% ST410 A2001

\documentclass[twoside,a4paper,12pt]{article}
\usepackage{fancyhdr,amsmath,graphicx}

\title{*****}

%NEW COMMANDS

%\newcommand{\eqref}[1]{(\ref{#1})}
\newcommand{\mat}[1]{\mbox{\boldmath{$#1$}}}
\newcommand{\worth}[1]{{\tiny \hfill[#1 marks]}}
\newcommand{\worthone}{{\tiny \hfill[1 mark]}}
\newcommand{\pc}[1]{$\{#1_t\}$}
\newcommand{\pcq}[1]{\{#1_t\}}

\newcommand{\mX}{\mat{X}}
\newcommand{\mY}{\mat{Y}}
\newcommand{\mT}{\mat{T}}
\newcommand{\mgam}{\mat{\gamma}}
\newcommand{\mGam}{\mat{\Gamma}}

\setlength{\floatsep}{0cm} \setcounter{totalnumber}{4}
\renewcommand{\textfraction}{0}

\newcommand{\eps}{\varepsilon}
\newcommand{\var}{{\rm Var}}
\newcommand{\cov}{{\rm Cov}}
\newcommand{\corr}{{\rm Corr}}
\newcommand{\nid}{{\rm NID}}
\newcommand{\iid}{{\rm IID}}
\newcommand{\norm}{{\rm N}}
\newcommand{\wn}{{\rm WN}}
\newcommand{\mse}{{\rm MSE}}
\newcommand{\diag}{{\rm diag}}
\newcommand{\Exp}{{\rm Exp}}

%FORMATTING
\setlength{\topmargin}{-1cm} \setlength{\textheight}{23.5cm}
\setlength{\textwidth}{16cm} \setlength{\oddsidemargin}{-0.5cm}
\setlength{\evensidemargin}{-0.5cm} \setlength{\unitlength}{0.8cm}

%\renewcommand{\baselinestretch}{1.2}
\setlength{\parindent}{0pt} \setlength{\parskip}{11pt}

\newcommand{\thepagetotal}{3}
\newcommand{\yyear}{2004}
\newcommand{\course}{ST402}
\newcommand{\coursenamea}{Principles and Methods}
\newcommand{\coursenameb}{of Statistical Practice}
\newcommand{\timeallow}{2}
\newcommand{\numques}{four}
\newcommand{\numans}{all four}
\newcommand{\ppercent}{25\%}
%
\pagestyle{fancyplain}
%
\lfoot[\fancyplain{{\small\copyright LSE \yyear/\course}}
{{\small\copyright LSE \yyear/\course}}]
{\fancyplain{{\fontfamily{phv}\selectfont\small\copyright LSE
\yyear/\course}} {{\small\copyright LSE \yyear/\course}}}
%
\rfoot[\fancyplain{}{Page \thepage\, of \thepagetotal}]
{\fancyplain{{\fontfamily{phv}\selectfont\small Page \thepage\, of
\thepagetotal}} {{\small Page \thepage\, of \thepagetotal}}}
%
\cfoot[]{} \lhead[]{} \rhead[]{} \lhead[]{}
%
\rhead[]{} \setlength{\headheight}{0pt} \setlength{\headsep}{0pt}
%
\setlength{\topmargin}{0pt}
\renewcommand{\headrulewidth}{0pt}
%
%DOCUMENT
\begin{document}
%
\thispagestyle{plain} \fontfamily{phv}\selectfont \noindent
\includegraphics[width=0.468in,height=0.468in]{lse.eps}
\vspace{0.10in}\\ \vspace{0.2in}\\
\noindent {\bfseries \yyear\,Examination}\vspace{0.9in}\\
\noindent {\fontsize{36pt}{2pt}\selectfont\bfseries \course}\vspace{0.3in}\\
%
\noindent {\bfseries \LARGE \coursenamea\vspace{0.1in}\\
\hspace{0.25in}
\coursenameb \vspace{0.1in}} \vspace{2in}\\
%
\noindent {\bfseries Instructions to candidates}\vspace{0.2in}\\
\noindent Time allowed: \timeallow\ hours\vspace{0.2in}\\
This paper contains \numques\ questions. Answer \textbf{\numans}\
questions.
All questions will be given equal weight \ppercent. \vspace{0.20in}\\
The numbers in square brackets beside each question indicate
approximate marks for each part of that question. \\
\noindent
\begin{tabbing}
You are supplied with:\hspace{0.3in}\=\parbox[t]{3.6in}{Graph
Paper\\ Murdoch and Barnes Statistical Tables\vspace{0.2in}}\\
You may also use:\>\parbox[t]{3.6in}{Electronic calculator (as
prescribed by the examination regulations).}
\end{tabbing}
\newpage
\fontfamily{cmr}\selectfont

\begin{enumerate}
%---------------------------------------------------------------------
% QUESTION 1
%
\item Consider the function of two variables
$$
f_{X,Y}(x,y) = \left\{\begin{array}{ll} \lambda(\theta+\lambda)
e^{-\theta x - \lambda y}, & 0 < x < y < \infty, \\
0, & \text{otherwise}.
\end{array} \right.
$$

\begin{enumerate}
\item Show that $f_{X,Y}$ is a density function. \worth{3}
%
\item Show that $X$ is exponentially distributed with mean
$1/(\theta+\lambda)$, that is, show that $X \sim
\Exp(\theta+\lambda)$. Write down the conditional density,
$f_{Y|X}(y|x)$. Derive the conditional expectation, $E(Y|X)$.
Hence or otherwise, find $\cov(X,Y)$. \newline ${}_{}$ \worth{8}
%
\item  Derive the conditional moment generating function,
$M_{Y|X}(t|X)$. Hence or otherwise, find the moment generating
function, $M_Y(t)$, and show that $Y$ can be expressed as a sum of
two independent random variables. [{\sl Note}: for this question,
you may use, without proof, the fact that if $V \sim \Exp(\phi)$
then $M_V(t) = \frac{\phi}{\phi-t}$ provided $|t|<\phi$] \worth{8}
%
\item Let $Z=X+Y$. Give an expression for the $r^{\rm th}$ cumulant of $Z$. \worth{6}
\end{enumerate}



