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$%
\begin{tabular}
[c]{p{1.8in}p{2.15in}p{1.8in}}%
\multicolumn{1}{l}{{\Large \textbf{Part A: Statistics}}} & \quad &
\multicolumn{1}{r}{{\Large \textbf{Problem Sheet 1}}}\\\hline
\end{tabular}
\ $

\vspace{0.25in}

\begin{enumerate}
\item[\textbf{1.}] Let $X_{1},\ldots,X_{n}$ be a random sample from a uniform
distribution with p.d.f.%
\[
f\left(  x\right)  =\left\{
\begin{array}
[c]{ll}%
1, & 0<x<1,\\
0, & \text{otherwise.}%
\end{array}
\right.
\]
Show that, if $X_{\left(  r\right)  }$ is the $r^{th}$ order statistic, then%
\[
E\left(  X_{\left(  r\right)  }\right)  =\frac{r}{n+1},\quad V\left(
X_{\left(  r\right)  }\right)  =\frac{r}{\left(  n+1\right)  \left(
n+2\right)  }\left(  1-\frac{r}{n+1}\right)  .
\]
Define the median of the random sample, distinguishing between the two cases
$n$ odd and $n$ even. Show that the median has expected value $\frac{1}{2}$ if
the random sample is drawn from a uniform distribution on $\left(  0,1\right)
$. Find its variance in the particular case when $n$ is odd. What is the
expected value of the median if the random sample is drawn from a uniform
distribution on $\left(  a,b\right)  $?

\item[\textbf{2.}] Let $X$ be a continuous random variable with cumulative
distribution function $F$ which is strictly increasing. If $Y=F\left(
X\right)  $, show that $Y$ is uniformly distributed on the interval $\left(
0,1\right)  $.\medskip\newline The \emph{Weibull distribution} with index
$\alpha$ has c.d.f.
\[
F(x)=\left\{
\begin{array}
[c]{ll}%
0, & x<0,\\
1-\exp\left(  -\left(  \left.  x\right/  \lambda\right)  ^{\,\alpha}\right)
, & x\geq0,
\end{array}
\right.  \quad\lambda,\alpha>0.
\]
It is typically used in industrial reliability studies in situations where
failure of a system comprising many similar components occurs when the weakest
component fails; it is also used in modelling survival.\medskip\medskip
\newline Explain why a probability plot for the Weibull distribution may be
based upon plotting the logarithm of the $r^{th}$ order statistic against
$\log\left[  -\log\left(  1-\dfrac{r}{n+1}\right)  \right]  $ and give the
slope and intercept of such a plot.\ \newline

\item[\textbf{3.}] Let $X_{1},X_{2},\ldots,X_{n}$ be independent identically
distributed random variables from a distribution with p.d.f.%
\[
f\left(  x,\theta_{1},\theta_{2}\right)  =\frac{1}{\theta_{2}}\exp\left(
-\frac{\left(  x-\theta_{1}\right)  }{\theta_{2}}\right)  ,\quad x>\theta
_{1},\quad\left(  \theta_{1}\theta_{2}\right)  \in\mathbb{R}\times
\mathbb{R}^{+}.
\]
Find maximum likelihood estimators of $\theta_{1}$ and $\theta_{2}$.

\item[\textbf{4.}] Find the likelihood for $\theta$ based on $X$, a binomially
distributed random variable with probability function
\[
p_{X}(x)={\binom{m}{x}}\theta^{x}(1-\theta)^{m-x},\quad x=0,1,\ldots m.
\]
Show that the maximum likelihood estimate of $\theta$ is $\widehat{\theta
}=x\left/  m\right.  $ and that the observed and expected information both
have maximum likelihood estimates equal to $m\left/  \left(  \widehat{\theta
}(1-\widehat{\theta})\right)  \right.  $.

