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\title{Maximum Likelihood Estimation\footnote{\copyright 2019 Joseph C Watkins}}
\author{Worksheet 17}
\date{}
\maketitle

\begin{enumerate}
\item For a parameter $\theta>0$, we model the accuracy of a dart player by the $Beta(\theta,1)$ density
$$f_X(x|\theta) = \left\{\begin{array}{ll}
0&\hbox{if}\ x<0, \\
\theta x^{\theta -1}\quad&\hbox{if}\ 0\le x < 1, \\
0&\hbox{if} \ 1\le x, 
\end{array}\right.$$
for a continuous random variable $X$, the distance the dart is from the center of the board.
\begin{enumerate}
\item For $n$ observations, give the likelihood function.
\item Find the maximum likelihood estimate for observations $x_1,\ldots, x_n$.
\item Give 10000 simulations of maximum likelihood estimates of $\hat\theta$ based on the values $\theta = 1/4,1/2,3/4$  and $n=50$. Give the mean and standard deviation of the simulation for both values of $\hat\theta$.
\item How do the means of your simulations compare to the actual values of $\theta$. 
\item How do the standard deviations from the simulations compare to the standard deviation estimate given by the Fisher information?
\end{enumerate}
\item Daily rainfall data, in millimeters, is modeled as having a $\Gamma(1/2,\beta)$ distribution. The density is 
$$f_X(x|\alpha,\beta) =  \left\{\begin{array}{ll}
0&\hbox{for}\ x<0,\\
\frac{\beta^{1/2}}{\sqrt{\pi}}x^{-1/2}e^{-\beta x}\quad&\hbox{for}\ x\ge 0.\\
\end{array}
\right.$$
\begin{enumerate}
\item Find the maximum likelihood estimator for $\beta$ based on rainfall amounts $x_1,x_2,\ldots,x_n$
\item Compute the Fisher information to estimate to variance of this estimator.
\item How does it compare to the method of moments estimator?
\end{enumerate}
\end{enumerate}
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