\documentclass[12pt]{article}

\usepackage{amsmath,amsthm,amssymb,amsfonts}
\usepackage{graphicx}

%\usepackage{undertilde}

\newcommand{\tmpcrap}{\dsp \vphantom{\frac{\frac{X}{Y}^{\frac{A}{B}}}{\frac{X}{Y}^{\frac{C}{D}}}}}

% Original margins

\addtolength{\textheight}{1.8truein}
\addtolength{\topmargin}{-.90truein}

\addtolength{\oddsidemargin}{-0.5truein}
\addtolength{\evensidemargin}{-0.5truein}
\addtolength{\textwidth}{1.0truein}


\newcommand{\dsp}{\displaystyle}

\newcommand{\argmin}{\mathop{\rm argmin}}

\newcommand{\real}{\ensuremath{\mathbb{R}}}


\newcommand{\comment}[1]{} 
\newcommand{\vsph}{\vspace{.50cm}}
\newcommand{\vsp}{\vspace{.2cm}}
\newcommand{\vecc}[1]{\text{\boldmath $#1$}}
%\newcommand{\utilde}[1]{\underset{\widetilde{}}{#1}}
\newcommand{\G}{\,|\,}
\newcommand{\field}[1]{\mathbb{#1}}

\def\EE{ {\rm I} \kern-.15em {\rm E} }

\newcounter{qnum} \newcounter{pnum}[qnum]


% \newcommand{\topic}[1]{\vsph \hrule \vsph \mbox{} \hfill { \it #1}
%   \hfill \mbox{} }

% \newcommand{\QQ}{\vsph \hrule \vsph \stepcounter{qnum} {\noindent \bf
%     Problem \theqnum.} \hspace{.2cm} }

\newcommand{\QQ}[1]{
\pagebreak[2] 
\vsph \hrule \vsph \stepcounter{qnum} {\noindent \bf
    Problem \theqnum.}\hspace{.03cm} (#1) \hspace{.1cm} }

\newcommand{\cont}{{\noindent \bf Problem \theqnum \ continued:\,}
  }

\newcommand{\sep}{\vsph \hrule \vsph}

\newcommand{\PP}{\stepcounter{pnum} \item[{\bf (\alph{pnum})}]\, }

\newenvironment{parts}{
  \begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
      \setlength{\leftmargin}{2.4em}\setlength{\labelsep}{0.8em}\setlength{\itemindent}{-.5em}}
  }{\end{list}}

\def\god{{\buildrel {\cal D} \over \longrightarrow}}
\def\goas{{\buildrel {\rm a.s.} \over \longrightarrow}}

\def\r{\rightarrow} \def\p{\stackrel{p}{\r}} \def\d{\stackrel{d}{\r}}
\def\argmax{\mathop{\rm argmax}}

\newcommand{\var}{\mbox{Var}}


\usepackage{amsfonts}

\newcommand{\RR}{\mathbb {R}}

\newcommand{\eps}{\varepsilon}


\newenvironment{clist}
{%\vspace{-1cm}
\begin{list}{$\bullet$}
{
%\setlength{\topsep}{0.0cm}
%\setlength{\itemsep}{0.0cm} 
%\setlength{\parsep}{0.0cm}}
\setlength{\topsep}{.1cm}
\setlength{\itemsep}{.1cm} 
\setlength{\parsep}{.1cm}}
}{\end{list}}


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

% For 5166 and 5198

\newcommand{\twobytwo}[4]{\mbox{%
\ensuremath{
 \begin{array}{r|r|r|}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{D} & 
              \multicolumn{1}{c}{\overline{D}} \\
\cline{2-3}
E  & #1 & #3 \\
\cline{2-3}
\overline{E} & #2 & #4\\
\cline{2-3}
 \end{array}
}%
}}
\newcommand{\OR}{\textnormal{OR}}
\newcommand{\E}{\textnormal{E}}
\newcommand{\V}{\textnormal{Var}}

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

\begin{document}

\mbox{}

\vspace{-.8in}

\begin{center}
  \bf \large

  Ph.D.\ Qualifying Exam\\
  Wednesday--Thursday, August 23--24, 2023

\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}


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

% xxx Day 1 problems begin here.


