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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.D.\ Qualifying Exam\\
  Monday--Tuesday, January 5--6, 2015

\end{center}

\vspace{.1in}

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

\vspace{.2in}


\QQ{5106} 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 $v$ .  Prove that $H$ is a
symmetric, orthogonal, and reflection matrix.  That is, $H$ satisfies
\begin{displaymath}
\text{i) $H = H^T$,} \quad  \text{ii) $HH^T = I_n$,}  \quad
\text{and iii) $\det(H) = -1$.}
\end{displaymath}


\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
$$\theta = (\alpha_1, \mu_1, \sigma_1, \alpha_2, \mu_2, \sigma_2).$$

\begin{parts}

\PP
Use the EM algorithm for iteratively estimating $\theta$. 
Let $\theta^{(m)}$ be the current values 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}


\QQ{5166} Let $\{X_1, X_2, \ldots, X_n\}$ be a random sample (iid)
from a $N(\mu, \sigma^2)$ distribution.  Please show the following.

\begin{parts}

\PP
 $\bar{X} \sim N(\mu, \sigma^2/n)$.

\PP
$\frac{1}{\sigma^2} \sum_{i=1}^n (X_i - \bar{X})^2 \sim \chi^2_{n-1}$.

\PP
$\bar{X}$ and $s^2$ are independent.
\end{parts}

\QQ{5166} Consider the following linear model involving two factors A
and B in an experiment:
\begin{eqnarray*}
Y_{ijk} & = & \mu +\alpha_i +e_{ik} + \beta_j + \epsilon_{ijk}, \hspace{0.1in}
\hspace{0.2in} i=1,\ldots, a; \hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1, \ldots, r,
\end{eqnarray*}
where $\mu$, $\{\alpha_{i}, i=1, \ldots, a\}$,
and $\{\beta_{j}, j=1, \ldots, b\}$ are unknown constants. The random errors
 $\{e_{ik}\}$ are iid  $N(0,\sigma_e^2)$, $\{\epsilon_{ijk}\}$ are iid
 $N(0, \sigma_\epsilon^2)$, and the two groups $\{e_{ik}\}$ and $\{\epsilon_{ijk}\}$ are
 independent.

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


\PP Find Cov($Y_{ijk}$, $Y_{lmn}$) and  Cov($\bar{Y}_{i\cdot\cdot}$, $\bar{Y}_{\cdot j \cdot}$).

\PP Let $S_A =br \sum_{i=1}^a(\bar{Y}_{i\cdot\cdot}-\bar{Y}_{\cdots})^2$.  Find E$(S_A)$.

\end{parts}


\QQ{5167} Consider 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.  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
In the model, suppose that the values of the predictor $x_i$ are replaced by $100x_i$ for $i=1, \ldots, n$.
How are $\{\hat{\beta}_0, \hat{\beta}_1, \hat{\sigma}^2\}$  affected by this change?
How is the $t$-test of $H_0: \beta_1=0$ affected by this change?

\PP
Let $\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i$ be the fitted values and $\hat{e}_i = y_i-\hat{y}_i$ be the residuals for
 $i=1, \ldots, n$.  What is the sample correlation between the $\hat{e}_i$'s and the $y_i$'s?
What is the sample correlation between the $\hat{e}_i$'s and the $\hat{y}_i$'s?

\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 $\sigma_i^2 = w_i\sigma^2$ where
$\{w_i, i=1, \cdots, n\}$ are known positive constants.
Define $\vecc{\beta}=(\beta_0, \beta_1, \ldots, \beta_p)'$.


\begin{parts}

\PP Let $\hat{\vecc{\beta}}$ be the least squares estimate of $\vecc{\beta}$ and $\tilde{\vecc{\beta}}$ be the weighted least squares estimate of
$\vecc{\beta}$, respectively. Give the expressions for $\hat{\vecc{\beta}}$ and $\tilde{\vecc{\beta}}$.

\PP Discuss the properties of $\tilde{\vecc{\beta}}$,  such as mean, covariance matrix, and
distribution of $\tilde{\vecc{\beta}}$.

\PP Compare the two estimates $\hat{\vecc{\beta}}$ and $\tilde{\vecc{\beta}}$.

