

\begin{slide}
\subsection{\bf Method of Moments---I}
{\Smallsize
\begin{itemize}
\item This is a simple and, in general, non-optimal method.  

\item Even here---for more than one parameter---there are two versions 

\begin{description}
\item{$\bullet$} (a) ignore correlations (used in practice but wrong) and 
\item{$\bullet$} (b) include correlations (best of a bad lot!).  Of
course, simplicity can lead to huge savings in computer time, and so it
should not be sneered at.
\end{description}

\item The basic idea of method of moments is simple: 

\begin{description}
\item{$\bullet$} Consider a set of observations $x_i$, then central
limit theorem implies
\[
\frac{1}{N} \sum^N_{i=1} f(x_i) \rightarrow \mu
\]
\item{$\bullet$} where
\[
\mu (\alpha ) = \int\limits p(x, \alpha) f(x) dx
\]
\end{description}

\item Here, $\mu$ is $\mu(\alpha )$ where $\alpha$ is some unknown
parameter.  So putting 
\[
\mu (\alpha) = \frac{1}{N} \sum^N_{i=1} f(x_i)
\]

\item gives one displaymath for $\alpha$ with solution $\alpha = \alpha^*$:

\begin{description}
\item{$\bullet$} $\alpha^*$ is the moment estimate of $\alpha$. 
\end{description}
\end{itemize}
}
\end{slide}

