\documentclass[11pt,a4paper]{article}%
\usepackage{amsfonts}
\usepackage{amsmath}%
\setcounter{MaxMatrixCols}{30}%
\usepackage{amssymb}%
\usepackage{graphicx}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.50.0.2890}
%TCIDATA{LastRevised=Monday, February 16, 2009 17:22:19}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%TCIDATA{Language=American English}
%BeginMSIPreambleData
\providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
%EndMSIPreambleData
\setlength{\oddsidemargin}{0mm}
\setlength{\evensidemargin}{0mm}
\setlength{\topmargin}{0mm}
\setlength{\headheight}{0mm}
\setlength{\headsep}{0mm}
\setlength{\textheight}{227mm}
\setlength{\textwidth}{158mm}
\setlength{\topskip}{\baselineskip}
\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 3}}}\\\hline
\end{tabular}
\ $

\vspace{0.25in}

\begin{enumerate}


\item[\textbf{1.}] Consider a random sample of size $n$ from a distribution
with probability density function $f\left(  x;\theta\right)  $, and consider a
test of the null hypothesis $H_{0}:\theta=\theta_{0}$ against the alternative
$H_{1}:\theta=\theta_{1}$. Define the terms \emph{type I error}, and
\emph{type II error}. State and prove the Neyman-Pearson lemma.\medskip
\ \newline Let $X_{1},\ldots,X_{n}$ be independent $N\left(  \theta,\sigma
_{0}^{2}\right)  $ random variables, where $\sigma_{0}^{2}$ is known. Find the
best critical region for a test of size $\alpha$ of $H_{0}:\theta=\theta_{0}$
against $H_{1}:\theta=\theta_{1}$, where $\theta_{1}>\theta_{0}$%
.\medskip\newline Show that the power function $Q\left(  \theta\right)  $ of
this test is given by%
\[
Q\left(  \theta\right)  =1-\Phi\left(  \frac{\sqrt{n}}{\sigma_{0}}\left(
\theta_{0}-\theta\right)  +z_{\alpha}\right)
\]
where $\Phi$ is the cumulative distribution function of the standard normal
distribution and $\Phi\left(  z_{\alpha}\right)  =1-\alpha$.\medskip\newline
If $\theta_{0}=0$, $\theta_{1}=0.5$ and $\sigma_{0}=1$, how large must $n$ be
if $\alpha=0.05$ and the power at $\theta_{1}$ is to be 0.975?

\item[\textbf{2.}] A telephone receptionist for a large partnership of
financial advisers is responsible for determining the precise nature of each
incoming enquiry and connecting the client with an appropriate adviser. The
number of inappropriate connections on any given day may be modelled by a
random variable $X$ which has a Poisson distribution with mean $\mu.$ If $Z$
is the number of inappropriate connections made over a period of $n$ days,
determine the distribution of $Z$ and find its expected value.\medskip
\ \newline Uhura, who has been such a receptionist for many years, has been
found to have a mean rate of $\mu_{U}=0.47$ inappropriate connections per day.
For several months she has been training Spock, a new receptionist, with
corresponding mean rate $\mu_{S}.$ At a meeting of senior partners, it was
conjectured that Spock was already as proficient as Uhura; accordingly they
resolved to keep a daily record of the number of inappropriate connections
made by him over his next 10 working days. Find a critical region of size 5\%
for a test of the hypothesis that Spock is as proficient as Uhura versus the
alternative that he is less proficient.\medskip\ \newline For what values of
$\mu_{S}$ does the probability of type II error fall below 10\%?\medskip
\ \newline\{Note that, if $\varphi_{\mu}(k)=\sum_{x=0}^{k}\mu^{x}e^{-\mu
}\left/  x!\right.  ,$ then $\varphi_{4.7}(8)=0.95,\ \varphi_{13}(8)=0.1$.\}

\item[\textbf{3.}] Explain how the likelihood ratio test can be used to carry
out a test of a hypothesis $H_{0}$ against an alternative $H_{1}$ (i) when
$H_{0}$ and $H_{1}$ are simple hypotheses, (ii) when both are composite, with
$H_{0}\subset H_{1}$. In case (ii) for large samples, which statistic is used
to calculate the $p$-value of the test? What is its asymptotic
distribution?\medskip\newline In studying the sex ratio in a population using
a sample of size $n$, it is usually assumed that each child is male with
probability $p$, independently of all other children. In such a study Renkonen
(1956) observed 19,711 male births out of a total of 38,562 births in American
families with two children each. Use the likelihood ratio statistic to test
the hypothesis $H_{0}:p=\frac{1}{2}$ against a suitable alternative, which you
should specify.\medskip\newline Renkonen also found 17,703 males out of 35,042
similar births in Finland. Use the generalised likelihood ratio test to test
the hypothesis that $p$ has the same value in each country versus a suitable alternative.

