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\begin{document}
$%
\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 2}}}\\\hline
\end{tabular}
\ $

\vspace{0.25in}

\begin{enumerate}
\item[\textbf{1.}] Independent random variables $X_{1},X_{2},\ldots,X_{n}$
have common probability density function $f\left(  x;\theta\right)  $
depending on the unknown parameter $\theta$. Define the terms \emph{likelihood
function} for $\theta$, \emph{maximum likelihood estimator} for $\theta$,
\emph{Fisher's information} for $\theta$.\medskip\newline What is the
connection between Fisher's information and the asymptotic distribution of the
maximum likelihood estimator?\medskip\newline Suppose $X_{1},X_{2}%
,\ldots,X_{n}$ are independent, each with density%
\[
f\left(  x;\theta\right)  =\frac{1}{\pi\left[  1+\left(  x-\theta\right)
^{2}\right]  },\quad x\in\mathbb{R},\ \theta\in\mathbb{R}.
\]
Derive an equation which must be satisfied by the maximum likelihood
estimator, $\widehat{\theta}$.\medskip\newline Prove that Fisher's information
for $\theta$ is $\frac{1}{2}n$. Explain how to calculate an approximate 95\%
confidence interval for $\theta$.

\item[\textbf{2.}] Let $\overline{X}$ be the mean of a random sample
$X_{1},X_{2},\ldots,X_{n}$ from a distribution with mean $\mu$ and variance
$V(\mu)$. Show that the variance of $h(\overline{X})$ is approximately
constant, where
\[
h(\mu)=\int^{\,\mu}V(u)^{-\left.  1\right/  2}du.
\]
Show that

\begin{enumerate}
\item[(i)] $V(\mu)=\mu$ for a Poisson distribution;

\item[(ii)] $V(\mu)=\left.  \mu(1-\mu)\right/  m$ for the proportion $\left.
X\right/  m$ of successes when $X\sim B(m,\mu)$;

\item[(iii)] $V(\mu)=\left.  \mu^{2}\right/  \nu$ when $X$ has a gamma
distribution with mean $\mu$ and index $\nu$.\medskip\ 
\end{enumerate}

Find variance stabilising transformations for these distributions.\medskip\ 

\item[\textbf{3.}] The following data are time intervals in days between
earthquakes which either registered magnitudes greater than 7.5 on the Richter
scale or produced over 1,000 fatalities. Recording starts on 16 December, 1902
and ends on 4 March, 1977, a total period of 27,107 days. There were 63
earthquakes in all, and therefore 62 recorded time intervals.%
\[%
\begin{tabular}
[c]{rrrrrr}\hline
840 & 294 & 454 & 667 & 556 & 304\\
1901 & 335 & 30 & 129 & 209 & 83\\
40 & 203 & 735 & 365 & 82 & 887\\
139 & 638 & 121 & 280 & 736 & 319\\
246 & 44 & 76 & 46 & 194 & 375\\
157 & 562 & 36 & 40 & 99 & 832\\
695 & 1354 & 384 & 9 & 599 & 263\\
1336 & 436 & 38 & 92 & 220 & 460\\
780 & 937 & 150 & 434 & 584 & 567\\
1617 & 33 & 710 & 402 & 759 & 328\\
145 & 721 &  &  &  & \\\hline
\end{tabular}
\]
Assuming the data to be from a random sample $X_{1},\ldots,X_{n}$ drawn from
an exponential distribution with parameter $\lambda$, obtain the maximum
likelihood estimator $\widehat{\lambda}$ of $\lambda$ and calculate the
maximum likelihood estimate.\medskip\newline Given that the moment generating
function of a gamma distribution with parameters $\left(  n,\lambda\right)  $
is%
\[
M_{n}\left(  t\right)  =\left(  \frac{\lambda}{\lambda-t}\right)  ^{n},
\]
show that $Y=\sum_{i=1}^{n}X_{i}$ has a gamma distribution. Show that%
\[
\left(  \frac{a}{n\overline{x}},\frac{b}{n\overline{x}}\right)  ,
\]
is an exact 95\% central confidence interval for $\lambda,$ where $%
%TCIMACRO{\dint _{0}^{a}}%
%BeginExpansion
{\displaystyle\int_{0}^{a}}
%EndExpansion
\dfrac{y^{n-1}e^{-y}dy}{\Gamma\left(  n\right)  }=%
%TCIMACRO{\dint _{b}^{\infty}}%
%BeginExpansion
{\displaystyle\int_{b}^{\infty}}
%EndExpansion
\dfrac{y^{n-1}e^{-y}dy}{\Gamma\left(  n\right)  }=0.025$.\medskip\newline
Obtain Fisher's information for $\lambda$ and use it to show that%
\[
\left(  0.0018,0.0030\right)
\]
is an approximate 95\% confidence interval for $\lambda$.

\item[\textbf{4.}] If $S^{2}$ is the variance of a random sample which is
drawn from a normal distribution $N\left(  \mu,\sigma^{2}\right)  $, then
$cS^{2}$ has a standard distribution for a particular value of the constant
$c$. Write down the distribution and specify the constant $c$.\medskip\newline
How many pairs of positive constants $a,b$ exist such that $P\left(
a<cS^{2}<b\right)  =0.95?$ For any such pair, deduce that the random interval%
\[
\left(  \frac{\left(  n-1\right)  S^{2}}{b},\frac{\left(  n-1\right)  S^{2}%
}{a}\right)
\]
will contain the variance $\sigma^{2}$ for approximately 95\% of a large
number of such random samples.