\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
$X_i \in \mathbb R^d$, the value $Y_i \in \mathbb Z^+$ follows a Poisson
distribution with parameter $\lambda_{X_i} = e^{\beta^T X_i}$, where
the parameter $\beta \in \mathbb R^d$.  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 
$$ \hat \beta = \argmax_{\beta}\, l(\beta)\,.$$
\PP  Prove that the Hessian matrix w.r.t.\ the log-likelihood function is semi-negative definite. 
\PP
Write out the Newton-Raphson algorithm to find  $\hat \beta$. 
\end{parts}


\QQ{5106} Let $X$ be a random variable which follows a Gamma
distribution with shape $\alpha$ and rate $\beta$.  For a constant
$a>0$, our goal is to use the tilted sampling to estimate the tail
probability:
 $$ \theta = P\{X>a\} = \int_a^\infty f(x; \alpha, \beta) dx, $$
where the Gamma density function is $\dsp f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}$ for $x>0$.  We simplify the calculation by assuming $\alpha = 3$ in this problem.  

\begin{parts}

\PP Compute $M(t) = \int_0^\infty e^{tx} f(x; 3, \beta)\, dx$.


\PP Derive an expression to find the optimal amount of tilt $t$ to
estimate $\theta$ for a given $a$.  (Hint: the mean of $X$ is
$\frac{\alpha}{\beta}$.)

\PP 
Based on the optimal $t$, write out the sampling method to estimate $\theta$.  

\end{parts}


\QQ{5166}  Consider the following linear model for a randomized block design:
\[y_{ti} = \mu + \tau_t + \beta_i + \epsilon_{ti}\,, \quad t=1, \ldots, k\,; \quad i=1, \ldots, n,\]
where $\mu$ is an overall mean, $\tau_t$ is the effect of $t$th
treatment, $\beta_i$ is the effect of $i$th block, and
$\{\epsilon_{ti}: t=1, \ldots, k; i=1, \ldots, n\}$ are assumed
to be iid $N(0, \sigma^2)$.
Define
 \[ Y = (Y_{11}, Y_{12}, \ldots, Y_{1n}, Y_{21}, Y_{22}, \ldots, Y_{2n},
 \ldots, Y_{k1}, Y_{k2}, \ldots, Y_{kn})'.\]

\begin{parts}
\PP
 Find the mean vector $\vecc{\mu} = E(Y)$ and the variance-covariance matrix $V= \mbox{Cov}(Y)$.
 What is the distribution of Y?

\PP Define
\begin{itemize}
\item $S_D$: Total Variation of the observations,
\item $S_T$: Sum of Squares for Treatments,
\item $S_B$: Sum of Squares for Blocks,
\item $S_R$: Sum of Squares for Experimental Errors.
\end{itemize}
 Write an ANOVA table for this experiment, including the Sources of Variation, Degrees of Freedom (df),
Sums of Squares, and Mean Squares. Derive the expectations for the Mean Squares.

\end{parts}

\QQ{5166} Consider a $2^{5-2}$ fractional factorial design with generators {\bf I=1234} and {\bf I=135}.
After analyzing the results from this design the researcher decides to perform a second $2^{5-2}$
design exactly the same as the first but switching the signs of column 2 from ``$+$"
to ``$-$" and ``$-$" to ``$+$"  (a fold-over design).

\begin{parts}

\PP
 What is the defining relation of the first design? What is the resolution of the first design?
 List the aliases of the main effects in this design.
\PP
  What is the defining relation of the second design? What is the resolution of the second design?
  List the aliases of the two-term interactions of factors {\bf 1, 2,} and {\bf 3} in this design.

\PP
 What is the defining relation of the combined design (designs 1 and 2 together)?
 What is the resolution of the combined design? (Hints: combine the defining relations of the two designs
 together and remove all the defining words that have both``$+$" and ``$-$" signs.)

 \PP
 Can you modify the first and second designs such that the combined design has a higher resolution?

