$$
\PP 
Find the expressions for $\dot{l}(\beta)$ and $\ddot{l}(\beta)$, the first and second derivatives of $l$ with respect to $\beta$.  Verify that $\ddot{l}(\beta) < 0$.

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

\QQ{5106}

Find the maximum likelihood estimate of $\theta$ where $\theta$ is a parameter in the multinomial distribution: 
$$ (x_1, x_2, x_3, x_4) \sim M(n; 0.1(1-\theta), 0.4(2-3\theta), 0.1(1+3\theta), \theta)$$
\begin{parts}

\PP 
Choose a variable for the missing data and use the EM algorithm for iteratively estimating $\theta$. 
Let $\theta^{(m)}$ be the current value of the unknown.  Derive the mathematical formula to update for $\theta^{(m+1)}$.   

\PP
Let $L(\theta)$ denote the likelihood with parameter $\theta$.  Prove that 
$$ L(\theta^{(m+1)}) \ge  L(\theta^{(m)}). $$
\end{parts}

\pagebreak

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

%%% fred %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\QQ{5326}  Let $X_1, X_2, X_3, \ldots$ be iid $\text{Uniform}(0,1)$.

\begin{parts}
  
\PP Find both $P(X_1^2 + X_2^2<1)$  and $P(X_2 - 2X_1>0)$.

\PP Find the density of $Y = \displaystyle \frac{X_2}{X_1}$.

\PP Find the moment generating function of $Z =
2(X_1+X_2+\cdots+X_{20})-5$.

\PP What is the approximate distribution of $Z =
2(X_1+X_2+\cdots+X_{20})-5$\,? (Give the name of the distribution and
the values of any parameters.)

\end{parts}

\QQ{5326} A fair dodecahedron (a 12-sided solid) has its sides labeled 1,
2, \ldots, 12.  The dodecahedron is rolled five times.  Let $(X_1, X_2,
X_3, X_4, X_5)$ denote the observed sequence of values.

\begin{parts}

\PP Find $P(X_1 < X_2 < X_3 < X_4 < X_5)$.

\PP Find $P(X_1 \leq X_2 \leq X_3 \leq X_4 \leq X_5)$.

\PP Find the probability that the sequence $(X_1,X_2,X_3,X_4,X_5)$ contains
three consecutive consecutive integers.  (In other words,
$(X_i,X_{i+1},X_{i+2}) = (y,y+1,y+2)$ for some values $i$ and $y$.
Some possible sequences $(X_1, \ldots, X_5)$ which satisfy this are
$(8,{\bf 6,7,8},3)$, $(2,{\bf 8,9,10,11})$, and $({\bf 4,5,6,7,8})$.)

\end{parts}

\QQ{5327} The following parts use the family of densities
\begin{displaymath}
  f(x|\theta) = \frac{\theta^2 \log x}{x^{\theta+1}}
\quad \text{for} \quad 1 \leq x \leq \infty\,, \quad \theta>0\,.
\end{displaymath}

\begin{parts}
  
  \PP Suppose $X_1, \ldots, X_n$ are iid $f(x|\theta)$.  Find the MLE
  for $\theta$.

  \PP Suppose that we have two independent random samples:
  $X_1,\ldots,X_n$ are iid $f(x|\theta)$, and $Y_1,\ldots,Y_m$ are iid
  $f(x|\beta)$.  Find the likelihood ratio test (LRT) of
  \begin{displaymath}
    H_0: \theta=\beta \quad \text{versus} \quad H_1: \theta \neq \beta\,.
  \end{displaymath}

  \PP Show that the test in part (b) can be based on the statistic
%  \begin{displaymath}
$ \displaystyle   U = \frac{\sum_{i=1}^n \log X_i}
    {\sum_{i=1}^n \log X_i + \sum_{i=1}^m \log Y_i}\,.$
%  \end{displaymath}


\end{parts}

\pagebreak

\QQ{5327} Let $X_1, X_2, \ldots, X_n$ be a random sample from the density
\begin{align*}
  &f(x|\theta)= \theta (\theta+1)\, x (1-x)^{\theta-1}
\quad \text{for} \quad 0 < x < 1\,,\quad \theta>0\,.
\end{align*}
Answer the following.  Justify your answers.

\begin{parts}
  
\PP Find a complete sufficient statistic for $\theta$.

\PP Is $\sum X_i$ sufficient for $\theta$?

% \PP What is the approximate distribution of the MLE $\hat{\theta}$
% when $n$ is large?

\PP Find $I(\theta)$, the Fisher information in a single observation
from $f(x|\theta)$.


\end{parts}


%% eric %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\QQ{6346} Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and
$(\mathcal{O}, \mathcal{G})$ be a measurable space.

\begin{parts}
  
\PP If $f:\Omega \rightarrow [0, \infty)$ is a non-negative simple function with $\int f d\mu < \infty$ and $B$ is a set with $\mu(B)=0$, show
$$
\int_B f d\mu = 0.
$$


\PP Extend part (a) to include non-negative measurable functions $f$.


\PP Suppose a non-negative random variable $Y:\Omega \rightarrow [0,
\infty)$ has $EY<\infty$, and $X:\Omega \rightarrow \mathcal{O}$ is a
measurable function.  Show that there exists a function $g$ such that
$$
\int_{X\in A}Y(\omega)d\mu(\omega)=\int_A g(\omega)d\mu_X(\omega)
\qquad \text{for any $A\in \mathcal{G}$.}
$$


\end{parts}


\QQ{6346} For the parts below, you may assume that $X_n$ and $X$
are random variables with all moments finite.

\begin{parts}
  
\PP  State Markov's inequality.


\PP  Define convergence in probability.


\PP  Define convergence in $L_p$.


\PP  Show $X_n \xrightarrow{L_p} X \Rightarrow  X_n \xrightarrow{P} X$ for finite $p$.


\PP  Prove Lyapunov's inequality:  For $0 < q < p$,
$$
\left[E\left(|X|^q\right)\right]^{1/q} \leq \left[E\left(|X|^p\right)\right]^{1/p}
$$


\PP  Show that finite second moment implies finite variance.

\end{parts}


\end{document}