\end{parts}

\pagebreak

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

\vspace{.2in}


\QQ{5326} Let $X$ be a random variable with density (pdf) given by
\begin{displaymath}
  f(x) =
  \begin{cases}
b\,e^x & \text{for $-1<x<2$}\\
0 & \text{otherwise}
  \end{cases}
\qquad \text{where $b = \displaystyle \frac{1}{e^2-e^{-1}}$\,.}
\end{displaymath}

\begin{parts}

\PP  Find the density (pdf) of the random variable $Y=X^2$.

\PP  Find the moment generating function (mgf) of $X$.

\PP Find the moment generating function of $\frac{1}{2}(X_1 - X_2)$
where $X_1$, $X_2$ are iid with the density $f(x)$ given above.\\
(If you could not answer part (b), then in your answer to part (c) just
let $M(t)$ denote the answer to part (b).)

\end{parts}

\QQ{5326} A type of flowering plant comes in two varieties, red and
blue.  Plants of the red variety produce only red flowers; plants of
the blue variety produce only blue flowers.  A large barrel of seeds
contains a mixture of the two varieties, with proportion $p$ of
the seeds being the red variety.  From this barrel, $k$ seeds are
selected at random and planted.  Each seed grows into a plant which
produces a random number of flowers having a Poisson distribution with
mean $\lambda$.  The plants are independent of each other.  Let $X$ be
the number ``red'' seeds planted, and let $Y$ be the total number red
flowers produced.  

\begin{parts}

\PP Find the following quantities: $EY$,
$\text{Var}(Y)$, and $\text{Cov}(X,Y)$.

\PP Find $P(Y=0)$ and $P(Y=1)$.  (Simplify your answers if you can.)

\end{parts}

\QQ{5327} Let $X_1, X_2, \ldots, X_n$ be iid
$N(\theta,\theta^2)$ (i.e., normal with mean $\theta$ and variance
$\theta^2$) with $\theta>0$ and $T=(\bar{x},s^2)$ where $\bar{x}$ and $s^2$ are the sample mean and sample variance of $X_1, \ldots, X_n$.  Answer the following.  Justify your answers.

\begin{parts}

  \PP Is $T$ minimal sufficient for $\theta$?

\PP Is $T$ complete?

\PP Is $\bar{x}$ an asymptotically efficient estimate of $\theta$?

\end{parts}

\pagebreak

\QQ{5327} Let $X_1, X_2, \ldots, X_n$ be iid $\text{Uniform}(0,\theta)$.

\begin{parts}

\PP Find $\hat{\theta}$, the maximum likelihood estimator (MLE) for $\theta$.

\PP Calculate the mean squared error (MSE) of $\hat{\theta}$.

\PP Find the best unbiased estimator of $\theta^{\text{\bf 2}}$.  Justify your answer.  (You may use results from lecture without proof.)

\end{parts}

\QQ{6346}  Suppose
\begin{itemize}
\item $X_1, X_2, \ldots$ are independent and identically distributed uniform$(0, 1)$ random variables
\item $X_{(n)}$ is the maximum of the $X_i, i = 1, 2, \ldots, n$
\item $Y$ is normal$(0, 1)$ 
\item $Y_i = -Y$ for $i = 1, 2, \ldots$
\end{itemize}

\begin{parts}

\PP Does $X_{(n)}\xrightarrow{P} 1$?  Prove your answer.

\PP Find the limiting distribution of $n(1-X_{(n)})$.  That is, give the distribution of $X$ where $n(1-X_{(n)}) \xrightarrow{D} X$.

\PP Does $Y_n\xrightarrow{P} Y$?  Prove your answer.

\PP Show $Y_n \xrightarrow{D} Y$.

\PP In general, what property would $Y$ need to have in order for  $Y_n \xrightarrow{D} Y \Rightarrow Y_n\xrightarrow{P} Y$?

\end{parts}

\QQ{6346} Let $(\Omega, \mathcal{F})$ be a measurable space.  For $A
\in \mathcal{F}$, define $\delta_x$ by
$$
\delta_x(A)=
\left\{
\begin{array}{ll}
1, & x \in A, \\
0, & x \notin A. \\
\end{array}
\right.
$$

\begin{parts}

\PP Show $\delta_x$ is a measure for all $x \in \Omega$.

\PP  Prove or give a counterexample:  $\delta_x \ll \mu$, where $\mu$ is the counting measure.

\PP  Prove or give a counterexample:  $\delta_x \ll \mu$, where $\mu$ is Lebesgue measure.

\PP Find $\int f(\omega)d\delta_x(\omega)$ for any real-valued non-negative measurable function $f$.

\end{parts}

\end{document}