\end{parts}


\QQ{5167} Consider independent observations $\{Y_i,X_i\}_{i=1}^n$, where $Y_i\in\mathbb{R}$ and $X_i\in\mathbb{R}$. 

\begin{parts}

\PP Assume that $E(Y_i\mid X_i)=\beta_0+\beta_1 X_i$ and $\text{Var}(Y_i\mid X_i)=\sigma^2$, where $\beta_0,\beta_1\in\mathbb{R}$ and $\sigma^2>0$ are all unknown. Derive the OLS estimator for $(\beta_0,\beta_1)$.

\PP Suppose we instead assume that $E(Y_i\mid X_i)=\beta_0+\beta_1 X_i$ and $\text{Var}(Y_i\mid X_i)=\sigma^2/d_i$\,, where $\beta_0,\beta_1\in\mathbb{R}$ and $\sigma^2>0$ are all unknown, but $d_i>0$ is known for $i=1,\ldots,n$. Describe how you would like to estimate $(\beta_0,\beta_1)$.

\PP If we are in doubt of the model assumption $E(Y_i\mid X_i)=\beta_0+\beta_1 X_i$\,, how can we test it?

\end{parts}

% xxx Finished proofing up to here.


\QQ{5167} Suppose $X$ is a continuous predictor, and $F$ is a factor
with two levels $\{+1,-1\}$, represented by two dummy variables
$U_+ = I(F = +1)$ and $U_-= I(F =-1)$ where $I(\cdot)$ is the
indicator function that takes values 0 and 1. The response $Y$ is a
continuous variable. Consider the model
$$E(Y \mid X, F) = \alpha + \beta X + \gamma U_+.$$

\begin{parts}

\PP Sketch the fitted regression lines (on a plot of $Y$ versus
$X$) for the two levels of $F$, and determine the value of $E(Y \mid X = 1, F = +1)-E(Y \mid X = 1, F =-1).$

\PP Interpret $\gamma$.

\PP Can we fit a model under the assumption $E(Y \mid X, F) = \alpha + \beta X + \gamma_+ U_++\gamma_- U_-$? Why or why not?

\end{parts}

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

\pagebreak

\mbox{}

\vspace{-1.7cm}

\QQ{5198} A study of the association between concussions in youth and
mental health problems identified in children aged 5-18 years who
presented to a medical facility with either a concussion (the
exposure) or an orthopedic injury (the comparison group). Children
seen for reasons associated with mental health in the year prior to
injury were excluded, and all children were covered by the same
national health insurance plan. There were 152,231 children who
presented with concussion; each was matched by age and sex to multiple
children with orthopedic injury. Among the 152,321 children who
presented with concussion, 53,863 experienced mental health problems
during the subsequent follow up of approximately 10 years. Of the
296,482 children with orthopedic injury, 80,076 showed mental health
problems during the follow up.

\begin{parts}
\PP Display these data in the usual $2\times 2$ format as shown:
\twobytwo{a=\qquad\quad}{c=\qquad\quad}{b=\qquad\quad}{d=\qquad\quad}

\PP In this study design, are relative risk (RR) and odds ratio (OR) both
  appropriate measures for the association between concussion and mental health?
  Explain.

\PP Using your selected measure of association (RR or OR), use these data to
  estimate the association and provide an approximate 95\% confidence
  interval. Interpret.

\PP The investigators excluded children with medical visits for reasons
  associated with mental health in the year prior to injury. Is this a strength
  or a weakness of the study? Explain.

\PP What advantage, if any, is the comparison group of children with
  orthopedic injury compared to a comparison group of randomly selected children
  without concussion?

\end{parts}

\vspace{-.25cm}

\QQ{5198} A case-control study investigated the association between vaccination against
influenza A and B viruses (``flu vaccine'') and multiple sclerosis (MS). The
investigators identified 110 subjects with MS and matched each to a control
subject according to age, sex and nationality. There were 50 pairs in which both
the MS case subject and paired control subject had received the flu vaccine; 10
pairs in which the MS case subject had the vaccine and the control subject did
not; 25 pairs in which the control subject but not the paired case subject had
received the vaccine; and 25 pairs in which neither subject in the pair had
received the vaccine.

\begin{parts}
\PP Formally test the association between flu vaccination and MS. Clearly state the null and alternative hypotheses and your conclusion.
  
