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\centerline{\bf{CALIFORNIA STATE UNIVERSITY, EAST BAY }}
\centerline{\bf{STATISTICS DEPARTMENT}} \centerline{ }
\centerline{\bf{Statistics 6502 Mathematical Statistics}}
\centerline{\bf{Winter 2013}}

\medskip
\noindent \centerline{\bf{Midterm 1 - Extra Credit}}

\begin{enumerate}

% *******************************************************************

\item {\bf ( 16 points )} Suppose $X_1,X_2,...,X_n$ is a random
sample from
\begin{eqnarray*}
  f(x|\theta) = e^{-(x-\theta)}
\end{eqnarray*}
for $x \ge \theta$ and $0$ otherwise.
\begin{enumerate}
\item Find the method of moments estimator (MME) of $\theta$, $\tilde{\theta}$.
Is it possible for the MME to underestimate $\theta$?
\item Is the MME unbiased?  Calculate $E[\tilde{\theta}]$.
\item Calculate $Var(\tilde{\theta})$.
\item Find the maximum likelihood estimator (MLE) of $\theta$, $\hat{\theta}$.
Is it possible for the MLE to underestimate $\theta$?
\item Find the probability density of the MLE.
\item Is the MLE unbiased?  Calculate $E[\hat{\theta}]$.
\item Calculate $Var(\hat{\theta})$.
\item Suggest a transformation of the MLE that makes it unbiased.
Is it possible for the transformed MLE to underestimate $\theta$?
\end{enumerate}

% *******************************************************************

\item
{\bf ( 8 points )} Suppose that certain electronic components
have lifetimes that are exponentially distributed: $f(x|\lambda) =
\lambda e^{-\lambda x}, x > 0$.  Ten new components are put on
test and observed for 120 hours.  The first three failures were
97, 105 and 110.
\begin{enumerate}
\item
What is the likelihood function of $\lambda$?
What is the maximum likelihood estimate of $\lambda$?
\item
What is the asymptotic distribution of the maximum likelihood
estimate?
\item
Give an approximate $100(1-\alpha)\%$ confidence interval for
$\lambda$ based on the the maximum likelihood estimator
$\hat{\lambda}$.
\item
Suggest a transformation that might improve the confidence
interval from part (c).
\end{enumerate}

\newpage

% *******************************************************************
\item {\bf ( 10 points )}
The Poisson distribution has been used by traffic engineers as a model for light traffic, based on the rationale that if the rate is approximately constant and the traffic is light (so individual cars move independently of each other), the distribution of counts of cars in a given time interval or space area should be nearly Poisson.  The following table shows the number of right turns during 300 3-minute intervals at a specific intersection.

\begin{center}
\begin{tabular}{c|c}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  $x$ & Frequency \\
  \hline
  0 & 14 \\
  1 & 30 \\
  2 & 36 \\
  3 & 68 \\
  4 & 43 \\
  5 & 43 \\
  6 & 30 \\
  7 & 14 \\
  8 & 10 \\
  9 & 6 \\
  10 & 4 \\
  11 & 1 \\
  12 & 1 \\
  13+ & 0 \\
  \hline
\end{tabular}
\end{center}

\begin{enumerate}
\item
Fit the Poisson distribution to these data using maximum likelihood estimation.
Fit the Poisson model
\begin{eqnarray*}
f(x|\lambda) = \frac{ {\lambda}^x}{x!} e^{-\lambda}
\end{eqnarray*}
where $x = 0,1,...$ and $\lambda > 0$.  Give a numerical answer for the
estimate of the parameter in the model.
\item
Comment on the fit by comparing the observed and the expected counts for $x=1, x=2$.
\item
Suppose you were not comfortable with the assumed parametric model.
\begin{enumerate}
\item
Compute the sample mean and sample variance of the data.
\item
Explain how you would use the nonparametric bootstrap to compute
confidence intervals for the mean $\mu$ and the variance $\sigma^2$.
\item
Explain how you would use the nonparametric confidence intervals to check
the assumption of the Poisson model.
\end{enumerate}
\end{enumerate}

% *******************************************************************

%\newpage



%\item
%{\bf ( 25 points )} % Hogg and Craig, pp. 396-397
%Consider the one random variable $X$ that has a binomial distribution
%with $n = 5$ and $p = \theta$.  Let $f(x|\theta)$ denote the p.d.f. of
%$X$ and let
%\begin{equation}
%H_0: \theta = \frac{1}{2}
%\end{equation}
%and
%\begin{equation}
%H_1: \theta = \frac{3}{4}.
%\end{equation}
%The following table gives the values of $f(x|\frac{1}{2})$ and
%$f(x|\frac{3}{4})$.

%\begin{tabular}{c|cccccc}
%  \hline
%  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%  x & 0 & 1 & 2 & 3 & 4 & 5 \\
%  \hline
%  $f(x|\frac{1}{2})$ & 1/32 & 5/32 & 10/32 & 10/32 & 5/32 & 1/32 \\
%  $f(x|\frac{3}{4})$ & 1/1024 & 15/1024 & 90/1024 & 270/1024 & 405/1024& 243/1024 \\
%  \hline
%\end{tabular}

%\begin{enumerate}
%\item
%Sketch a picture of the null and alternative probability mass functions.
%\item
%Compare the likelihood ratio, $\Lambda$, for each possible value of $X$ and order
%the values of $x$ according to $\Lambda$.
%\item
%What is the likelihood ratio test of $H_0$ versus $H_1$ at level $\alpha = 1/32$?
%Compute the power of this test.
%\item
%What is the likelihood ratio test of $H_0$ versus $H_1$ at level $\alpha = 6/32$?
%Compute the power of this test.
%\item
%Make the ROC for this test by computing the power for each simple versus simple test
%of a defined level.   (Hint: Recall that the ROC plots power versus level of the test.)
%\end{enumerate}

%\item
%{\bf ( 15 points )} You are given a single observation from either
%the uniform distribution on $(-2, 2)$ or the standard normal
%distribution.
%\begin{enumerate}
%\item
%Accurately sketch a picture of the null and alternative densities
%on one set of axes.
%\item
%Find the most powerful test of size $\alpha = 0.2$ for
%\begin{equation}
%  H_0: X \sim Uniform(-2,2)
%\end{equation}
%versus
%\begin{equation}
%  H_1: X \sim Normal(0,1^2).
%\end{equation}
%(Hint:  Be careful when determining the rejection region of the
%test.)
%\item
%Calculate the power, $\pi$, of the test.  Calculate the
%probability of Type II Error, $\beta$, for the the test.
%\end{enumerate}

\end{enumerate}

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