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\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.D.\ Qualifying Exam\\
  Friday--Saturday, January 6--7, 2017

\end{center}

\vspace{.1in}

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

\vspace{.2in}

\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}


\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}
 Consider the following linear model for a randomized block design:
\[y_{ti} = \mu + \beta_i + \tau_t +\epsilon_{ti}\,,\; t=1, \ldots, k;\; i=1, \ldots, n,\]
where $\mu$ is an overall mean, $\tau_t$ is the effect of $t$th treatment, $\beta_i$ is the effect of $i$th block,
$\{\epsilon_{ti}: t=1, \ldots, k; i=1, \ldots, n\}$ are assumed to be i.i.d.\ $N(0, \sigma^2)$.

\begin{parts}

\PP
 The least squares estimate of $\beta_i$ is $\hat{\beta}_i = \bar{y}_{\cdot i}-\bar{y}_{\cdot\cdot}\,$.
 Find the expectation and variance of $\hat{\beta}_i$.

\PP Show the decomposition of variation for the experiment: $S_D = S_B+ S_T  + S_R$ where
\begin{itemize}
\item $S_D$: Total Variation of the observations,
\item $S_B$: Sum of Squares for Blocks,
\item $S_T$: Sum of Squares for Treatments,
\item $S_R$: Sum of Squares for Experimental Errors.
\end{itemize}

\PP Find the expectation of $S_B$.
\end{parts}


\QQ{5166}
 Consider the following unbalanced one-way random-effects 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 i.i.d.\  $N(0,\sigma_\alpha^2)$,
$\{\epsilon_{ij}, i=1, \ldots, k; j=1, \ldots, n_i\}$ are i.i.d.\
 $N(0, \sigma_\epsilon^2)$, and the $\alpha_{i}$'s and $\epsilon_{ij}$'s are
 independent. Define
 \[ \mbox{SSA} = \sum_{i=1}^kn_i(\bar{y}_{i\cdot}-\bar{y}_{\cdot\cdot})^2, \hspace{0.3in}  \mbox{SSE} = \sum_{i=1}^k\sum_{j=1}^{n_i}(y_{ij}-\bar{y}_{i\cdot})^2,\]
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  Cov$(y_{ij}, y_{i'j'} )$.  What is the distribution of $\bar{y}_{\cdot\cdot}$?

\PP Show that
$\dsp \mbox{SSA} = \sum_{i=1}^k\frac{y_{i\cdot}^2}{n_i} -
\frac{y_{\cdot\cdot}^2}{\sum_{i=1}^kn_i}$.
Show that SSA and SSE are independent.

\PP Find unbiased estimates for the variances $\sigma_\alpha^2$ and $\sigma_\epsilon^2$.


\end{parts}


\QQ{5167}
Suppose that $\{(x_1, y_1), \ldots, (x_n, y_n)\}$ is a sample from a bivariate normal distribution, i.e.,
\[ \left( \begin{array}{c} x_i \\ y_i \end{array} \right) \sim
N\left(\left(\begin{array}{c} \mu_1 \\  \mu_2 \end{array} \right),
\left(\begin{array}{cc} \sigma^2_1 & \rho \sigma_1\sigma_2\\
\rho \sigma_1\sigma_2 & \sigma_2^2 \end{array} \right)\right), \hspace{0.05in} i=1, \cdots, n.\]

\begin{parts}

\PP
 Show that the conditional distribution of $y_i$ given $x_i$ is normal and
\[ y_i | x_i \sim
N\left(\mu_2+\rho \frac{\sigma_2}{\sigma_1}(x_i -\mu_1), \hspace{0.05in} \sigma_2^2(1-\rho^2)
\right), \hspace{0.05in} i=1, \cdots, n.\]

\PP
  Define
  \begin{eqnarray}\label{param}
  \beta_1 = \rho \frac{\sigma_2}{\sigma_1}, & \beta_0 = \mu_2 - \beta_1 \mu_1, & \sigma^2 =\sigma_2^2(1-\rho^2).
  \end{eqnarray}
  Then $y_i$ given $x_i$ follows the simple regression model:
  \[ y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \hspace{0.05in} i=1, \cdots, n,\]
where $\{\epsilon_i, \hspace{0.05in} i=1, \cdots, n\}$ are i.i.d.\ $N(0, \sigma^2)$. The moment estimates of $\beta_0$,
$\beta_1$, and $\sigma^2$, denoted by $\tilde{\beta}_0$, $\tilde{\beta}_1$, and $\tilde{\sigma}^2$, respectively,
are obtained by simply substituting the sample means ($\bar{x}$ and $\bar{y}$), sample variances ($\hat{\sigma}^2_1$
and $\hat{\sigma}^2_2$), and sample correlation $\hat{\rho}$ into (\ref{param}). Are the moment estimates the same as
the least squares estimates $\hat{\beta}_0$, $\hat{\beta}_1$, and $\hat{\sigma}^2$?

\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:
\[\xi_i= \epsilon_i - \theta_1\epsilon_{i-1} - \theta_2\epsilon_{i-2},\]
where $\{\epsilon_i, i=-1, 0, 1, \ldots, n\}$ are i.i.d.\ $N(0,\sigma^2)$
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 Suppose that $\theta_1=0.5$ and $\theta_2 = -2$. Let $\hat{\vecc{\beta}}$ be the weighted  least squares estimate of $\vecc{\beta}$. 
Give the expression of $\hat{\vecc{\beta}}$ in this case.  Define
 $\hat{Y}=X\hat{\vecc{\beta}}$,  and $\hat{\vecc{\xi}} = Y-\hat{Y}$.  Are $\hat{Y}$ and $\hat{\vecc{\xi}}$ independent? Show your reasons.