\PP Find the estimated odds ratio for vaccine versus no vaccine and give a corresponding (approximate) 95\% confidence interval. Interpret in the context of this study.

\PP The investigators noted that 30\% of the MS case subjects had a family
  history of MS whereas only 12\% of control subjects had this family
  history. Explain how to adjust for family history as a confounder when
  assessing the association between flu vaccine and MS in this study. Describe
  how to perform your recommended analysis.

\end{parts}  



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
% BEGIN FORMULA SHEET
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{center}
\large \bf  Formula Sheet for STA 5198 Problems
\end{center}


\noindent Some known formulae based on the usual $2 \times 2 $
table $\twobytwo{a}{c}{b}{d}$.
%\[
%\twobytwo{a}{c}{b}{d}.
%\]

\begin{align}
\mbox{RR} &= \mbox{relative risk}\nonumber\\
\mbox{OR} &= \mbox{odds ratio}\nonumber\\
\mbox{AR} &= \mbox{attributable risk} = \frac{R - R_{\overline{E}}}{R}, \text{ where $R$ is the overall risk}
\label{eq:AR}
\end{align}
\begin{align}
\widehat{\mbox{Var}}\left(\log \widehat{\mbox{RR}}\right) &\approx
\frac{1}{a} - \frac{1}{a+b} +
\frac{1}{c} - \frac{1}{c+d} \\
\widehat{\mbox{Var}}\left(\log \widehat{\mbox{OR}}\right) &\approx \frac{1}{a} + \frac{1}{b} +
\frac{1}{c} + \frac{1}{d}
\end{align}
Approximate $100(1-\alpha)\%$ confidence interval for $\mbox{AR}$ is
given by
\begin{equation}
\frac{(ad - bc)\exp(\pm u)}{nc + (ad-bc)\exp(\pm u)},
\end{equation}
where
\[
u = \frac{z_{\alpha/2}(a + c)(c + d)}{ad-bc} \sqrt{\frac{ad(n-c) +
    c^2b}{nc(a+c)(c+d)}}.
\]
\hrule
\begin{itemize}
\item Mantel-Haenszel estimate of the odds ratio: 
\begin{equation}\left(\sum_i \frac{a_i d_i}{n_i}\right)/\left( \sum_i \frac{b_i c_i}{n_i}\right)
\end{equation}
\item Cochran-Mantel-Haenszel test for significance of the E-D association adjusted for confounder:
\begin{equation}
X^2 = \frac{\left(\sum a_i - \sum E(a_i)\right)^2}{\sum \V(a_i)}
\end{equation}
\[
\E(a_i) = \frac{D_i E_i}{n_i}, \qquad \V(a_i) = \frac{D_i \overline{D}_i E_i \overline{E}_i}{n_i^2(n_i -1)}
\]
\item Breslow-Day test for homogeneity of the relative risks (or odds ratios) across strata:
\begin{equation}
X^2 = \sum\frac{(a_i - \E(a_i))^2}{\V(a_i)},
\end{equation}
where now $\E(a_i)$ and $\V(a_i)$ are computed using the Mantel-Haenszel estimate of the common relative risk across strata.

\item $\chi^2$ test for association in 2-way table with observed counts $\{O_{ij}\}$:
\begin{equation}
X^2 = \sum_{i,j} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}, \qquad E_{ij} = \frac{O_{i\cdot}}{n}\frac{O_{\cdot j}}{n}n
\end{equation}

\item $\chi^2$ test for trend in $\ell \times 2$ table with exposure scores $x_1, x_2, \ldots, x_\ell$:
\begin{equation}
X^2_{(L)} = \frac{(T_1 - \frac{n_D}{n} T_2)^2}{V},
\end{equation}
where $T_1 = \sum a_i x_i$, $T_2 = \sum m_i x_i$, $T_3 = \sum m_i x_i^2$ and $V = n_Dn_{\bar{D}}(nT_3 - T_2^2)/[n^2(n-1)]$


