

\subsection{\bf Comparison of Moment Method}
\subsection{\bf with Maximum Likelihood---I}

\begin{slide}
\begin{itemize}
\item Now, Maximum Likelihood Method gives result with error estimate
given by
\[
\frac{1}{\lbrack N\sigma^2 \rbrack} \big\vert_{m.l.m.} = 
\int\limits^{+1}_{-1} \frac{dx}{p} \left( \frac{\partial p}{\partial a}
\right)^2
\]
\item and from our general theory, this general formula gives in our
general case:
\begin{eqnarray*}
\frac{1}{\lbrack N\sigma^2 \rbrack} \big\vert_{m.l.m.}
& = & \frac{1}{2} \int\limits^{+1}_{-1} dx \frac{x^2}{(1+ax)} \\
& = & \frac{1}{2a^2} \left\lbrace \ln \left\lbrack \frac{1+a}{1-a}
\right\rbrack - 2a \right\rbrace \\
& = & \frac{1}{3} + \frac{a^2}{5} + \frac{a^3}{7} + \ldots \\
& = & \frac{1}{3}\ \mbox{when}\ a=0 
\end{eqnarray*}
\end{itemize}
\end{slide}