\item[\textbf{4.}] A random variable $X$ has a distribution given by
\[
P\left(  X=i\right)  =\pi_{i}\quad\left(  i=1,2,\ldots,p\right)  ,
\]
where $\sum_{i=1}^{p}\pi_{i}=1$. In a sample of size $n$ from a population
with distribution $X$, the frequency of outcome $i$ is $n_{i}$, where
$n_{i}\geq0$ and $\sum_{i=1}^{p}n_{i}=n$. Find the maximum likelihood
estimators of $\pi_{i}$.\smallskip\ \newline The seeds of the Indian creeper
plant \textit{Pharbitis nil} are valued for their purgative properties. The
leaves can be variegated or unvariegated and, at the same time, faded or
unfaded. In an experiment reported by Bailey (1961), plants were crossed in
such a way that their offspring would produce variegated leaves with
probability $\frac{1}{4}$ and faded leaves with probability $\frac{1}{4}$.
What probability model would you propose under the assumption that the
properties of variegated appearance and faded appearance are
independent?\smallskip\ \newline Of 290 plants which were observed, 31 had
variegated faded leaves, 37 had variegated unfaded leaves, 35 had unvariegated
faded leaves and 187 had unvariegated unfaded leaves. Use a $\chi^{2}$ test of
goodness-of-fit to show that the data offer strong evidence that your
independence model is inappropriate.\smallskip\ \newline A genetic theory
which allows for an effect called \textit{genetic linkage} assumes a
probability model for the above observations with respective probabilities
\[
\frac{1}{16}+\theta,\frac{3}{16}-\theta,\frac{3}{16}-\theta,\frac{9}%
{16}+\theta.
\]
Find $\widehat{\theta}$, the maximum likelihood estimate of $\theta
$.\smallskip\ \newline Let $H_{0}$ be the null hypothesis that the genetic
linkage model is appropriate with general alternative $H_{1}$. If $L_{0}$ is
the supremum of the likelihood under $H_{0}$ and if $L_{1}$ is the supremum of
the likelihood under $H_{1}$, show that
\[
\Lambda=2\sum_{i=1}^{4}n_{i}\log\left(  \frac{n_{i}}{n\pi_{i}\left(
\widehat{\theta}\right)  }\right)  ,
\]
where $\Lambda=-2\left(  \log L_{0}-\log L_{1}\right)  $. Write down the
approximate distribution of $\Lambda$.\smallskip\ \newline What can you infer
about the plausibility of the genetic linkage model?\medskip\newline[You may
remember that you met the genetic linkage model in Problem Sheet 1, Question
6. You might also like to try the test on those data.]

\item[\textbf{5.}] The ordered pairs of random variables $\left(  X_{k}%
,Y_{k}\right)  ,\ k=1,2,\ldots,N,$ are independent and
\[
P\left(  \left(  X_{k},Y_{k}\right)  =\left(  i,j\right)  \right)
=\theta_{ij},\quad i=1,2,\ldots,r;\ j=1,2,\ldots,c,
\]
where $\sum_{i,j}\theta_{ij}=1$. The frequency of the outcome $\left(
i,j\right)  $ is $n_{ij}$, where $n_{ij}>0$.\medskip\medskip

Find the maximum likelihood estimators for $\theta_{ij}$ assuming that

\begin{enumerate}
\item[(i)] $\theta_{ij}=\phi_{i}\rho_{j},\quad i=1,2,\ldots,r;\ j=1,2,\ldots
,c,$ where $\sum_{i}\phi_{i}=\sum_{j}\rho_{j}=1$, and

\item[(ii)] without this assumption.
\end{enumerate}

Hence find test statistics for the null hypothesis that the $X_{k}$s and
$Y_{k}$s are independent using

\begin{enumerate}
\item the likelihood ratio method,

\item Pearson's $\chi^{2}$ statistic.
\end{enumerate}

What can you say about the distributions of the two statistics for large
values of $n$?\smallskip\ \newline Some individuals are carriers of the
bacterium \emph{streptococcus pyogenes}. To investigate whether there is a
relationship between carrier status and tonsil size in schoolchildren, 1398
children were examined and classified according to their carrier status and
tonsil size. The data appear below. Is there an association?%
\[%
\begin{tabular}
[c]{lcc}\hline
Tonsil size & \multicolumn{2}{c}{Carrier status}\\
& Carrier & Non-carrier\\\hline
Normal & 19 & 497\\
Large & 29 & 560\\
Very large & 24 & 269\\\hline
\end{tabular}
\]

\end{enumerate}


\end{document}