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\begin{document}

\mbox{}

%\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.D.\ Qualifying Exam\\
  Thursday--Friday, January 2--3, 2020

\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} Let $Y$ be a discrete random variable with nonnegative
integer values.  For any $y \in \{0, 1, 2, \cdots\}$,
$$ Prob(Y=y) = \alpha_1 P_1(y; \lambda_1) +  \alpha_2 P_2(y; \lambda_2),$$
where $P_1$ and $P_2$ are two Poisson probability mass functions with means $\lambda_1$ and $\lambda_2$, 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
$$\theta = (\alpha_1, \lambda_1, \alpha_2, \lambda_2).$$
Use the EM algorithm for iteratively estimating $\theta$. 
Let $\theta^{(m)}$ be the estimated value at the $m$-step.  Derive the mathematical formula to update it for $\theta^{(m+1)}.$   

\QQ{5106} Consider a data set of observations $\{x_n\}$ where
$n = 1, \cdots, N$, 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 $U_1, \cdots, U_M$ of the data covariance matrix $S$
corresponding to the $M$ largest eigenvalues
$\lambda_1, \cdots, \lambda_M$.

\pagebreak

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

\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 $\tau_t$ is $\hat{\tau}_t = \bar{y}_{t\cdot }-\bar{y}_{\cdot\cdot}$.
 Find the expectation and variance of $\hat{\tau}_t$.

\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 Show that under the Null Hypothesis $H_0: \tau_t=0$ for $t=1,\ldots, k$,  $S_T$ and $S_R$ are independent.
\end{parts}

\QQ{5166} Two treatments, A and B, are compared in an experimental
design. For each treatment, measurements are recorded successively in
time.  Suppose that the measurements follow the models:
\[ X_i(t) = \mu_i + \xi_i(t), \hspace{0.2in} i=1,2;\]
where the  processes $\{\xi_i(t), i=1,2;  t= -1, 0, 1,\ldots, \}$ are generated by the  models:
\[\xi_1(t) - 0.8 \xi_1(t-1) = a_1(t),  \hspace{0.2in} \xi_2(t)=a_2(t) +2.5a_2(t-1),\]
where $\{a_i(t), i=1,2;  t=0, \pm 1,  \pm 2 \ldots,\}$  are i.i.d.\ $N(0,\sigma^2)$ variables.\\

\noindent Let $\bar{X}_i = \sum_{t=1}^nX_i(t)/n$ for $i=1, 2$.

\begin{parts}

\PP
 Show that  $\xi_1(t) = \sum_{j=0}^{\infty} (0.8)^j a_1(t-j)$ is a solution of the equation $$\xi_1(t) -0.8 \xi_1(t-1) = a_1(t).$$ Is the process $\{\xi_1(t), t= -1, 0, 1,\ldots, \}$ stationary?

\PP
  Based on the expression in (a) for $\xi_1(t)$,  calculate  autocorrelations $$\gamma_k = \mbox{Cov}(\xi_1(t), \xi_1(t+k))$$ for $k \geq 0$.  Show that
$\gamma_k = 0.8 \gamma_{k-1}$ for any $k\geq 0$.

\PP
What is the distribution of  $\bar{X}_2$ ?

\end{parts}

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


\QQ{5167}
Consider the following simple linear regression model of $Y$ on $X_1$: 
\begin{equation}\label{eq1}
Y=\alpha+\beta X_1+e.
\end{equation}
Suppose $n=60$, $\bar X_1=-3.3$, $\bar Y=5.5$, $SXX=100.0$,
$SYY=123.4$, 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 $\beta>1$.

\PP What assumptions are required for the test in part (b) (in
addition to (1))?

\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 $i\neq j$. Is the sample mean $\overline Y=\sum_{i=1}^n Y_i/n$ still the MLE for $\mu$? What is the variance of $\overline Y$? Does the variance go to zero when $n\rightarrow\infty$ in this case, and why?
\end{parts}

Some {\sc R} output that might be helpful. 
\begin{verbatim}
> qt(c(0.95,0.975,0.99,0.995),57)
[1] 1.672029 2.002465 2.393568 2.664870
> qt(c(0.95,0.975,0.99,0.995),58)
[1] 1.671553 2.001717 2.392377 2.663287
> qt(c(0.95,0.975,0.99,0.995),59)
[1] 1.671093 2.000995 2.391229 2.661759
\end{verbatim}

\pagebreak

\QQ{5167}
Suppose we have univariate response $Y$ and 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 and sample covariance 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 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 regression coefficient estimator of $X_p$ regressed on $(X_1,\ldots,X_{p-1})$. Discuss how to obtain $\widehat{\Theta}$ when the sample covariance is not positive definite due to small-$n$-large-$p$ (hint: use lasso regression).
\end{parts}

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

% Increment problem counter for missing biostat problems.

