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\begin{document}

\mbox{}

%\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.D.\ Qualifying Exam\\
  Wednesday--Thursday, August 21--22, 2019

\end{center}

\vspace{.1in}

{\bf $\bullet$ Begin your solution to each problem on a new sheet of paper.

\vsp

$\bullet$ Statistics PhD students should do the 5106 problems.

\vsp

$\bullet$ Biostatistics PhD students should do the 5198 problems.

\vsp

$\bullet$ All students should do the 5166 and 5167 problems.
}

\vsp

\vspace{.2in}


\QQ{5106} Consider a data set of observations $\{x_n\}$ where
$n = 1, \ldots, N$, and $x_n$ is a Euclidean variable with
dimensionality $D$.  Our goal is to project the data onto a space
having dimensionality $M < D$ while maximizing the variance of the
projected data.  Prove that the optimal linear projection for which
the variance of the projected data is maximized is defined by the $M$
eigenvectors $U_1, \ldots, U_M$ of the data covariance matrix $S$
corresponding to the $M$ largest eigenvalues
$\lambda_1, \ldots, \lambda_M$.

\QQ{5106} Find the maximum likelihood estimate of $\theta$ where
$\theta$ is the 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}

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

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

\begin{parts}

  \PP Show that $\sum_{i=1}^n (X_i - \bar{X})^2$ can be expressed in
  the form: $\sum_{i=1}^n (X_i - \bar{X})^2= \sum_{i=2}^nY_i^2 $ where
  $\{Y_2, Y_3, \ldots, Y_n\}$ are iid random variables with the
  $N(0, \sigma^2)$ distribution.

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

  \PP Let $s^2_n= \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2 $.
  What is the limit distribution of $\sqrt{n-1} (s^2_n - \sigma^2)$
  when $n$ increases to $\infty$?

\end{parts}

\pagebreak

\mbox{}

\vspace{-1cm}

\QQ{5166} The values in the following table are the burning times
(rounded to the nearest tenth of a minute) of random samples of two
kinds of emergency flares:

\vspace{.1in}

\begin{tabular}{lcccccccccc} \hline
 Observation ID         & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9& 10\\[3pt] \hline
 Brand A: $y_1$  & 15.2 & 13.1 & 15.3 & 28.1 & 2.5  & 13.7   & 15.8  & 17.2   & 22.3 &  12.5 \\[3pt] \hline 
    Brand B: $y_2$  & 16.6 & 18.1 & 14.8 & 18.6 & 10.0 & 26.5   & 11.3  & 13.5   & 12.9 &  42.5 \\[3pt] \hline
  \end{tabular}

\vspace{.1in}

\begin{parts}

  \PP Assume that the two random samples $\{y_{1i}, i=1,\ldots, 10\}$
  and $\{y_{2i}, i=1,\ldots, 10\}$ are independent, normally
  distributed, and with means $\mu_1$ and $\mu_2$ and the same
  variance $\sigma^2$. State your null and alternative hypotheses, and
  perform a test on the means of the two populations.  Test using
  $\alpha=0.05$.

  \PP Identify any potential outliers in the two samples and give your
  justifications.

  \PP State your null and alternative hypotheses, and test the two
  distributions using the Wilcoxon Rank-Sum test that is robust
  against outliers. Test using $\alpha=0.05$.
 
\end{parts}



\QQ{5167}
Consider the following simple linear regression model of $Y$ on $X_1$: 
\begin{equation*}
Y=\alpha+\beta X_1+e.
\end{equation*}
Suppose $n=45$, $\bar X_1=-3.3$, $\bar Y=5.5$, $SXX=100.0$, $SYY=123.4$, and $SXY=88.8$. 

\begin{parts}
\PP  Find the following: RSS, $R^2$, $\hat\alpha,\hat\beta,\hat{\sigma}^2\equiv\widehat{var}(e)$.

\PP Test the hypothesis (at level 0.05) that $\beta=1$ versus $\beta\neq1$.