\vspace{0.5cm}
%---------------------------------------------------------------------
% QUESTION 2
%
\item Consider the transformation to random variables $X$ and $Y$
given by $(U,V) = T(X,Y)$. The joint density of $U$ and $V$ is
\begin{equation} \label{chngvar:eq}
f_{U,V}(u,v) = f_{X,Y}(x(u,v), y(u,v)) |J(u,v)|
\end{equation}
where $u$ and $v$ are in the range of $T$ and $J(u,v)$ is the
Jacobian of the inverse transformation.
\begin{enumerate}
%(a)
\item Let $U = X/Y$ and $V=Y$. Using equation \eqref{chngvar:eq} show that
$$
f_{U,V}(u,v) = f_{X,Y}(uv,v)|v|
$$
where $u$ and $v$ are in the range of the transformation.
\worth{4}
%(b)
\item Let $X$ and $Y$ be independent standard normal random
variables.
\begin{enumerate}
\item
Show that if $U = X/Y$ then $U$ has a Cauchy distribution, that
is,
$$
f_U(u) = \frac{1}{\pi(1+u^2)}, \quad \text{for} \ \ -\infty < u <
\infty.
$$
What can you conclude about the mean of the ratio of two standard
\newline normals? \worth{10}
\item Find the distribution function of $|Y|$
and hence derive the density of $|Y|$. \newline ${}_{}$\worth{5}
\item Derive the density of $X/|Y|$. \worth{6}
\end{enumerate}
\end{enumerate}

\newpage

%---------------------------------------------------------------------
% QUESTION 3
%
\item
\begin{enumerate}
\item
Let $(X_1, \cdots, X_n)$ be a random sample from a Poisson
distribution with mean $\lambda >0$. Let $\theta = \lambda^2$.
\begin{enumerate}
\item Find the maximum likelihood estimator $\hat \theta$ for $\theta$.
\worth{5}
\item Find the bias of the estimator $\hat \theta$. \worth{4}
\end{enumerate}

\item
Let $ (X_1, \cdots, X_n)$ be a random sample from a distribution
with density function $f(x, \theta)$, $T  = T(X_1, \cdots, X_n)$
be a statistic, and $\tilde \theta = \tilde \theta (X_1, \cdots,
X_n)$ be an estimator for $\theta$.
\begin{enumerate}
\item
Give the condition under which $T$  is a sufficient statistic for
$\theta$. \worth{4}
\item
Show that $E \{ (\tilde \theta - \theta)^2 \} = \mbox{Var}(\tilde
\theta) + [ E(\tilde \theta) - \theta]^2$. \worth{3}
\item
Let $T  $ be a sufficient statistic for $\theta$. Explain why
$\hat \theta \equiv E ( \tilde \theta | T) $ is a statistic. Show
that $\hat \theta$ is a better estimator for $\theta$ than $\tilde
\theta$ in the sense that
\[
E \{ (\hat \theta - \theta)^2 \} \le E \{ (\tilde \theta -
\theta)^2 \}.
\]
[{\sl Note}: you may use, without  proof, the equality $E(X) = E
\{ E(X|T) \}$ for any random variables $X$ and $T$.] \worth{9}
\end{enumerate}
\end{enumerate}
%---------------------------------------------------------------------
% QUESTION 4
%
\item
Let $X_1, \cdots, X_n$ be a random sample from an exponential
distribution with the density function $f(x; \theta) = \theta
e^{-\theta x} I(x>0)$, where $\theta >0$ is an unknown parameter.
\begin{enumerate}
\item Find the maximum likelihood estimator for $\theta$. \worth{4}
\item
Show that the moment generating function for $X_1$ is $ M_{X_1}(t)
=(1- t/\theta)^{-1}$ for $t \in (0,\; \theta)$. \worth{3}
\item Find the moment generating function of $Y\equiv  X_1 + \cdots + X_n$,
and show that  $T \equiv 2 \theta Y$ is a pivotal function.
\worth{4}
\item
Find an exact 95\% confidence interval for $\theta$ when $n=10$
and $Y = 19.4$. Will you reject the null hypothesis $H_0: \theta =
2$ at the significance level $\alpha =0.05$. Give reasons. [{\sl
Note}: for this part of the question, you may use, without proof,
the fact that the $\chi^2$ distribution with $k$ degrees of
freedom has moment generating function $(1 - 2t)^{-k/2}$. ]
\worth{7}

\item Suppose now the last
$n-m$ (where $m<n$) observations are censored in the sense that,
for $j=m+1, \cdots, n$, we only known $X_j > u_j$ for some $u_j>0$
while $X_j$ itself is unknown. Outline an EM algorithm for
estimating $\theta$. \worth{7}
\end{enumerate}
\end{enumerate}


\end{document}
