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\newcommand{\yyear}{2003}
\newcommand{\course}{ST402}
\newcommand{\coursenamea}{Principles and Methods}
\newcommand{\coursenameb}{of Statistical Practice}
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%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

\begin{enumerate}
% (a)
\item The density of the random variable $X|\Theta = \theta$ is given
by
$$ f_{X|\Theta}(x|\theta) = \left\{\begin{array}{ll}
1 - \theta, & -1 < x < 0, \\
\theta,     & 0 \leq x < 1, \\
0,          & \text{otherwise}.
\end{array} \right.
$$
\begin{enumerate}
% i.
\item Verify that, if $0 \leq \theta \leq 1$, then $f_{X|\Theta}$
is a density function. \worth{5}
% ii.
\item Give an expression for the conditional expectation,
$E(X|\Theta)$. \worth{5}
\end{enumerate}

% (b)
\item Now suppose that the density of $\Theta$ is given by
\begin{equation} \label{thetadens:eq}
f_{\Theta}(\theta) = \left\{\begin{array}{ll}
 \frac{(a+b+1)!}{a! b!} \ \theta^a (1-\theta)^b, & 0 \leq \theta \leq 1, \\
0, & \text{otherwise}.
\end{array}\right.
\end{equation}
where $a$ and $b$ are positive integers.
\begin{enumerate}
% i.
\item Using the fact that $f_{\Theta}$ defined by \eqref{thetadens:eq} is
a density function, write down an expression for $\int_0^1
\theta^a (1-\theta)^b d\theta$. \worth{2}
% ii.
\item Evaluate $E(\Theta)$. Hence or otherwise, evaluate $E(X)$.
\worth{5}
% iii.
\item Evaluate $\cov(X, \Theta)$. \worth{8}
\end{enumerate}

\end{enumerate}

\vspace{0.5cm}
%---------------------------------------------------------------------
% QUESTION 2
%
\item
\begin{enumerate}
%(a)
\item Derive the cumulant generating function of a standard normal
random variable. \newline ${}_{}$ \worth{5}

%(b)
\item Consider a sequence of independent identically distributed
random variables $Y_1,\ldots,Y_n$ with common mean $E(Y) = \mu$
and common variance $\var(Y) = \sigma^2$. Let $S_n = \sum_{i=1}^n
Y_i$ and $Z_n = (S_n - n \mu)/ \sqrt{n \sigma^2}$.

\begin{enumerate}
% i.
\item Show that the cumulant generating function of $S_n$ is given
by
$$
K_{S_n}(t) = n K_Y(t)
$$
where $K_Y$ is the cumulant generating function of $Y$. \worth{4}
% ii.
\item Derive an expression for the cumulant generating function
of $Z_n$, that is $K_{Z_n}(t)$, in terms of the cumulant
generating function of $Y$. \worth{5}
% iii.
\item Using the power series expansion, show that
$K_{Z_n}(t) \longrightarrow t^2/2$ as $n \longrightarrow \infty$.
What does this result imply about the sample mean, $\bar{Y} =
\frac{1}{n} S_n$? \worth{5}
\end{enumerate}

% (c)
\item Now suppose that $Y_1,\ldots,Y_n$ are normal independent
identically distributed random variables with mean 0 and variance
$\sigma^2$, that is, $Y \sim N(0,\sigma^2)$. Write down the moment
generating function of $Y$. Hence or otherwise, find the expected
value and the variance of the estimator of $\sigma^2$ given by
$$
\widehat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n Y_i^2.
$$
\worth{6}
\end{enumerate}

\newpage
%--------------------------------------------------------------------
% QUESTION 3
%
\item \label{q3}
\begin{enumerate}
\item
Let $X_1, \cdots, X_n$ be a random sample from a population with
the density function $f(x, \theta)$, where $\theta$ is an unknown
parameter. Define the likelihood function of $\theta$, and further
define the maximum likelihood estimator of $\theta$. Write down
the asymptotic distribution of the maximum likelihood estimator,
assuming the required regularity conditions. Argue why the maximum
likelihood estimator may be viewed as an approximate minimum
variance unbiased estimator when the sample size $n$ is large.
\worth{8}

\item
Let $X_1, \cdots, X_n$ be a random sample from the density
function
\[
f(x) = \left\{
\begin{array}{ll}
\theta^2 x e^{-\theta x} \quad \quad & x>0,\\
0 &  x\le 0,
\end{array} \right.
\]
where $\theta>0$ is an unknown parameter.
\begin{enumerate}
\item
Find the maximum likelihood estimator for $\theta$. \worth{4}
\item
Suppose now that only the last $n-m$ $(m<n)$ observations of the
sample are known explicitly, while for the first $m$ only their
sum, $Z = X_1 + \cdots + X_m$, is known. Determine the maximum
likelihood estimator of $\theta$ based on the observations $Z,
X_{m+1}, \cdots, X_n$. \worth{5}
\item
Suppose now that the first $m$ observations are censored in the
sense that we only known $X_j \in [u_j, v_j]$ for some given $v_j
> u_j >0$ while $X_j $ itself is unknown, $j=1, \cdots, m$.
Outline an EM algorithm for estimating the parameter $\theta$.
\worth{8}
\end{enumerate}

\end{enumerate}

\vspace{0.5cm}
%--------------------------------------------------------------------
% QUESTION 4
%
\item \label{q4}
\begin{enumerate}
\item
Let $X_1, \cdots, X_n$ be a random sample from a Poisson
distribution with unknown mean $\mu >0$.
\begin{enumerate}
\item Fine the maximum likelihood estimator $\widehat \theta$ for
$\theta \equiv e^{-\mu}$. \worth{5}
\item
Show that $\widehat \theta$ is a biased estimator. \worth{5}
\item
Outline a bootstrap procedure for estimating the bias of $\widehat
\theta$. \worth{5}
\end{enumerate}
\item
Let $X_1, \cdots, X_n$ be a random sample from the normal
distribution $N(0, \nu)$.
\begin{enumerate}
\item
Find the most powerful test (MPT) for testing the null hypothesis
$H_0:\, \nu =1$ against the alternative hypothesis $H_1:\, \nu
=4$. \worth{5}

\item
Identify, with reasons, the composite hypotheses for which the
above MPT is in fact the {\sl uniformly} most powerful test.
\worth{5}

\end{enumerate}
\end{enumerate}

\end{enumerate}
\end{document}