\item[\textbf{5.}] Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample from a
gamma distribution with probability density function
\[
f_{X}(x;\lambda,\alpha)=\Gamma(\alpha)^{-1}\lambda^{\alpha\,}x^{\alpha
-1}e^{-\lambda x},\quad x>0,\quad\lambda,\alpha>0,
\]
so that $E(X)=\alpha\left/  \lambda\right.  $. Show that the expected
information matrix for $\lambda$ and $\alpha$ is
\[
n\left(
\begin{array}
[c]{ll}%
\alpha\left/  \lambda^{2}\right.  & -1\left/  \lambda\right. \\
-1\left/  \lambda\right.  & \dfrac{\partial^{2}\log\Gamma(\alpha)}%
{\partial\alpha^{2}}%
\end{array}
\right)  .
\]


\item[\textbf{6.}] (Harder question.) According to genetic linkage theory,
observed frequencies of four phenotypes resulting from crossing tomato plants
are in the ratio $\frac{9}{16}+\theta:\frac{3}{16}-\theta:\frac{3}{16}%
-\theta:\frac{1}{16}+\theta$. In 1931, J.W. MacArthur reported the following
frequencies:
\[%
\begin{tabular}
[c]{ll}\hline
& \textbf{Observed}\\
\textbf{Phenotype} & \textbf{frequency}\\\hline
Tall, cut-leaf & \multicolumn{1}{r}{926}\\
Tall, potato-leaf & \multicolumn{1}{r}{288}\\
Dwarf, cut-leaf & \multicolumn{1}{r}{293}\\
Dwarf, potato-leaf & \multicolumn{1}{r}{104}\\
& \multicolumn{1}{r}{}\\
\textbf{Total} & \multicolumn{1}{r}{1611}\\\hline
\end{tabular}
\
\]
Write down the likelihood of $\theta$ given these observations. Find the
maximum likelihood estimate of $\theta$, use it to calculate expected
frequencies for the four phenotypes and compare them with the observed
frequencies. Does genetic linkage theory look plausible?\bigskip
\end{enumerate}

\begin{center}
\textbf{Optional Computer Exercises\medskip}
\end{center}

\begin{enumerate}
\item[\textbf{1.}] The data set \emph{memories.txt}, which you will find on
the website, contains recall times (in seconds) of pleasant and unpleasant
memories [Dunn, G. and Master, D. (1982), \emph{Psychological Medicine},
\textbf{12}]. Download the data set into one of your directories (say,
\texttt{c:%
%TCIMACRO{\TEXTsymbol{\backslash}}%
%BeginExpansion
$\backslash$%
%EndExpansion
Data}), read it into R and get it ready for use with the commands\medskip
\newline\texttt{%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
memories
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- read.table("c:/Data/memories.txt", header = TRUE)\newline%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
attach(memories)\medskip\newline}If you enter \texttt{memories} you will be
able to look at the data.\medskip\newline Try looking at boxplots of these
data with the command \texttt{boxplot(memories)}. What conclusions can you
draw?\medskip\newline The apparent skewness revealed by the boxplots along
with the fact that these are measurements of waiting times might lead you to
suspect that the data are exponentially distributed. Order the vector
\texttt{Pleasant} and assign it to \texttt{y}, construct a vector \texttt{x}
and plot \texttt{y} against \texttt{x} with the sequence of commands
below.\medskip\newline\texttt{y
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- sort(Pleasant)\newline x
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- -log(1 - c(1:20)/21)\newline plot(x,y)\medskip\newline}Explain why this plot
allows you to assess whether the pleasant memories are exponentially
distributed. What do you conclude? Repeat this procedure for the unpleasant
memories and state your conclusions about the adequacy of an exponential model.

\item[\textbf{2.}] Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample from a
truncated Poisson distribution with probability mass function
\[
p(x;\theta)=\frac{\theta^{x}e^{-\theta}}{x!(1-e^{-\theta})},\quad
x=1,2,\ldots,\quad\theta>0.
\]
Find the likelihood and obtain the score function and the observed
information.\medskip\ \newline On a spring afternoon in Portland, Oregon, data
on the sizes of different groups observed in public places were collected by
Coleman and James:
\[%
\begin{tabular}
[c]{rcccccc}%
Group size: & 1 & 2 & 3 & 4 & 5 & 6\\
Frequency: & 1486 & 694 & 195 & 37 & 10 & 1
\end{tabular}
\
\]
A possible model for these data is a truncated Poisson distribution. Show that
the log-likelihood is maximised by the value of $\theta$ which maximises
\[
3663\log\theta-2423\theta-2423\log(1-e^{-\theta}).
\]
Using the sample mean as a starting value, apply the Newton-Raphson method to
obtain the maximum likelihood estimate of $\theta$ to 3 places of decimals.
Obtain an approximate 95\% confidence interval for $\theta$.
\end{enumerate}


\end{document}