

\subsection{\bf Proof of Maximum Likelihood}
\subsection{\bf Theorem Part (a)---II}

\begin{slide}
\begin{itemize}
\item The error in this is $O\left\lbrace (\alpha^* - \alpha_0)^2
\frac{L_3}{L_2} \right\rbrace$ which is $O\left( \frac{1}{n} \right)$
as we will soon show that $(\alpha^* - \alpha_0 )^2$ is $O\left(
\frac{1}{n} \right)$.

\item Now $L_1$ and $L_2$ are random variables and Equation~(17)
defines $\alpha^*$ as a random variable.

\item We can calculate from (17):
\begin{eqnarray*}
\langle \alpha^* \rangle & = & \alpha_0 - \langle L_1/L_2 \rangle \\
& = & \alpha_0 - \frac{\left\langle \frac{\partial}{\partial\alpha}
\ln P(x , \alpha_0 ) \right\rangle}{\left\langle 
\frac{\partial^2}{\partial\alpha^2} \ln P(x , \alpha_0 ) \right\rangle} 
\end{eqnarray*}
where we use theorem on $F/G$, proved at end of Section~2.  
\end{itemize}
\end{slide}