\PP What assumptions are required for the test in part (b)?

\PP Find the 95\% prediction interval for a new observation $X_1^\star=-2$. 

\PP Consider the following regression model, 
$$
Y_i = \mu + \epsilon_i,\quad i=1,\dots,n,
$$
where $\mathrm{var}(\epsilon_i)=\sigma^2$ for all $i$, and
$\mathrm{Corr}(\epsilon_i,\epsilon_j)=\rho>0$ for $i\neq j$. How would
you estimate $\mu$? What is the variance of your estimator; and does
it go to zero when $n\rightarrow\infty$?
\end{parts}

Some {\sc R} output that might be helpful. 
\begin{verbatim}
> qt(c(0.95,0.975,0.99,0.995),42)
[1] 1.681952 2.018082 2.418470 2.698066
> qt(c(0.95,0.975,0.99,0.995),43)
[1] 1.681071 2.016692 2.416250 2.695102
> qt(c(0.95,0.975,0.99,0.995),44)
[1] 1.680230 2.015368 2.414134 2.692278
\end{verbatim}


\QQ{5167}
Suppose we have a univariate response $Y$ and a multivariate predictor $X\in\mathbb{R}^p$. We are interested in the following regression model,
$$
Y=\beta_0+\beta^TX+\varepsilon=\widetilde{\beta}^T\widetilde{X}+\varepsilon,
$$ 
where $\widetilde{\beta}=(\beta_0,\beta^T)^T\in\mathbb{R}^{p+1}$ and $\widetilde{X}=(1,X^T)^T\in\mathbb{R}^{p+1}$.  Let $Y_n\in\mathbb{R}^{n\times 1}$, $X_n\in\mathbb{R}^{n\times p}$ and $\widetilde{X}_n=(1_n,X_n)\in\mathbb{R}^{n\times (p+1)}$ be the data matrices,  where $1_n$ is a vector of ones. 
You may use this result to answer the questions: Suppose $M=\begin{pmatrix} 
A & B \\
B^T & C 
\end{pmatrix}
$ is a $m\times m$ symmetric positive definite matrix, then its inversion can be written as
$$M^{-1}=\begin{pmatrix} 
A^{-1}-A^{-1}BDB^TA^{-1} & -A^{-1}BD \\
-DB^TA^{-1} & D 
\end{pmatrix},$$
where $D=(C-B^TA^{-1}B)^{-1}$.
\begin{parts}
\PP Write down the ordinary least squares estimates for $\widetilde{\beta}$, $\beta_0$ and $\beta$ using the given notation.


\PP Let $\overline{X}$ and $S$ be the sample mean vector and sample
covariance matrix of $X$. Show that
$$(\widetilde{X}_n^T\widetilde{X}_n)^{-1}=
\begin{pmatrix} 
n^{-1}+(n-1)^{-1}\overline{X}^TS^{-1}\overline{X} & -(n-1)^{-1}\overline{X}^TS^{-1} \\
-(n-1)^{-1}S^{-1}\overline{X} & (n-1)^{-1}S^{-1} 
\end{pmatrix}
$$

\PP Suppose we want to estimate the inverse of the covariance matrix
denoted as
$\Theta\equiv\{\mathrm{Cov}(X)\}^{-1}\in\mathbb{R}^{p\times p}$. When
$p$ is very large, directly computing the inversion of the sample
covariance matrix $S$ is not practical. When $\Theta$ is sparse, the
graphical lasso method uses a regression ``trick'' to obtain the
sparse estimator of the inverse covariance. The algorithm updates one
column at a time for $\Theta$. To see this, we decompose $\Theta$
as $$\Theta=\begin{pmatrix}
  \Theta_{11} & \theta_{12}\\
  \theta_{12}^T & \theta_{22}
\end{pmatrix},
$$ 
where $\theta_{12}\in\mathbb{R}^{(p-1)\times 1}$. Suppose
$\widehat{\Theta}=S^{-1}$, show that
$\widehat{\theta_{12}}/\widehat{\theta_{22}}$ is the least squares
estimator of the regression coefficient vector for $X_p$ regressed on
$(X_1,\ldots,X_{p-1})$. Discuss how to use lasso regression to obtain
$\widehat{\Theta}$.
\end{parts}