\stepcounter{qnum}
\stepcounter{qnum}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 not related.

\begin{parts}
  
  \PP Define $Y= \tan X$ where $X$ is uniformly distributed on the
  interval $(-\pi/2,\pi/2)$.  Find the cumulative distribution
  function (cdf) and the density (pdf) of $Y$.

Note: $\dsp \frac{d}{du} \tan^{-1}(u) = \frac{1}{1+u^2}$

\PP Let $Z$ have the Cauchy density given by
\begin{displaymath}
  f_Z(z) = \frac{1}{\pi(1+z^2)}\,,\quad -\infty<z<\infty.
\end{displaymath}
Show that $Z \stackrel{d}{=} 1/Z$, that is, $Z$ and $1/Z$
have the same distribution.

\PP Find the density (pdf) of the ratio $S/T$ where $S,T$ are i.i.d.\
$N(0,1)$ (standard normal).

\end{parts}

\QQ{5326} An urn contains $R$ red balls and $G$ green balls.  Balls
are drawn at random from this urn until a red ball is drawn.  What is
the probability that {\bf more} than $k$ draws are required?  (Assume
$k \geq 1$ is some given integer.)  Answer this question in each of
the following situations.  Simplify your answers.

\begin{parts}

\PP Draws are done {\bf WITH} replacement.

\PP Draws are done with{\bf OUT} replacement.  (In this case assume
that $1 \leq k \leq G$.)

\PP The urn is a Polya urn: after a ball is drawn, it is returned to
the urn and another ball of the same color is added to the urn.

\end{parts}

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

\pagebreak

\vspace{.1in}

\hrule

\vspace{.2in}

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} & density $f(x)$&$\mu=E(X)$&Var(X) \\ 
  \hline 
Normal & $\mathrm{N}(\mu, \sigma^2)$&
$\frac{1}{\sqrt{2 \pi} \sigma}e^\frac{-(x-\mu)^2}{2 \sigma^2},\quad-\infty<x<\infty$&$\mu $&$\sigma^2$\\
  \hline
  \end{tabular}

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


\QQ{5327} Consider independent observations $(X_i,Y_i),i=1,\ldots,n$, where 
$$
P(Y_i=k)=\pi_k\,, \qquad X_i\mid Y_i=k\sim N(\mu_k,\sigma^2),
$$
with $Y_i \in \{1,2\}$,
$0<\pi_1,\pi_2<1$, $\pi_1+\pi_2=1$, $\mu_k\in\mathbb{R}$, $\sigma^2>0$.

 \begin{parts}
   
 \PP Construct consistent estimators for $\pi_1,\pi_2,\mu_1,\mu_2$ and $\sigma^2$. Which of them (if any) are biased?

 \PP Verify that $(X_i,Y_i)$ satisfies the following equation:
 $$
 \log\dfrac{P(Y_i=2\mid X_i=x)}{P(Y_i=1\mid X_i=x)}=\beta_0+\beta x.
 $$
 Derive the explicit expressions of $\beta_0$ and $\beta$.
 
 \PP Find a consistent estimate for $\beta_0$. Prove the consistency.

 \end{parts}



\QQ{5327} Consider independent observations $X_i,i=1,\ldots,n$, where
 $X_i\sim N(\mu,1)$ with $|\mu|\le 1$.

\begin{parts}

\PP Let $\bar{X}$ be the sample mean. Is $\bar{X}$ admissible? Give a rigorous proof to your conclusion.

\PP Find the likelihood ratio test for 
$$
H_0: \mu=0 \quad \text{vs.} \quad H_1: \mu\ne 0, |\mu|\le 1.
$$

\PP If $n$ is very large, is the knowledge of $|\mu|\le 1$ important for testing the hypothesis in part (b)?

\end{parts}


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

\pagebreak

\QQ{6346}  Answer the following.

\begin{parts}

\PP Give the definition of a submartingale.

\PP State Doob's decomposition theorem and prove that the decomposition is unique.

\PP Suppose $X_n$ is a submartingale and its Doob decomposition is $X_n=X_0+Y_n+A_n$ where $Y_n$ is the martingale. Show $E(A_n)<\infty$.
  
\end{parts}


\QQ{6346} Let $E_1, E_2, \ldots $ be a sequence of events and $\mu$ a
probability measure.

\begin{parts}
  
\PP Show that
  $\mu\left( \bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty
  \mu(E_i)$.

\PP If $\mu(E_i)=0$ for all $i$, prove that
  $\mu\left( \bigcup_{i=1}^\infty E_i\right) = 0$.

\PP If $\mu(E_i)=1$ for all $i$, prove that
  $\mu\left( \bigcap_{i=1}^\infty E_i\right) = 1$.

\PP If $E_i \subset E_{i+1}$, prove that $\mu(\lim_{i\rightarrow \infty} E_i) = \lim_{i\rightarrow \infty} \mu(E_i)$.

\end{parts}

\end{document}


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