

\subsection{\bf Theorem on Asymptotic Validity}
\subsection{\bf of Maximum Likelihood Method}
\subsection{\bf Conditions---First Assertion}
\begin{slide}
\begin{itemize} 
\item (a) If we fix $n$ and conduct many such series of $n$ observations,
the mean of $\alpha^*$ satisfies:
\[
\langle\alpha^* \rangle = \alpha_0 + O \left( \frac{1}{n} \right)
\]

\item Note $\langle\alpha^* \rangle$ is not $=\alpha_0$ exactly
($\alpha_0$ is true theoretical parameter).

\item If for {\em any\/} estimate $\langle\alpha_{\rm estimate} \rangle
= \alpha_0$, then we say estimate is unbiased:

\item if $\langle \alpha_{\rm estimate} \rangle = \alpha_0 + b
(\alpha_0 )$ then we have a biased estimate with bias $b (\alpha_0 )$.

\item So maximum likelihood estimate is biased for finite $n$ and is
only asymptotically unbiased.

\item However, bias is smaller than Standard Deviation and so it is of
no real importance.

\item Bias is discussed in detail by Cramer.
\end{itemize}
\end{slide}