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

\pagebreak

\QQ{5198} Imagine we conducted a study to investigate whether waist-to-hip ratio (WHR) is causally related to an elevation in serum concentrations of C-reactive protein (CRP), a marker of inflammation. We recruited 150 subjects with elevated markers of inflammation from the five primary-care clinics in a small town outside Tallahassee. We then selected controls using a random sample of 150 subjects from the community (all were confirmed not to have elevated concentrations of C-reactive protein). 

The gold standard for measuring WHR is in-home assessment by trained clinical personnel. Using this gold standard we could determine the ``true'' distribution of high and low waist-to-hip ratio in cases and controls to be as follows:
\begin{center}
	\begin{tabular}{l|p{1.5cm}|p{1.5cm}}
		& CRP & No CRP \\
		\hline
		High WHR & 67 & 32 \\
		\hline
		Low WHR & 83 & 118 \\
		\hline
		Total & 150 & 150 \\
	\end{tabular}
\end{center}


\begin{parts}
  \PP Specify the study design's name. Calculate an appropriate measure of the ``true'' association using these data, derive its 95\% confidence interval, and interpret the result.\\


\PP In reality, we could not use the gold standard for our study (it
was impractical and too expensive).  Instead we had to rely on
self-reports of waist and hip measurements to estimate waist-to-hip
ratio.  This method was found to have 75\% sensitivity and 95\%
specificity in both cases and controls, relative to the gold
standard. Use these values to answer the questions in the following parts.

Complete a table having the format given below showing how the cases
and controls would be reclassified using self-reports, given the
sensitivity and specificity of that method (round to whole
numbers). Show the derivations.
\begin{center}
	\begin{tabular}{l|p{1.5cm}|p{1.5cm}}
		& CRP & No CRP \\
		\hline
		High WHR & a =  & b =  \\
		\hline
		Low WHR & c =  & d =  \\
		\hline
		Total &  &  \\
	\end{tabular}
\end{center}


\PP Calculate the appropriate measure of association we would have
observed if we used the self-report method. Derive the 95\% confidence
interval, and interpret the result.


\PP Describe how the observed value is different from the true value.

\end{parts}


\pagebreak


\QQ{5198} The standardized mean difference (SMD) is frequently used
for contrasting two groups on continuous outcomes. Specifically,
suppose that $\mu_0$ and $\mu_1$ are the population means of certain
continuous outcome measures in two groups (say, control and treatment)
in a study, and $\sigma$ is the true standard deviation of the
measures, which is common for both groups. Then, the SMD is calculated
as $(\mu_1 - \mu_0) / \sigma$. In practice, it can be estimated by
replacing $\mu_0$ and $\mu_1$ with the sample means $\bar{y}_0$ and
$\bar{y}_1$ and replacing $\sigma$ with the pooled sample standard
deviation $s_\text{p}$.

\begin{parts}
  
  \PP Suppose we observe the following data displaying the BMI (body
  mass index) for some patients in two groups:
\begin{center}
	\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|}
		\hline
		Group & \multicolumn{10}{l|}{BMI} \\
		\hline
		Control & 32.1 & 29.6 & 29.1 & 25.4 & 29.5 & 31.4 & 33.2 & 32.9 & & \\
		\hline
		Treatment & 37.8 & 26.2 & 31.5 & 31.7 & 30.4 & 27.3 & 35.8 & 17.0 & 26.6 & 29.7 \\
		\hline
	\end{tabular}
\end{center}
Use these data to estimate the SMD.


\PP The most widely accepted definition of obesity is a BMI of 30 or
higher. Calculate the odds ratio (OR) for obesity using the data in
(a).


