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{\bf \Large Assignment in STK4011/STK9011-f17 }

Posted:   Friday November 24th  at 9 am.

Deadline: Tuesday November 28th  at 2pm.


This is text for the final assignments in STK4011/STK9011-f17.
It consists of three  problems. Both handwritten reports and answers
using a word processing system are acceptable. Each student must deliver a separate
and individually formulated report.  That means that you should not compare your answers, 
and discussion of the problems should only involve general viewpoints  on how the problems can be solved.  
There is more information at:
http://www.uio.no/english/studies/examinations/compulsory-activities/mn-math-mandatory.html

There are two ways to deliver the reports. You can choose what is most convenient.

1. You can deliver {\bf two} paper copies with your candidate number at the Math department's  expedition  at Ullev{\aa}l. Remember to indicate whether you attend STK4011 or STK9011.

2. Use the electronic filing system, delivery,  for uploading exam papers.

\vspace{0.25 cm} \noindent \textbf{Problem 1}\\

Let $X, Y $ and  $Z$ be independent  and uniformly distributed on the interval $[0,1]$.
\begin{itemize}
\item[a)] Use the convolution formula to find the density of $X+Y$.
\item[b)] Verify the results from part a) by using moment generating functions. 
\item[c)] Use the convolution formula to find the density of $X+Y+Z$.
\end{itemize}

\vspace{0.25 cm} \noindent \textbf{Problem 2}\\

The Pareto distribution is often used to model phenomena with heavy  tails such as income.
The density is
\begin{equation}
f(x\vert \alpha, \beta)= \left \{ \begin{array}{cl}  \frac{\beta\alpha^{\beta}}{x^{\beta+1}} & \mbox{ if } \alpha \leq x <\infty
\;\;\alpha, \beta >0 \label{eq:Pareto}
\\
0 & \mbox{  else }\end{array} \right. 
\end{equation}
Assume that $X_1,\ldots, X_n$ is a random sample of random variables which are Pareto distributed so the probability density is $f(x\vert \alpha, \beta)$.
\begin{itemize}
\item[a)] Show that the cumulative distribution function equals 
\[
F_X(x)= 1- (\frac{\alpha}{x})^{\beta} \mbox{   for }    x \geq \alpha.  
\]

\item[b)]  Find $E[X^k]$ when X is Pareto distributed, i.e has density $(\ref{eq:Pareto}).$

\item[c)] Show  that the Pareto distribution belongs to the
exponential family of distributions when $\alpha$ is a known constant.  Also show that $\log X - \log \alpha$ has an exponential distribution with scale parameter $1/\beta.$

\item[d)] Show  that the maximum likelihood estimators $\hat{\alpha}$  and $\hat{\beta}$ of $\beta$  
and  $\alpha$ are 
\[
\hat{\beta}= \frac{n}{\Sigma_{i=1}^n \log (X_i/X_{(1)})} \mbox{ and } \hat{\alpha} = X_{(1)}
\]
where $X_{(1)}= \min_{i} X_i$.
\item[e)]  Show that $ \hat{\beta}$ and $\hat{\alpha} $ are independently  distributed
and that $2n\beta/\hat{\beta}$ is $\chi^2$ distributed with $2(n-1)$ degrees of freedom and
$\hat{\alpha}$ is Pareto distributed. 

[Hint: To show independence argue that it is sufficient to
consider independent and exponentially distributed random variables $U_1,\ldots, U_n$.
Then use the transformation $V_1=U_{(1)}$ and $V_i= (n-i+1)(U_{(i)}-U_{(1)}), i=2,\ldots,n$ 
where $(U_{(1)},\ldots,U_{(n)})$ is the order statistic.
Before treating the general case it may be helpful to consider the situation where n=3 or 4.]