\item $\kappa$ statistic for agreement in square 2-way table with observed counts $\{O_{ij}\}$ and expected counts (assuming independence of rows and columns) $\{E_{ij}\}$:
\begin{equation}
\kappa = \frac{p_O - p_E}{1 - p_E}, \qquad p_O = \sum O_{ii}/n, \ \ p_E = \sum E_{ii}/n
\end{equation}

\item Some normal quantiles.  Here $Z \sim N(0, 1)$.
%\[
%\begin{array}{|r|cccccc|}
%\hline
%z & 2.33 & 1.96 & 1.64 & 1.28 & 1.04 & 0.84\\
%\hline
%P(Z \le z) & 0.99 & 0.975 & 0.95 & 0.90 & 0.85 & 0.80 \\
%\hline
%\end{array}
%\]
\[
\begin{array}{|r|cccccc|}
\hline
z & 0.84 & 1.04 & 1.28 & 1.64& 1.96 & 2.33 \\
\hline
P(Z \le z) & 0.80 & 0.85 & 0.90 & 0.95 & 0.975 & 0.99 \\
\hline
\end{array}
\]

\item 
Critical values of the $\chi^2_{\nu}$ distribution.  For $W \sim \chi^2_{\nu}$, the entries are $q$ such that $\Pr(W \ge q) = p$.
	\[
	\begin{array}{r|rrrr}
	\multicolumn{1}{c}{}& \multicolumn{4}{c}{\mbox{Probability}\ \  p} \\
	\multicolumn{1}{c|}{\nu = \mbox{df}} & 0.10 & 0.05 & 0.025 & 0.01\\
		\hline
1 & 2.7 & 3.8 & 5.0 & 6.6\\
2 & 4.6 & 6.0 & 7.4 & 9.2\\
3 & 6.3 & 7.8 & 9.3 & 11.3\\
4 & 7.8 & 9.5 & 11.1 & 13.3\\
5 & 9.2 & 11.1 & 12.8 & 15.1\\
6 & 10.6 & 12.6 & 14.4 & 16.8\\
7 & 12.0 & 14.1 & 16.0 & 18.5\\
8 & 13.4 & 15.5 & 17.5 & 20.1\\
9 & 14.7 & 16.9 & 19.0 & 21.7\\
10 & 16.0 & 18.3 & 20.5 & 23.2\\
11 & 17.3 & 19.7 & 21.9 & 24.7\\
12 & 18.5 & 21.0 & 23.3 & 26.2
	\end{array}
      \]
\label{chisq.table}
	
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% END FORMULA SHEET
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

\pagebreak

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

%\vspace{-0.6in}


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

\vspace{.2in}


\QQ{5326} A Polya urn initially contains 1 red ball and $G$ green
balls.  An infinite sequence of balls is drawn from this urn.  (Since
this is a Polya urn, after each ball is drawn, it is returned to the
urn and another ball of the same color is added to the urn.)\\ [-6pt]

\noindent Answer the following questions.  Justify your answers.
Simplify your answers when possible.  In these questions let $A_i$ be
the event that the $i$-th ball drawn from the urn is red, and let $X$
be the random number of draws needed to get the first red ball.

\begin{parts}

\PP Find $P(A_2)$.

\PP Find $P(A_1 \,|\, A_2)$.

\PP Find $P(X=k)$ where $k$ is an arbitrary positive integer.

\PP  Find $EX$.

\PP Find the probability that the ball drawn immediately after the
first red ball is also red.  (Note: this is {\large \bf not} a
conditional probability.)
  