\PP Most clinicians find SMDs difficult to understand, while ORs are
easy to interpret. Transforming SMDs to ORs often requires certain
cut-offs of the continuous measures (like BMI at 30) to define the
binary outcomes (like obesity). One strategy of making such a
transformation independent of cut-offs is to assume that the continuous
measures follow logistic distributions (rather than normal
distributions), whose cumulative distribution functions are
$\frac{1}{1 + e^{-(x - \mu_0)/s}}$ and
$\frac{1}{1 + e^{-(x - \mu_1)/s}}$ in the two groups. Here, $s$ is a
scale parameter. Derive the means and variances of these two logistic
distributions. You may directly use the result that
$\int_{-\infty}^{\infty} \frac{x^2 e^{-x}}{(1 + e^{-x})^2} dx =
\frac{\pi^2}{3}$.

\PP Assuming that the continuous measures follow logistic
distributions, show that log(OR) = $\frac{\pi}{\sqrt{3}}$ SMD for any
cut-off point.

\end{parts}


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

% End Day 1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin Day 2.
 
\mbox{}

\vspace{-0.6in}


{\bf Begin your solution to each problem on a new sheet of paper.}

\QQ{5326}  Answer the following.  The parts are {\bf not} related.

\begin{parts}
  
  \PP Suppose the random variable $X$ has density
  \begin{displaymath}
f_X(x)=\frac{1}{\pi} \frac{1}{1+x^2}\,, \quad -\infty<x<\infty.
  \end{displaymath}
  Find $M_X(t)$, the moment generating function (mgf)
of $X$.  Give a careful description of $M_X(t)$ for all real values $t$.
Prove your answer.

\PP For each of the two functions given below, answer the following: Does
a distribution exist whose mgf $M(t)$ is the given function?  If yes, specify
this distribution.  If no, prove it.
\begin{align*}
&\frac{1}{3}\left( e^{3t} + e^{5t} + e^{7t} \right)\\[6pt]
&\frac{1}{3}\left( 2e^{3t} + 2e^{5t} - e^{7t} \right)
\end{align*}

\PP Suppose $Y \sim \text{Geometric}(\beta)$ and
$Z\,|\,Y \sim \text{Binomial}(Y,p)$.  Find $M_Z(t)$, the mgf of $Z$.  Simplify
your answer.  (Hint: one approach is to use iterated expectations.)\\[6pt]
Note: The pmf of $Y$ is $f_Y(y) = (1-\beta)^{y-1}\beta$
for $y=1, 2, 3,\ldots$ where $0<\beta<1$.

\end{parts}

\QQ{5326} Suppose that $n$ balls are placed at random into $n$ cells.
(The balls are independent of each other, and the
cells are equally likely.)  Let $X$ be the number of empty cells
remaining after the balls are placed.  

\begin{parts}

\PP Find $P(X=0)$.

\PP Find $P(X=1)$.

\PP Find $EX$.

\PP Find $\text{Var}(X)$.

\end{parts}

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

\vspace{.2cm}

\hrule

\vspace{.4cm}

\pagebreak

The following table will be useful for solving the STA 5327 problems.

\begin{tabular}{|c|c|c|c|c|}

  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \hline
  \textbf{Name} & \textbf{Notation} & $f(x)$&$\mu=E(X)$&Var(X) \\
  \hline
  Poisson & $\mathrm{Poisson}(\lambda)$ &
 $\tmpcrap \frac{e^{-\lambda}\lambda^x}{x!}$, $\quad x=0, 1, 2, \ldots.$&$\lambda$&$\lambda$\\
  \hline 
Exponential & $\tmpcrap \mathrm{Exp}(\beta)$ &
$\frac{1}{\beta}e^{-x/\beta},\quad x>0$&$\beta$&$\beta^2$\\
  \hline
Normal & $\mathrm{N}(\mu, \sigma^2)$&
$\tmpcrap \frac{1}{\sqrt{2 \pi} \sigma}e^\frac{-(x-\mu)^2}{2 \sigma^2},\quad-\infty<x<\infty$&$\mu $&$\sigma^2$\\
  \hline
Chi squared & 
$\tmpcrap \chi^2(p)$&$\frac{1}{\Gamma(p/2)2^{p/2}}x^{p/2-1}e^{-x/2},\quad x>0$ &$p$&$2p$ \\
  \hline
  \end{tabular}