\item[f)] Show that $X_{(1)}$ and $\Sigma_{i=1}^n \log (X_i/X_{(1)})$ are sufficient statistics 
for $\alpha$ and $\beta$
based on a random sample where the probability density is Pareto , i.e. has a probability distribution defined in  equation ($\ref{eq:Pareto})$.
\end{itemize}
It can also be shown the statistics $X_{(1)}$ and $\Sigma_{i=1}^n \log (X_i/X_{(1)})$ are complete  statistics 
for $\alpha$ and $\beta$ based on a random sample where the probability density is Pareto. 
Assuming that:
\begin{itemize}
\item[g)] Find the UMVUE or best unbiased estimator for $\beta$.
\item[h)] Also show that $ X_{(1)}[1- \frac{1}{(n-1)}\frac{1}{\hat{\beta}}] $
is the  UMVUE or best unbiased estimator for $\alpha$.

\end{itemize}

\vspace{0.25 cm} \noindent \textbf{Problem 3}\\

Let $X_1,\ldots,X_n$ be a random sample from the logistic distribution which has cumulative 
distribution function
\[
F_X(x)= \frac{1}{1+\exp(-(x-\theta))}, \; -\infty< x< \infty,\; -q\infty< \theta< \infty.
\]
Here $\theta$ is an unknown parameter.
\begin{itemize}
\item[a)] Show that the empirical  mean 
\[
\theta_n^{*}=  \frac{1}{n}\Sigma_{i=1}^{n}X_i
\]
is the method of moment estimator for $\theta$. What is the approximate distribution  of  $\theta_n^{*}$ for large n?
Here you can use without proof that $Var(X_1)= \pi^{2}/3$.
\item[b)] Find an equation satisfied by the maximum likelihood estimator $\hat{\theta}_n$. Show that 
$\hat{\theta}_n$ exists and is unique. 
\item[c)] What is the approximate distribution  of  $\hat{\theta}_n$ for large n?  What is the asymptotic relative efficiency, ARE, of $\theta_n^{*}$ with respect to $\hat{\theta}_n$ ?
\item[d)] Based on one observation, $X_1$ say, describe the uniformly most powerful test with level  $\alpha$
for the null hypothesis
\[
 H_0: \theta=\theta_0 \mbox{    versus the alternative hypothesis   } H_1: \theta>\theta_0.
 \] 
 \item[e)] Explain why the test from part d) also is the
 uniformly most powerful test with level $\alpha$ for the null hypothesis
 \[
 H_0: \theta  \leq \theta_0 \mbox{    versus the alternative hypothesis   } H_1: \theta>\theta_0.
 \] 
\end{itemize}


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%\item[c)]  If $X \sim f(x\vert \alpha, \beta)$, show that $Y= \log(X)- log(\alpha)$ 
%has probability density
%\[
%f(x\vert \beta)= \left \{ \begin{array}{cl} \beta \exp(-\beta y) & \mbox{ if } 0 < y
%\\
%0 & else \end{array} \right. 
%\]
%and use this to find the Cramer-Rao lower bound for an unbiased estimator 
%of $\theta= 1/\beta$.
%\end{itemize}
%Assume that $X_1,\ldots, X_n$ is a random sample of random variables which are Pareto distributed so the probability density is $f(x\vert \alpha, \beta)$.
%begin{itemize}
%\%item[d)] Show  that the maximum likelihood estimator $\hat{\theta}$ of $\theta$ is  
%\[
%\hat{\theta}= \frac{1}{n}\Sigma_{i=1}^n \log (X_i) - \log( \alpha).
%\]
%and show that $\hat{\theta}$ is a best linear unbiased,BUE (or uniform minimum variance unbiased estimator ,UMVUE).
%\item[e)] Find an exact $1-\alpha$ confidence interval based on observations 
%$x_1,\ldots, x_n$ from the random sample $X_1,\ldots, X_n$.
%\item[f)] Also derive  an approximative  $1-\alpha$ confidence interval based on observations 
%$x_1,\ldots, x_n$ from the random sample $X_1,\ldots, X_n$.
%\end{itemize}
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