\end{parts}

%\sep

\pagebreak

\noindent The following facts about the bivariate normal distribution
are {\bf not} needed to solve the next problem, but might be used to
reduce the work in some parts.  You may freely use any of
these facts without proof.\\

\noindent Let $(X,Y)$ have a bivariate normal distribution with
parameters $\mu_X$, $\mu_Y$, $\sigma^2_X$, $\sigma^2_Y$, $\rho$. \\

\noindent The joint density (pdf) of $(X,Y)$ is defined for all
$(x,y) \in \mathbb{R}^2$ by
\begin{align*}
g(x,y)&=  \left(2\pi \sigma_X \sigma_Y \sqrt{1-\rho^2}\right)^{-1}\\
& \quad\times \exp\left(\frac{-1\phantom{x}}{2(1-\rho^2)}
\left(\left(\frac{x-\mu_X}{\sigma_X}\right)^2 
- 2\rho\left(\frac{x-\mu_X}{\sigma_X}\right)\left(\frac{y-\mu_Y}{\sigma_Y}\right)
+\left(\frac{y-\mu_Y}{\sigma_Y}\right)^2\right)\right)\,.
\end{align*}
The marginal distributions are $X \sim N(\mu_X,\sigma^2_X)$, 
$Y \sim N(\mu_Y,\sigma^2_Y)$. \\[6pt] 
The conditional distributions are:
\begin{align*}
  Y\,|\,X=x &\sim N\left(\mu_Y + \rho \sigma_Y \left(\frac{x-\mu_X}{\sigma_X}\right)\,,\,
\sigma^2_Y(1-\rho^2)\right)\,,\\
X \,|\, Y=y &\sim N\left(\mu_X + \rho \sigma_X \left(\frac{y-\mu_Y}{\sigma_Y}\right)\,,\,
\sigma^2_X(1-\rho^2)\right)\,.
\end{align*}

\QQ{5326} Let $(X,Y)$ have a joint density which is defined for all
$(x,y) \in \mathbb{R}^2$ by
\begin{displaymath}
  f(x,y) =
  \begin{cases}
    C \exp\left( -\left(x^2-xy+ \dsp \frac{y^2}{2}\right)\right) &\text{for $y \geq \sqrt{2}$,}\\
    0 & \text{for $y < \sqrt{2}$\,.}
  \end{cases}
\end{displaymath}
For $y \geq \sqrt{2}$, this density is proportional to a bivariate normal density $g(x,y)$ with
$\mu_X=\mu_Y=0$, $\sigma^2_X = 1$, $\sigma^2_Y = 2$, and
$\rho=1/\sqrt{2}$.

Answer the following.  Justify your answers.  Simplify your answers
and give explicit formulas for all answers.  If your answer is valid
only in a certain range, that should be part of your answer.  Some
answers may involve $\Phi$, the $N(0,1)$ cdf defined by
$\Phi(z) = \dsp \int_{-\infty}^z
\frac{e^{-u^2/2}}{\sqrt{2\pi}}\,du$\,.

\begin{parts}

\PP Find the value of $C$.  (If you cannot find this value, just
  leave it as $C$ when doing the later parts.)

\PP Find the marginal density of $Y$.

\PP Find the marginal density of $X$.
  
\PP Find $E(X | Y=y)$ for $y > \sqrt{2}$.

\PP Find $E(Y | X=x)$ for $-\infty < x < \infty$.

\end{parts}


% \sep

\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} & $f(x)$&E(X)&Var(X) \\ 
  \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 & $\chi^2(p)$&$\tmpcrap \frac{1}{\Gamma(p/2)2^{p/2}}\,x^{p/2-1}e^{-x/2},\quad x>0$ &$p$&$2p$  \\
  \hline
  \end{tabular}\\[4pt]

\vspace{0.5cm}

\QQ{5327} We consider $X_1,\ldots,X_n$ i.i.d.\ from $N(\mu,\sigma^2)$, where the parameter space is $\Theta=\{(\mu,\sigma^2): \sigma^2>0,\, |\mu|\le\sigma/10\}$.

\begin{parts}

\PP Find the method of moments (MOM) estimator for $\mu$. Call it $\hat \mu$.

\PP Find the mean squared error (MSE) for the MOM estimator $\hat\mu$. Recall that $\text{MSE}_{\mu}(\hat\mu)=E(\hat\mu-\mu)^2$.