\item[\textbf{5.}] One of the data sets you have seen in the lecture notes
concerns the silver content of Byzantine coins; it appears in Table 2.3. The
silver content for the first and fourth coinages of King Manuel I, Comnemus
($1143-1180$) is reproduced in the table below.%
\[%
\begin{tabular}
[c]{cc}%
\multicolumn{2}{c}{Silver content of coins}\\\hline
First coinage & Fourth coinage\\\hline
5.9 & 5.3\\
6.8 & 5.6\\
6.4 & 5.5\\
7.0 & 5.1\\
6.6 & 6.2\\
7.7 & 5.8\\
7.2 & 5.8\\
6.9 & \\
6.2 & \\\hline
\end{tabular}
\]
The respective sample means and variances from these data are $\overline
{x}_{1}=6.744$, $S_{1}^{2}=0.295$, $\overline{x}_{4}=5.614$, $S_{4}^{2}=0.131$.

\begin{enumerate}
\item[(i)] Assuming the data to be normally distributed, obtain a suitable
test statistic for testing the null hypothesis that the mean silver content of
the two coinages is the same. What further assumptions are required for the
test to be valid?

\item[(ii)] Carry out the test with the alternative hypothesis that the mean
silver content is different for the two coinages. How would you modify the
test if the alternative hypothesis were to be that there is \textit{less}
silver in the fourth coinage?
\end{enumerate}

\item[6.] (Extension question.) In the theory of random sampling from a
distribution involving unknown parameters, a \emph{pivot quantity} is a
function of a random sample and of the unknown parameters that has a known
standard distribution. Confidence intervals for functions of the unknown
parameters may sometimes be calculated by using appropriate pivot quantities.
Given independent random samples of $n_{i}$ $\left(  n_{i}\geq
2;\ i=1,2\right)  $ observations from two normal distributions with means
$\mu_{i}$ and variances $\sigma_{i}^{2}$, specify which pivot quantities you
would use to obtain confidence intervals for each of the following:

\begin{enumerate}
\item[(i)] $\mu_{1}$ if $\sigma_{1}^{2}$ is known;

\item[(ii)] $\sigma_{1}^{2}$;

\item[(iii)] $\mu_{1}$;

\item[(iv)] $\sigma_{1}^{2}$ if $\sigma_{1}^{2}=\sigma_{2}^{2}$;

\item[(v)] $\mu_{1}-\mu_{2}$ if $\sigma_{1}^{2}=\sigma_{2}^{2}$.
\end{enumerate}

\noindent Name the distribution of the pivot quantity in each of the five
cases or, alternatively, write down its probability density function. Now use
an appropriate pivot quantity to obtain a 99\% confidence interval for the
difference between the mean silver contents, $\mu_{1}-\mu_{4\text{,}}$ in
question 5. How could you use this to carry out the test in part (ii) of the
previous question?



\textbf{Optional Computer Exercises\medskip}

\item[\textbf{1.}] Referring back to question 3, you will find a file
\emph{quakes.txt} containing the earthquake data on the website. If you have
downloaded it into a directory called, say, c:%
%TCIMACRO{\TEXTsymbol{\backslash}}%
%BeginExpansion
$\backslash$%
%EndExpansion
data, you can load it into R with\medskip\newline%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
\texttt{quakes
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- read.table("c:/data/quakes.txt", header = TRUE)\medskip\newline}Use a
probability plot to check the exponential model; are you satisfied that it is
plausible? Quantiles for lower and upper tail probability $p$ for a gamma
distribution with parameters $\left(  n,1\right)  $ can be calculated using
the functions \texttt{qgamma(}$p$\texttt{, }$n$\texttt{, lower.tail=TRUE)} and
\texttt{qgamma(}$p$\texttt{, }$n$\texttt{, lower.tail=FALSE).} Calculate the
exact confidence interval suggested in question 3, and compare your result
with that for the approximate confidence intrval.

\item[\textbf{2.}] The data file \emph{engreek.txt} contains the scores on
each of 32 English sentences awarded by native English speaking teachers and
by Greek teachers as a result of errors made by Greek-Cypriot learners of
English. The scores are assessments made by the teachers of the quantity of
errors made by the students. The data are paired for each of 32 sentences, so
differences are to be tested for a mean of zero.\medskip\newline If you have
downloaded the data set into a directory called, say, c:%
%TCIMACRO{\TEXTsymbol{\backslash}}%
%BeginExpansion
$\backslash$%
%EndExpansion
data, you can load it into R and obtain a vector of differences with\medskip
\newline\texttt{%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
eg
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- read.table("c:/data/engreek.txt", header = TRUE)\newline%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
attach(eg)\newline%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
diff
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- Greek - English}\medskip\newline Before attempting a $t$-test, you should
check for normality. Try\medskip\newline\texttt{%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
qqnorm(diff)\medskip\newline}If you are not entirely happy about the
normality, you could try a few transformations. The commands below show how
you would try a power $\frac{3}{2}$ transformation, preserving the sign of the
difference.\medskip\newline\texttt{%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
sign
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- diff/abs(diff)\newline%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
tdiff
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
- sign*(abs(diff)\symbol{94}(3/2))\newline%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
qqnorm(tdiff)}\medskip\newline When you are satisfied that your data, whether
transformed or otherwise, are plausibly normal, perform a $t$-test of the null
hypothesis that the difference is zero (to see how to do this in R, try
entering \texttt{?t.test}). What should your alternative hypothesis be? What
conclusions can you draw?
\end{enumerate}


\end{document}