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% Survival Analysis Notes ************************************************
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% File Name:      SurvivalAnalysisNotes.tex
% Submitted:      Dr. Eric Suess,Statistics 6502, Spring 2007
% Submitted Date: February 1st, 2002
% Accepted Date:  February 1st, 2002
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% This is a first draft of the lecture notes "Survival Analysis" in LaTeX.
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\begin{document}
\pagestyle{headings} \markright{Statistics 6502, Spring 2011}

\large
\begin{center}
{\bf Exercises}
\end{center}

{\bf Maximum Likelihood Estimation}

\begin{enumerate}
\item
Find the m.l.e. of $\theta$ based on a random sample $X_1,
X_2,...,X_n$ from each of the following p.d.f.'s.
\begin{enumerate}
\item
\begin{eqnarray}
f(x|\theta) &=& \theta x^{\theta-1} \qquad 0 < x < 1, \quad 0 < \theta \nonumber\\
&=& 0 \qquad\qquad otherwise \nonumber
\end{eqnarray}
\item
\begin{eqnarray}
f(x|\theta) &=& (\theta+1) \ x^{-\theta-2} \qquad 1 < x, \quad 0 < \theta \nonumber\\
&=& 0 \qquad\qquad\qquad\quad \   otherwise \nonumber
\end{eqnarray}
\item
\begin{eqnarray}
f(x|\theta) &=& \theta^2x \ e^{-\theta x} \qquad 0 < x, \quad 0 < \theta \nonumber\\
&=& 0 \qquad\qquad\quad \ otherwise \nonumber
\end{eqnarray}
\item
\begin{eqnarray}
f(x|\theta) &=& \theta(1-\theta)^{x-1} \qquad x = 1,2,..., \quad 0 < \theta < 1 \nonumber\\
&=& 0 \qquad\qquad\qquad otherwise \nonumber
\end{eqnarray}
\end{enumerate}

\item
Find the asymptotic variance of the m.l.e. in each part of
question 1.

\item
Consider two independent random samples $X_1, X_2,...,X_n \sim
N(\mu,\sigma_1^2)$ and \newline $Y_1,Y_2,...,Y_m \sim N(\mu,\sigma_2^2)$.
\begin{enumerate}
\item
Using the data from the two random samples find the m.l.e. of
$\mu$, $\sigma_1^2$, and $\sigma_2^2$.
\item
Find the asymptotic variance of $\mu$.
\end{enumerate}

\end{enumerate}

\end{document}
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