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\begin{center}
\Huge \textbf{Exercise 7}
\end{center}
\vspace{1cm}



\begin{enumerate}
\item
Suppose $X_1, \cdots, X_n$ are a random sample from
Cauchy distribution with the density function
\[
f(x, \theta) = {\frac{1}{\pi} \{ 1 + (x- \theta)^2 \}^{-1} }.
\]
Find the equation of which the MLE of
$\theta$ is the solution.
Write down the Newton-Raphson algorithm for calculating
the MLE numerically. How would you choose an initial value
for the algorithm?

\item
Let $X_1, \cdots, X_n$ be a random sample from
an exponential distribution with the density function
$
f(x, \lambda) = \lambda e^{- \lambda x} I(x\ge 0)$,
where $\lambda>0$ is an unknown parameter.
Suppose the last $n-m$ $(m<n)$ observations are {\sl censored}
in the sense that we only known $X_j \in [u_j, v_j]$  for some
given $ v_j > u_j > 0$ while $X_j$ itself is unknown, $j = m+1, \cdots, n$.
Outline an EM algorithm for estimating the parameter $\lambda>0$.

\item
Let $X_1, \cdots, X_n$ be a random sample from Bernoulli distribution, i.e.
$P(X_1=1) = p = 1- P(X_1 =0)$, where $p\in (0, 1)$ is unknown. Let $\theta = p^2$.
\begin{enumerate}
\item
Find the Cram\'er-Rao lower bound for the variance of unbiased
 estimators for $\theta$.
\item
Find the MLE  $\widehat \theta$ for the parameter $ \theta$.
\item
Show that $E(\widehat \theta) \ne \theta $.
\item
Outline a bootstrap procedure for estimating the bias of $\widehat\theta $.
\end{enumerate}

\item
Given a random sample of $n$ observations from the N(0,$\theta$)
distribution, construct a most powerful test of the hypothesis
${H}_0:\theta= 1$ against  ${ H}_1: \theta =2$.  Find the
power of the test of the size 0.05 when $n=10$.

\item
Let $(X_1, \cdots, X_n)$ be a random sample from $N(0, \sigma^2)$. Find the
UMPT for testing hypotheses
$H_0: \, \sigma^2 \le \sigma_0^2$ against $H_1: \, \sigma^2 > \sigma_0^2$,
where $\sigma_0^2>0$ is a given constant.

\end{enumerate}