\PP Consider the estimator 
$$
\tilde\mu=\frac{1}{n}\sum_{i=1}^n\frac{X_i}{1+\lambda}\,,
$$
where $\lambda\ge 0$ is a constant.  For what values of $\lambda$ do we have
$$
\text{MSE}_{\mu}(\tilde\mu)\le\text{MSE}_{\mu}(\hat\mu)
$$
for all $|\mu|\le\sigma/10$, where $\hat\mu$ is the MOM estimator?

\end{parts}


\QQ{5327} We consider $X_1,\ldots,X_n$ i.i.d.\ from $N(\mu,\sigma^2)$, where the parameter space is $\Theta=\{(\mu,\sigma^2): \mu\in\mathbb{R},\sigma^2>0\}$.


\begin{parts}

\PP Assume that $\mu$ 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). (Hint: If $Z\sim N(0,1)$,
then $Z^2\sim \chi^2(1)$.)

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

\end{parts}


\pagebreak

\QQ{6346} Let $(\Omega,\mathcal{B}, P)$ be a probability space and let
$\mathcal{G}$ be a sub-sigma-field of $\mathcal{B}$. Suppose $X$ and
$Y$ are $L_{2}$-integrable random variables on
$(\Omega,\mathcal{B}, P)$; that is, $E(|X|^{2})$ and $E(|Y|^{2})$ are
both finite. Also, assume that $Y$ is $\mathcal{G}$-measurable. Let
$V(\cdot)$ denote the variance; that is,
$V(X) = E\left[\left\lbrace X - E(X)\right\rbrace^{2}\right]$ and
$V(X\vert \mathcal{G}) = E\left[\left\lbrace X - E(X\vert
    \mathcal{G})\right\rbrace^{2}\vert \mathcal{G}\right]$.

\begin{parts}

\PP Prove that $V(X\vert \mathcal{G}) \le E\left\lbrace (X - Y)^{2}\vert \mathcal{G}\right\rbrace$ almost surely.

\PP Use Part (a) and a particular specification of $Y$ to prove that $E[V(X\vert \mathcal{G})] \le V(X)$. 

\PP Define $f$ to be a contraction on $\mathbb{R}$. That is, $f: \mathbb{R}\rightarrow \mathbb{R}$ with
\begin{equation}\label{eq:contraction}
|f(x) - f(y)| \le |x-y|
\end{equation}
for all $x,y \in \mathbb{R}$ with strict inequality for some pair $x,y$. Use Part (a) and Equation (\ref{eq:contraction}) to prove that $V[f(X)\vert \mathcal{G}] \le V(X\vert \mathcal{G})$ almost surely.

\end{parts}


\QQ{6346} Let $Y_{1},\ldots, Y_{n}$ be a random sample from $F$. Suppose you are interested in performing inferences under a different distribution, $G$. Let $F$ and $G$ have densities with respect to Lebesgue measure, namely $f$ and $g$. Define the function $w(y) = g(y)/f(y)$, and assume that $f(y)>0$ whenever $g(y)>0$.

\begin{parts}

\PP Define the estimator
\begin{equation}\label{eq:thetahat}
\widehat{\theta} \equiv \frac{1}{n}\sum_{i = 1}^{n}w(Y_{i}) h(Y_{i}).
\end{equation}
Prove that $\widehat{\theta}$ converges almost surely to $E_{G}\left\lbrace h(Y)\right\rbrace$ as $n\rightarrow \infty$, provided the expectation exists. Here, $E_{G}$ is the expectation operator with respect to $G$.

\PP Provide the formal statement for the classical Central Limit Theorem (CLT) for independent and identically distributed (iid) random variables. What condition do we require for $\widehat{\theta}$ in (\ref{eq:thetahat}) to converge asymptotically to a normal distribution? What is this asymptotic distribution for $\widehat{\theta}$?

\PP Let $F$ be the standard Cauchy distribution with median zero and scale one, and let $G$ be a standard $t$-distribution with 2 degrees of freedom. Let $h(y) = y$. Does $\widehat{\theta}$ follow a CLT? Use Part (b) to justify your answer. Note $f(y) = \frac{1}{\pi(1+y^{2})}$ and $g(y) = (2+y^{2})^{-3/2}$.
	\end{parts}



\end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
