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\begin{center}
\Huge \textbf{Exercise 6}
\end{center}
\vspace{1cm}



\begin{enumerate}
\item
Let $Y_1, \cdots, Y_n$ be a random sample from a Poisson
distribution with mean $\theta>0$ unknown.
\begin{enumerate}
\item
Show that the sum $Y\equiv Y_1 + \cdots + Y_n$ is a sufficient statistic.
Find its mean and variance.  What is the distribution of $Y$?
\item
Obtain the MLE for $\theta$.
\item
Suppose now that only the first $m$ ($m< n$) observations of the sample
are known explicitly, while for the other $n-m$ only their sum, $Z$ say, is
known, determine the MLE of $\theta$.
\item
Compare the answers to (b) and (c), and comment.
\end{enumerate}

\item Find the maximum likelihood estimator of $\lambda$ given a random
sample from the gamma distribution with density function
\[
f(x) = \frac{1}{\Gamma (r)}\exp(-\lambda x)x^{r-1}\lambda^r.
\]
Assume  that $r$ is known.


\item Find the maximum likelihood estimator for $\theta$ from a random sample from the population with density function
\[
f(y;\theta) = \frac{2y}{\theta ^2}\ \   0 <y \le \theta, \theta > 0.
\]
Do not use calculus.  Draw a picture of the likelihood.

\item Suppose that we have a random sample of size n from the logistic distribution with density function
\[
f(y;\mu) = \frac{\exp (y-\mu)}{\left[1 + \exp (y-\mu)\right]^2}.
\]
Find the likelihood equation for $\mu$, and write the iteration
equations for the Fisher's score method.  (Hint: write quantities as much as
possible in terms of the density function and the distribution function
for the logistic.)


\end{enumerate}