\end{parts}

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

\pagebreak

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

\vspace{.2in}


\QQ{5326} Suppose that a Geiger counter is turned on at time zero and
that clicks on this Geiger counter occur according to a Poisson
process with a rate of $\lambda$ clicks per second.

\begin{parts}

  \PP Let $S_t$ denote the random number of clicks during the time
  interval $(0,t)$.  State a formula for $P(S_t =k)$ valid for
  nonnegative integers $k$. (No proof is needed.  Just state the
  answer.)

  \PP Let $T_1 < T_2 < T_3 < \cdots$ be the times of the first click,
  second click, third click, \ldots, and $X_1 = T_1$, $X_2 = T_2 - T_1$,
  $X_3 = T_3-T_2$, \ldots be the times between clicks (the
  interarrival times).  What is the joint distribution of
  $X_1, X_2, X_3, \ldots$?  In particular, give an explicit formula
  for the joint density of $(X_1, X_2, X_3)$.  (No proof is needed.
  Just state the answers.)

  \PP Use the facts in parts (a) and (b) {\bf or any other approach}
  to prove that
\begin{displaymath}
\int_{t}^{\infty} \frac{\lambda^r}{\Gamma(r)} z^{r-1} e^{-\lambda z}\,dz
= \sum_{y=0}^{r-1} \frac{(\lambda t)^y e^{-\lambda t}}{y!}\,, \quad
r=1, 2, 3, \ldots
\end{displaymath}
where $\lambda>0$ and $t>0$.  Give a detailed argument.

\PP Let $S_t$ be as defined in (a) for any value of $t$.  Find
$P(S_2 \leq 1, S_5 \leq 2, S_{10} =3)$.

\end{parts}

\QQ{5326}  Suppose the random variables $(X,Y)$ have the joint density
\begin{displaymath}
  f(x,y)= x^2 y e^{-x(y+1)} \quad \text{for $x>0, y>0$
\quad (and $f(x,y)=0$ otherwise).}
\end{displaymath}
Answer the following.  Carefully specify the support of any density or
joint density.

\begin{parts}

\PP Find the marginal density of $Y$.

\PP Find the joint density of $(U,V)$ where $U=X(Y+1)$ and $V=X$.

\PP Find the density of $U=X(Y+1)$.

\end{parts}

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

\pagebreak

\QQ{5327}
Let $X_1, X_2,\ldots ,X_n$ be i.i.d.\  from a distribution with pdf given by 
\begin{eqnarray*}
f(x \mid \theta) =
\begin{cases}
\theta^{-1}x^{(1-\theta)/\theta} & \text{if $0 \leq x \leq 1$}\\
 0 & \text{otherwise}, 
 \end{cases}
\end{eqnarray*}
where $\theta > 0$ is the unknown parameter. 

\begin{parts}

\PP  Show that  $T =  - 2 \sum_{i=1}^n \log (X_i)$ is a minimal sufficient statistic for $\theta$.  

\PP Find the distribution of $Y = -2 \log X_1$. 

\PP Using Basu's theorem or otherwise, find $\mathbb{E} [Y \mid T]$, 
the conditional expectation of the random variable $Y$ given $T$.  

\end{parts}

\QQ{5327} Suppose that $X_1, X_2, \ldots, X_n$ are independent and
identically distributed $\mbox{Poisson}(\lambda)$ random variables for
an unknown parameter $\lambda >0$.  Instead of observing the random
variables $X_i$, we only observe the events $X_i =0$ or $X_i > 0$ for
$i=1, \ldots, n$.

\begin{parts}

  \PP Find the maximum likelihood estimate (MLE) of $\lambda$ and
  discuss when the MLE is not finite.

  \PP Compute the probability that the MLE is not finite based on a
  sample of size $n$, assuming that the true value of $\lambda$ is
  $\lambda_0 > 0$.

\end{parts}

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

\pagebreak

\QQ{6346} Let $(\Omega, \mathcal{F}, \mu)$ be a measure space.  A
statement about elements $\omega\in \Omega$ is true locally almost
everywhere ($\mu$) if
\begin{itemize}
\item[i.] the statement is true for all $\omega \in A$, where $A\in\mathcal{F}$, and
\item[ii.] $\mu(A^\prime\cap F)=0$ for any $F\in\mathcal{F}$ with $\mu(F)<\infty$.

\noindent (Note: $A^\prime$ is the complement of $A$.)

\end{itemize}

\begin{parts}

  \PP Show that if something is true almost everywhere ($\mu$) then it
  is true locally almost everywhere ($\mu$).

  \PP If $\mu$ is $\sigma$-finite, show that if something is true
  locally almost everywhere ($\mu$) then it is true almost everywhere
  ($\mu$).

\end{parts}

  \QQ{6346} Let $(\Omega=[0, 1], \mathcal{F}=\mathcal{B}[0, 1], \mu)$
  be a probability space where $\mu$ is Lebesgue measure on $[0, 1]$
  and $\mathcal{B}[0, 1]$ is the restriction of the Borel
  $\sigma$-field to $[0, 1]$.  Let $X_n$ be a sequence of random
  variables given by
$$
X_n(\omega)=
\left\{
\begin{array}{ll}
n^2, & \omega \in [0, 1/n], \\
0,   & \omega \in (1/n, 1]. \\
\end{array}
\right.
$$

\begin{parts}

\PP Show that $X_n \xrightarrow{P} X$ by using the definition of convergence in probability, and give $X$.


\PP Show that $X_n \rightarrow X$ a.s. for the same $X$ as in part (a).

\PP Prove or disprove: For $p\geq 1$, $X_n \xrightarrow{L^p} X$ for the same $X$ as in parts (a) and (b).

\end{parts}

\end{document}