\QQ{5327} Consider independent observations
  $X_1,\ldots,X_n\sim \text{Poisson}(\lambda)$ and
  $Y_1,\ldots,Y_m\sim \text{Exp}(\lambda)$ .

\begin{parts}
    
\PP Find a two-dimensional sufficient statistic for $\lambda$.

\PP Define
$\bar{X}=\frac{1}{n}\sum_{i=1}^nX_i$ and $\bar{Y}=\frac{1}{m}\sum_{i=1}^m
Y_i$. Consider estimators of the form
$\hat{\lambda}(w)=w\bar{X}+(1-w)\bar{Y}, 0\le w\le 1$. Derive a
formula for the $MSE_{\lambda}(w)=E(\hat{\lambda}(w)-\lambda)^2$.  

\PP Is it possible to find a $w^*$ such that
$MSE_{\lambda}(w^*)\le MSE_{\lambda}(w)$ for all $\lambda>0$ and
$0\le w\le 1$? If so, find this $w^*$. If not, explain why.

\end{parts}


\QQ{5327} Consider independent pairs of observations
$(X_i,Y_i), i=1,\ldots,n$, drawn according to the following model:
$$
Y_i=\beta X_i+\epsilon_i,
$$
where
$\beta\in\mathbb{R}, X_i\sim N(0,\sigma^2),\epsilon_i\sim
N(0,\sigma^2)$, and $\epsilon_i$ is independent of $X_i$.

\begin{parts}

\PP Assume that $\beta$ is known, and find the MLE for
  $\sigma^2$. Call this estimator $\hat\sigma_1^2$.

\PP Find the asymptotic distribution of $\log{\hat\sigma_1^2}$, where
$\hat{\sigma}_1^2$ is found in part (a).

\PP Now assume that $\beta$ is unknown, and find the MLE for
$(\beta,\sigma^2)$. In finding the MLE for this question, you can use
the fact that the stationary point of this likelihood function is the
global maximizer.

\end{parts}

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

\pagebreak

\QQ{6346} Let $(\Omega, \mathcal{F})$ be a measurable space.

\begin{parts}
  

\PP Give the definition of a measure $\mu$.


\PP Give the definition of a probability measure $\mu$.

\PP Suppose $\mu_i, i=1, 2, \ldots$, are measures on
$(\Omega, \mathcal{F})$.  Let $a_1, a_2, \ldots$ be a sequence of
finite positive numbers. For $A\in \mathcal{F}$, define
$$
\nu(A)=\sum_{i=1}^\infty a_i\mu_i(A) 
$$
Prove or give a counterexample: $\nu$ is a measure on $(\Omega, \mathcal{F})$.

\PP Suppose the measures $\mu_i$ in part (b) above are {\bf
  probability} measures on $(\Omega, \mathcal{F})$.  Define $\nu$ as
above. Prove or give a counterexample: $\nu$ is a {\bf probability}
measure on $(\Omega, \mathcal{F})$.

\end{parts}



\QQ{6346} An urn initially contains one red and one green ball.
Conduct a series of draws.  At each draw, select a ball at random,
note its color, and return that selected ball and an additional ball
of the same color to the urn.  Let $X_i$ be the fraction of red balls
in the urn after $i$ draws.

\begin{parts}
  
\PP Give the definition of a discrete-time martingale.

\PP Prove or disprove: The sequence $X_1, X_2, X_3, \ldots$ is a
martingale.

\PP Find $E(X_k)$.

\end{parts}


\end{document}


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